UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN ASTROFÍSICA
INSTITUTO DE ASTRONOMÍA
ASTROFÍSICA OBSERVACIONAL
PULSATING STARS IN GLOBULAR CLUSTERS AND THE STUDY OF THE
STELLAR EVOLUTION ON THE HORIZONTAL BRANCH
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (ASTROFÍSICA)
PRESENTA:
DAN JOSUÉ DERAS BADILLO
TUTOR:
DR. ARMANDO ARELLANO FERRO
INSTITUTO DE ASTRONOMÍA
CIUDAD UNIVERSITARIA, Cd. Mx.  SEPTIEMBRE 2020
 
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Universidad Nacional Autónoma de México
Doctoral Thesis
Pulsating stars in globular clusters and the study of the
stellar evolution on the Horizontal Branch
Author:
M. en C. Dan Josué Deras Badillo
Supervisor:
Dr. Armando Arellano Ferro
A thesis submitted in fulfillment of the requirements
for the degree of Doctor of Philosophy
in the
Instituto de Astronomía, UNAM
September 8, 2020

iii
“He determines the number of the stars; he gives to all of them their names.”
Psalms 147:4

v
Resumen
En nuestra galaxia existen asociaciones que contienen entre decenas de miles hasta cientos de miles de estrellas
ligadas gravitacionalmente. Las estrellas que las conforman, se encuentran distribuidas en un volumen de unos
100 pc3. Éstas reciben el nombre de Cúmulos Globulares (GCs) y son los sistemas estelares más viejos en la
Galaxia; ésto los hace fundamentales en el estudio de las primeras etapas de la formación de la misma. La
determinación de algunos parámetros físicos como su metalicidad y luminosidad medias, así como su distancia
y su edad, son de interés fundamental para conjeturar sobre su origen, su evolución, su estructura y dinámica y
su relación con la formación del halo de nuestra galaxia.
La gran mayoría de los cúmulos globulares posee una población notable de estrellas variables. Sus estrellas
variables más emblemáticas son las RR Lyrae (RRL), que son estrellas de baja masa (M ≈ 0.6 M⊙ - 0.8 M⊙)
que pertenecen a la Rama Horizontal (HB) y caen dentro de la Franja de Inestabilidad (IS) en el diagrama
Hertzsprung-Russell (HR). Por estar en la HB, su luminosidad (magnitud absoluta) es más o menos constante
y es independiente de la temperatura y el período. Ésto las ha hecho indicadores de distancias cósmicas funda-
mentales desde inicios del siglo XX (Shapley, 1918). Las estrellas de la HB son estrellas evolucionadas desde
la Rama de las Gigantes Rojas (RGB), inicialmente con una masa M ≈ 2 M⊙, en la Secuencia Principal, y que
pierden masa en la RGB durante la etapa de He-flash, hasta que se instalan en la HB de Edad Zero (ZAHB),
más al rojo o al azul dependiendo de su masa y la composición química primordial (sobre todo He y CNO).
Ahí comienzan a quemar He en el núcleo mientras que la quema de H se mueve hacia un cascarón exterior al
núcleo. Cuando el He en el ńucleo se agota, evolucionan hacia la Rama Asintótica de las Gigantes (AGB). Esto
quiere decir que la IS es cruzada casi horizontalmente en ambas direcciones, lo que produce que las RRL estén
sujetas a cambios seculares en el período. La distribución de estrellas RRab y RRc es por lo tanto un resultado
de varios factores: las condiciones químicas primordiales de la generación estelar que dio origen a las RRL, su
masa restante después de la perdida durante la fase RGB, y de su estado evolutivo en sí. Si como indican algunos
resultados en cúmulos globulares, pueden coexistir más de una generación estelar en el mismo cúmulo, con abun-
dancias de He y CNO ligeramente diferentes, la distribución de modos RRab y RRc y los cambios seculares en
los períodos dará información relacionada con estas circunstancias. Su distribución sobre la HB y sus períodos
de pulsación también son de particular interés ya que ayudan a distinguir entre los tipos de Oosterhoff (OoI u
OoII). Los GCs tipo OoI pueden presentar o no una mezcla de RRL tipo RRab y RRc en la zona intramodo, que
es donde se superponen la IS del modo fundamental radial de pulsación con la IS del primer sobretono radial
de pulsación; mientras que los OoII presentan una clara segregación entre estrellas RRab y RRc. También se ha
observado que los GCs OoI son más metálicos que los OoII. Los modelos evolutivos desde la Secuencia Prin-
cipal (MS) hasta la HB predicen que una estrella de aproximadamente 2 masas solares, se podría convertir en
una RRL en aproximadamente 1012 o 1013 años, lo que es consistente con la edad típica de los GC. (e.g. Yepez
et al. (2019) para el caso de Pal 13). Ésto las hace indicadores de la presencia de poblaciones estelares viejas.
Las RRL pueden pulsar en el modo fundamental radial (RRab) con períodos entre 0.4 d y 0.8 d o en el primer
sobretono radial (RRc) con períodos entre 0.2 d y 0.4 d. En algunos GCs también se encuentran RRL que pulsan
simultáneamente en los dos modos (RRd).
vi Resumen
Más recientemente se han establecido calibraciones que, a partir de la descomposición de Fourier de las curvas
de luz de estrellas RRL, hacen posible determinar su metalicidad [Fe/H] y su luminosidad individual y por lo
tanto los valores medios para el cúmulo que las contiene (Jurcsik et al. 1996; Jurcsik 1998; Morgan et al. 2007;
Nemec et al. 2013). Asimismo, es posible estimar para cada RRL sus parámetros físicos M/M⊙, R/R⊙, Teff
y L/L⊙, lo que permite conjeturar sobre la estructura evolutiva de la HB (Arellano Ferro et al., 2016).
Nuestra finalidad última en este proyecto es estudiar la distribución de modos de pulsación en algunos cúmulos
específicos, identificar estrellas de estados evolutivos diferentes y estudiar los cambios seculares en los períodos
de las RRL. Ésto nos ayudará a entender el origen de los dos grupos de Oosterhoff en los GCs y la dependencia
de la luminosidad de su HB con respecto a su metalicidad media, con el consecuente impacto en la escala de
distancias desde el punto de vista de su población de estrellas RRL. Durante el proceso se hace un análisis
de membresía para diferenciar entre las estrellas en el campo y las que pertenecen a cada cúmulo. Esto se
hace usando los movimientos propios de las estrellas, medidos por la misión Gaia Collaboration et al. (2018)
(Gaia-DR2). También se obtienen la metalicidad, el enrojecimiento y la distancia medias de cada GC estudiado,
se realiza una búsqueda sistemática de nuevas variables y se clasifican según sus propiedades de pulsación u
orbitales en el caso de las binarias eclipsantes.
En este trabajo nos proponemos estudiar los cúmulos globulares NGC 6171 (M107), NGC 6205 (M13) y NGC
6712, determinar sus parámetros físicos, explorar la estructura de la HB estudiando la distribución detallada de
sus RRL. Se compararán con la estructura estudiada por nuestro grupo en otros cúmulos, identificando, según
su luminosidad, segregación entre modos de pulsación, dirección y tasa de variación de período a las estrellas
auténticamente más evolucionadas hacia la AGB; y aquellas en camino a la ZAHB y por lo tanto se discutirá sobre
la posible presencia de más de una población estelar. Nuestra estrategia observacional consiste en la obtención
de series temporales de imágenes CCD de cada cúmulo y su manejo (reducción) por medio de la diferenciación
de imágenes. Esto permite obtener fotometría de alta calidad aún en regiones densamente pobladas, como son
las regiones centrales de los cúmulos globulares. Para lograr ésto hemos empleado la técnica de Differential
Imaging Analysis (DIA) y su implementación en la paquetería DanDIA (Bramich, 2008), con la que nuestro
grupo tiene amplia experiencia. La fotometría de alta calidad permite ubicar con gran precisión a cada estrella
variable sobre el Diagrama Color-Magnitud (CMD). Además de actualizar el censo de las estrellas variables en
estos cúmulos, se ofrecerán estimaciones independientes de metalicidad y su distancia usando la descomposición
de Fourier de las curvas de luz de las estrellas RRL, las relaciones Período-Luminosidad (P-L) para las RRL y
las SX Phe y la luminosidad de sus variables semi-regulares (SR) en la punta de la rama de las gigantes.
Los resultados principales del cúmulo globular NGC 6171 están divididos en dos artículos descritos en Chapter
6 y Chapter 7. En el primer artículo se hizo un análisis de fotometría en los filtros VI de CCD de series de tiempo.
Con esto se determinaron los parámetros físicos individuales para estrellas RRL monomodales presentes en el
campo de visión de nuestras imágenes. El método seguido fue la descomposición de Fourier de sus curvas
de luz y calibraciones semi-empíricas. Esto llevó a la estimación de los valores medios de la metalicidad y
la distancia al cúmulo paterno. De las estrellas RRab encontramos [Fe/H]ZW = -1.33 ± 0.12 y una distancia
d = 5.3 ± 0.3 kpc y de las estrellas RRc encontramos [Fe/H]ZW = -1.02 ± 0.19 y una distancia d = 5.3 ± 0.2 kpc.
Métodos independientes tales como las relaciones P-L para estrellas RRab y SX Phe permitieron la estimación
de una distancia promedio ponderada al cúmulo de 5.4 ± 0.1 kpc. Se estimó el enrojecimiento del cúmulo como
E(B − V) = 0.45 ± 0.03 a partir de las curvas de color de las estrellas RRab. El CMD es consistente con una
edad de 11 Gyrs. La distribución de las estrellas RRab y RRc parece estar bien segregada alrededor del borde
rojo del primer sobretono de la franja de inestabilidad. Esto coloca a NGC 6171 entre los cúmulos tipo OoI
donde los modos de pulsación están claramente separados sobre la Rama Horizontal. Finalmente, exploramos
las estrellas localizadas en la punta de la RGB en búsqueda de variabilidad y encontramos dos nuevas variables
irregulares de tipo Lb.
Resumen vii
En el segundo artículo se usó una base de tiempo de 82 años para construir diagramas O − C para 22 estrellas
RRL. Este método dio como resultado el descubrimiento de cambios seculares de período en cuatro variables
para las cuales hemos estimado sus tasas de cambio de período β. Por otro lado, el 82% de la muestra de estrellas
han mantenido estables sus períodos en los últimos 82 años. Toda la base de datos fue utilizada para refinar los
períodos de las estrellas que presentan períodos estables. Entre las estrellas que muestran cambios de período
tres de ellas (V10, V12 y V16) tienen períodos decrecientes que son más grandes de lo esperado si se toma en
cuenta la evolución estelar. El promedio global de los cambios de período medidos en el cúmulo es básicamente
cero, a pesar de estos casos individuales con tasa de cambio de período significativa, en consonancia con las
predicciones teóricas para cúmulos con ramas horizontales más rojas.
En el tercer artículo se realizó la serie temporal fotométrica CCD en los filtros VI del cúmulo globular M13
(NGC 6205). De la descomposición de Fourier de las curvas de luz de las estrellas RRab y RRc se encontró
una metalicidad promedio de [Fe/H]zw = -1.58 ± 0.09. La distancia al cúmulo se estimó en 7.1 ± 0.1 kpc
por medio de métodos independientes relacionados con las familias de estrellas variables como las RRL, SX
Phe y W Virginis, de la luminosidad de la ZAHB teórica y de la solución órbital de una estrella binaria de
contacto recién descubierta. Los modos de pulsación de las RRL están segregados alrededor del borde rojo del
primer sobretono de la IS en este cúmulo tipo OoII. Se realizó un análisis de pertenencia de 52,800 estrellas
en el campo del cúmulo basado en los movimientos propios proporcionados por Gaia-DR2 el cual permitió la
identificación de 23,070 posibles miembros del cúmulo, para los cuales contamos con la fotometría VI de 7,630.
La identificación de estrellas miembro permitió la construcción de un CMD limpio y un ajuste apropiado de una
isócrona y una ZAHB consistentes con un enrojecimiento, una edad y una distancia de 0.02 mag, 12.6 Gyrs y
7.1 kpc, respectivamente. Reportamos siete nuevas variables: una RRc, dos SX Phe, tres SR y una binaria de
contacto. V31 presenta una naturaleza de doble modo (o RRd) y confirmamos a V36 como tipo RRd. También
se reportan quince candidatas a estrellas variables. El análisis de dieciocho estrellas en el campo del cúmulo,
reportadas como RRL de la base de datos de Gaia-DR2 revela que al menos siete no son variables. Los detalles
se comentan en Chapter 8.
En el cuarto artículo se estudió la población de estrellas variables del cúmulo globular NGC 6712 para usar
sus diferentes familias como indicadores de los parámetros físicos medios del cúmulo de interés estructural y
evolutivo. Se empleó la descomposición de Fourier de las curvas de luz de las RRL para derivar la metalicidad del
cúmulo:
⟨
[Fe/H]
⟩
ZW
= −1.25± 0.02 para las RRab y
⟨
[Fe/H]
⟩
ZW
= −1.10± 0.16 para las RRc. Se estimó el
enrojecimiento del cúmulo como E(B − V) = 0.35 ± 0.04 a partir de las curvas de color de las estrellas RRab.
La distancia al cúmulo se estimó usando cuatro diferentes métodos que dieron una distancia media ponderada
de
⟨
d
⟩
= 8.1 ± 0.3 kpc. La distribución de las estrellas RRab y RRc sobre la HB muestra una clara segregación
alrededor del borde rojo del primer sobretono de la IS, lo que parece una característica común en cúmulos de
tipo OoI con una HB muy roja. Se llevó a cabo un análisis de pertenencia de 60,447 estrellas en nuestro campo
de visión (FoV) usando los datos de Gaia-DR2 y se encontraron 1529 miembros probables y poseemos curvas
de luz para 1100 de ellos. Esto ayudó a producir un CMD consistente con una edad de 12 Gyrs. Esto también
permitió descubrir estrellas vecinas cercanas, dentro de la Point Spread Function (PSF), para varias estrellas
variables. También se determinó que basados sólo en el análisis de movimientos propios de las estrellas en
nuestro FoV, no es posible diferenciar entre las estrellas de campo que participan en la rotación Galáctica de las
que están siendo auténticamente influidas por perturbaciones de marea debido a sus constantes pasos a través
del bulbo de la Galaxia. Finalmente, se reportan dieciséis nuevas variables de tipo EW (9) y de tipo SR (7). Los
detalles se comentan en Chapter 9.
La estructura de la tesis es como sigue: En Chapter 1 se resumen algunas de las características de los cúmulos
elegidos para este trabajo. En Chapter 2 se da una breve descripción de la evolución de una estrella de baja
masa. En Chapter 3 se discuten las estrellas RRL como una clase de estrella variable. En Chapter 4 se discuten
viii Resumen
algunas de las propiedades de los GCs. En Chapter 5 se describen las calibraciones semi empíricas basadas en la
descomposición de Fourier de las curvas de luz de las RRL para la estimación de parámetros físicos de relevancia
astrofísica, para las estrellas de manera individual, y por lo tanto del valor medio del GC al que pertenecen. En
los Chapters 6 - 9, se presentan los artículos publicados derivados de esta tesis. En Chapter 10, se resumen el
trabajo realizado y las conclusiones generales.
ix
Acknowledgements
• A mis padres: Elizabeth y Humberto. Si ustedes no me hubieran concebido, seguramente yo no estaría
aquí.
• A mis hermanos: Israel, Elizabeth y Rolando. Este logro también les pertenece a ustedes.
• A las familias Angulo Badillo, García Badillo y Deras Borbolla. Su apoyo y sus oraciones siempre han
sido fundamentales para mi desarrollo como profesionista y como ser humano.
• Al Dr. Armando Arellano Ferro, por tener siempre su puerta abierta y su paciencia desde el inicio hasta
la culminación de esta aventura.
• A mi comité tutor: Dr. Xavier Hernández, Dr. Julio Ramírez y Dr. Lester Fox; por sus oportunos comen-
tarios y críticas constructivas para tener un mejor desempeño durante el doctorado.
• A mis sinodales: Dr. Lester Fox-Machado, Dr. William Schuster, Dr. Miguel Chávez, Dr. Leonardo
Sánchez y Dr. Alejandro Ruelas; y a mi amiga la Dra. Vera Passegger. A todos ellos por tomarse el
tiempo de leer esta tesis y de hacer sugerencias y correcciones para mejorarla.
• A Karla Ortíz, por las interminables charlas, los omelettes, las tortas de pollo rostizado, los tacos de Torres
Adalid, los desayunos en Aurelia y Moheli, las alitas, los cortes de cabello con Carmen, los mojitos en
el 3 Condesas y definitivamente por cada una de las chelas que compartimos. En fin, nunca terminaré de
agradecer tu apoyo incondicional el cual hizo posible esta tesis.
• A Bertha y Heike por su infinita paciencia, eficiencia y apoyo con toda la burocracia del posgrado.
• A todo el personal del Observatorio Astronómico Nacional, Sierra de San Pedro Mártir, B.C., por su
excelente servicio y amabilidad, siempre dispuestos a ayudar en lo que se necesitara y por haberme hecho
sentir como en casa en cada ocasión que los visité.

xi
Contents
Resumen v
Acknowledgements ix
List of Figures xiii
List of Tables xv
List of Abbreviations xvii
1 Understanding globular clusters through VI CCD photometry of RR Lyrae stars 1
1.1 NGC 6171 (M107) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 NGC 6205 (M13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Stellar structure and evolution of low-mass stars 5
3 RR Lyrae stars 11
3.1 Driving mechanisms of pulsation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 The κ and γ mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Radial and nonradial modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 The Blazhko effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Globular clusters 17
4.1 On the origin of GCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 On the evidence of multiple stellar populations in GCs . . . . . . . . . . . . . . . . . . . . . . 19
4.3 The Oosterhoff dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 The color-magnitude diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.1 The main sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.2 The red giant branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4.3 The horizontal branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4.4 The asymptotic giant branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Fourier light curve decomposition of RR Lyrae stars and their physical parameters 25
5.1 Finding the best period of pulsation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Semi-empirical calibrations for metallicity, absolute magnitude and temperature . . . . . . . . . 26
5.3 Period changes in RR Lyrae stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Physical parameters of RR Lyrae stars in NGC 6171 (M107) 35
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
xii Contents
6.2 Observations and transformation to the standard system . . . . . . . . . . . . . . . . . . . . . . 35
6.3 On the reddening of NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.4 Physical parameters of RRL stars in NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.5 The color-magnitude diagram of NGC 6171 and its HB . . . . . . . . . . . . . . . . . . . . . . 39
6.6 Contributions made by the student to this article . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Period changes in the RR Lyrae stars of NGC 6171 (M107) 53
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Times of maximum brightness and the O − C diagrams . . . . . . . . . . . . . . . . . . . . . . 54
7.4 Period changes and the evolution on the HB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.5 Contributions made by the student to this article . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8 A new study of the variable star population in the Hercules globular cluster (M13; NGC 6205) 67
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.2 Observations and transformation to the standard system . . . . . . . . . . . . . . . . . . . . . . 67
8.3 The physical parameters of RRL stars in NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . 67
8.4 Membership determination with Gaia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.5 The color-magnitude diagram of NGC 6205 and its HB . . . . . . . . . . . . . . . . . . . . . . 70
8.6 Contributions made by the student to this article . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9 NGC 6712: The variable star population of a tidally disrupted globular cluster 91
9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.2 Observations and transformation to the standard system . . . . . . . . . . . . . . . . . . . . . . 91
9.3 Physical parameters of RRL stars in NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.4 On the reddening of NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.5 Membership determination with Gaia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.6 The color-magnitude diagram of NGC 6712 and its HB . . . . . . . . . . . . . . . . . . . . . . 95
9.7 Contributions made by the student to this article . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10 Conclusions 119
10.1 Physical parameters of RR Lyrae stars in NGC 6171 (M107) . . . . . . . . . . . . . . . . . . . 119
10.2 Period changes of RR Lyrae stars of NGC 6171 (M107) . . . . . . . . . . . . . . . . . . . . . . 120
10.3 A new study of the variable star population in the Hercules gloubular cluster (M13; NGC 6205) 120
10.4 NGC 6712: The variable star population of a tidally disrupted globular cluster . . . . . . . . . . 121
10.5 On the search for new variable stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.6 On the determination of membership of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.7 On the classification of Galactic GCs and their Mv-[Fe/H] relation . . . . . . . . . . . . . . . . 123
A The standard photometric system 127
B DanDIA 129
C Star membership determination using Gaia-DR2 131
C.1 First Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
C.2 Second Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
xiii
List of Figures
2.1 Evolution of a low-mass star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 Variable stars on the HRD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 The different light curve shapes of RRL stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Bailey diagram of M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Hysteresis of RRL stars on the IS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5 Blazhko effect in two RRab stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 GC M13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 MSP in NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 MSP in NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 MSP in NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Oosterhoff dichotomy in GCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.6 Scheme of a CMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1 Distribution of RRL of NGC 6171 on the HB . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.1 RRL in NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 CMD of NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.1 O − C residuals of RRL stars in NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.1 RRL in NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.2 Star membership determination of NGC 6205 from Gaia-DR2 . . . . . . . . . . . . . . . . . . 71
8.3 CMD of NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.1 RRL stars in NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.2 VPD of NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.3 CMD of NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
10.1 The HB structure parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.2 Mv - Fe/H relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.1 The Johnson-Kron-Cousins Photometric System . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xv
List of Tables
5.1 Summary of calibrations for metallicity, absolute magnitude and temperature for RRab stars . . 30
5.2 Summary of calibrations for metallicity, absolute magnitude and temperature for RRc stars . . . 30
6.1 Data of variable stars in NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.2 Physical parameters of RRL stars in NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.3 Distance comparison to NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.1 Photometric sources of NGC 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 New periods and period change rates for RRLs in NGC 6171 . . . . . . . . . . . . . . . . . . . 55
8.1 Data of variable stars in NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.2 Physical parameters or RRL stars in NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.3 Distance comparison to NGC 6205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
9.1 Data of member variable stars in NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.2 Data of known, unclassified and newly discovered field variable stars (FV) in NGC 6712 . . . . 93
9.3 Physical parameters of RRL stars in NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . 95
9.4 Distance comparison to NGC 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xvii
List of Abbreviations
AGB Asymptotic Giant Branch
C Cluster
CMD Color Magnitude Diagram
CVSGC Catalogue of Variable Stars in Globular Clusters
FORE First Overtone Red Edge
FoV Field Of View
FV Field Variable
GC Globular Cluster
HRD Hertzsprung-Russell Diagram
HB Horizontal Branch
HST Hubble Space Telescope
IAC Instituto de Astrofísica de Canarias
IAO Indian Astronomical Observatory
IS Instability Strip
M Messier
MS Main Sequence
MSP Multiple Stellar Populations
MW Milky Way
NGC New General Catalogue
OoI Oosterhoff type I cluster
OoII Oosterhoff type II cluster
P-L Period-Luminosity Relation
RGB Red Giant Branch
RRL RR Lyrae
SGB Sub Giant Branch
TO Turn Off
UVES Ultraviolet-Visual Echelle Spectrograph
V Variable
VPD Vector Point Diagram
ZAHB Zero Age Horizontal Branch
ZAMS Zero Age Main Sequence
ZW Zinn-West

xix
Para mi familia

1
Chapter 1
Understanding globular clusters through VI
CCD photometry of RR Lyrae stars
In the Galaxy exist associations of stars which can contain from several hundred thousands up to a million
which are gravitationally bound. Their conforming stars are distributed in a volume of about 100 pc3. These
associations receive the name of Globular Clusters (GCs) and are the oldest stellar systems in the Galaxy. The
determination of some of their physical parameters such as their mean metallicity and luminosity, as well as their
distance and age, makes them important "laboratories" of fundamental interest. These parameters enable us to
speculate on their origin, evolution, structure, and dynamics and their relation with the formation of the halo of
the Galaxy.
GCs contain an assortment of stars including variable stars of the likes of quiescent Low-Mass X-Ray Binaries
(LMXBS), cataclysmic variables (CVs), millisecond radio pulsars, Cepheids, Blue Stragglers, SX Phe, and
chromospherically active binaries among others. Of particular interest for our intended purposes, RRL stars play
a key role in acting as indicators of the mean properties of GCs. These are pulsating stars in which their surface is
displaced in a radial direction due to a particular driving mechanism. This pulsation causes the brightness of the
RRL to fluctuate periodically, typically between 0.2 d and 0.8 d. The measurement of light in general is known
as photometry and the measurement of this fluctuation as a function of time is known as a light curve. The
description of a light curve can be achieved in terms of the sum of several sinusoidal harmonics, with the same
period but different shifts and amplitudes. Our physical intuition suggests that different shapes of light curves
may contain information regarding to different physical properties and processes taking place in the interior of
the star. The challenge has been (for the past few decades now) to be able to relate the amplitudes and the shifts
of the harmonics in the light curves to physical properties such as the luminosity or absolute magnitude, the
metallicity or the content of heavy elements in the atmosphere of the RRL stars. Several groups have worked
on hydrodynamical models to come up with semi-empirical calibrations of the pulsations of RRL stars and their
light curves from the study of several hundreds of them. With the advent of Gaia-DR2 (Gaia Collaboration et al.,
2018), the number of detected RRL stars in GCs and in the field of the Galaxy skyrocketed from a few hundreds
at the beginning of last century to over hundred thousands, as Clementini et al. (2018) reported over 140,000
RRL through period-amplitude and period-luminosity relations in the G band. As impressive as this number is,
it is also important to carry out careful individual studies of these RRL stars to confirm their variability as noted
by Deras et al. (2019) who crossed-matched the light curves of 18 RRL stars with variability flag detected by
Gaia with their own VI photometry of GC M13, and found that "only four previously known RRL stars were
confirmed with our data, no variability was found in seven stars, six stars were too faint for our observations or
we were unable to build a reliable light curve, and one is out of the FoV of our images."
As intrinsically bright as they are, in practice RRL stars in GCs are very faint as seen from Earth due to their great
distances. This presents a very difficult technical problem as to how to measure their light. At the beginning of the
2 Chapter 1. Understanding globular clusters through VI CCD photometry of RR Lyrae stars
last century, photographic plates were used to capture the light from GCs. Nevertheless, this technique required
very large telescopes, very sensitive emulsions and extremely long exposure times, which greatly hampered the
study of individual stars in these stellar systems at the time. Given the pressing need of astronomers to measure
ever fainter stars in the least amount of time possible and with a good exposure times, the Charged Coupled
Devices (CCDs) were developed. These devices greatly improved the collection of light from the RRL stars
and helped to determine the distances and chemical composition of GCs and even more distant objects such
as galaxies. On a historical note, at the beginning of the last century, the belief that the Milky Way (MW)
represented the entirety of the known Universe was still very much widespread. Harlow Shapley was the first
one to observe variable stars in GCs which he first wrongly assumed to be Cepheids. Since he knew that these
stars follow a well-known period-luminosity relation (discovered by Henrietta Leavitt in 1912) he was able to
determine the distances to GCs (albeit overestimated, since at the time, he was not aware of the presence of the
interstellar medium and the effect of extinction and how they affects distance measurements). Later, it turned
out that the stars that he had mistaken for Cepheids, were actually RRL stars which are fainter. Shapley then
made the assumption that the GC distribution around the Galactic center was symmetric, which allowed him to
estimate the overall size of the Galaxy, as well as the approximate position of the Solar System with respect to
it. As a result, he completely overturned the notion that the location of the Solar System was close to the inner
regions of the MW (Shapley, 1918).
With the CCDs now in the picture, we can take time series of images of a particular GC and measure the mag-
nitudes and colors of those stars with variable nature. The challenge is now to collect as much light as possible
from all the stars in the field of view of the CCD in the GC. This posses another technical issue: how to resolve
individual stars in the inner regions of GCs, where the brightnesses of the stars overlap due to a higher spatial
density. To solve this problem, numerical techniques of image differentiation have been developed which allow
to isolate blended stars. They correct for possible differences in the sensitivity of the CCD or optical defects
such as focus deficiency or turbulent atmospheric conditions. Depending on which physical processes we want
to focus on (which are dependent on the wavelength of the measured light), we need to select a specific filter
which can vary from the ultraviolet to the near infrared. The difference of any two filters, say V and I for exam-
ple, is a measure of the color of the star, which in turn is directly associated to its surface temperature; while its
magnitude depends on the intrinsic brightness as the distance at which it is located.
Once we have measured the magnitude and the color of a great number of individual stars in the GC, we can plot
the color vs. magnitude. This plot is known as the Color Magnitude Diagram (CMD) (see Fig. 4.6 in Chapter 4).
It is noteworthy that the distribution of stars on the CMD is not uniform at all, although there are well-defined
regions and sequences such as the Main Sequence (MS), the Red Giant Branch (RGB), the Turn Off point (TO),
the Horizontal Branch (HB) and the Asymptotic Giant Branch (AGB). These are a direct consequence of stellar
evolution. The CMD tells us how the stars have changed their brightness and temperatures during their evolution,
and allows us to study groups of stars of particular interest, such as the RRL stars.
In summary, detailed CCD observations of GCs enable us to recover the light curves of all sorts of variable stars
present in them. The decomposition into harmonics of the light curves of RRL stars in particular, allow us to
estimate their absolute magnitudes and metal content. In consequence, they are useful in determining the mean
distance and mean metallicity of the parental cluster by means of semi-empirical calibrations.
With the aim of studying and comparing possible differences in the evolutionary processes and in the structure
of the HB, we have chosen a sample of three GCs to be observed in VI filters: one very metal-poor (NGC 6205),
one with intermediate metallicity (NGC 6712) and one very metal-rich (NGC 6171). By accurately estimating
the pulsating periods of the RRL stars and their position on the Period-Amplitude or Bailey Diagram, we will
be able to determine their Oosterhoff classification either be it as OoI for metal-rich clusters or OoII for metal-
poor clusters. The distribution of RRab and RRc stars as well as their segregation with respect to their pulsation
1.1. NGC 6171 (M107) 3
modes or the simultaneous occupancy in the bimodal or "either-or" region, along with their possible secular
period changes, can in principle shed some light on the possible presence of more than one stellar population.
A brief description of each GC is given in the following sections.
1.1 NGC 6171 (M107)
NGC 6171 (M107, 0629-129) was discovered by Pierre Mecháin in 1782 and is a relatively metal-rich cluster.
It is located in the constellation of Ophiuchus at 6.4 kpc from the Sun and at 3.3 kpc from the Galactic center.
Its metallicity according to Jurcsik (1995) is [Fe/H] = -0.68 which makes it fall on the metal-rich side of GCs
and with a reddening of E(B −V) = 0.33. The CMD of this cluster shows a red HB with modes the of pulsation
between the RRab and RRc stars well split, a typical feature of an OoI-type cluster. Its RGB shows a notable split
at the height of the Sub Giant Branch (SGB), suggesting the presence of two different stellar populations (Piotto
et al., 2015). Prof. Christine Clement kindly provided us with observational data from the early decades of the
20th century. These data will add over 40 years to the data time-base and will enable us to study the secular
period changes in the population of RRL stars.
1.2 NGC 6205 (M13)
NGC 6205 (M13, C1639+365) was discovered by Edmond Halley in 1714 and is one of the most prominent and
better studied GCs in the celestial northern hemisphere. It is located in the direction of the Hercules constellation
in the outer halo of the Galaxy at about 7.1 kpc from the Sun and 8.4 kpc from the Galactic center. Given its
location, it presents a low reddening value (E(B − V) = 0.02). There is only one known RRab (V8) with a
period P = 0.750303 d which suggests an OoII-type cluster. This is relevant given the fact that its metallicity
is low among other globular clusters ([Fe/H] = -1.53 as given by Harris (1996) (2010 edition)). A remarkable
feature of its CMD is its predominantly blue HB with the modes of pulsation between RRab and RRc stars well
split. Its RGB branch has been found to be split into three subbranches, showing a difference in the abundance
of light elements such as O and Na, between them. This implies the presence of at least three different stellar
populations (Monelli et al., 2013).
1.3 NGC 6712
This GC was discovered by Guillaume Le Gentil in 1749 and it is located in the constellation of Scutum, a little
bit south from the Galactic disc, at a distance of 6.9 kpc from the Sun and just 3.5 kpc from the Galactic center
which places it within the Galactic Bulge. Some of the implications of its position is that its reddening is high
(E(B − V) = 0.45) and possibly differential. Its CMD shows a remarkable contamination due to field stars and
also a sparsely populated but predominantly red HB. The RGB of this cluster also shows a bifurcation signaling
the presence of two stellar populations (Monelli et al., 2013). Its metallicity is intermediate at [Fe/H]=-1.02
(Harris, 1996) (2010 edition).

5
Chapter 2
Stellar structure and evolution of low-mass
stars
It is the main goal of this thesis to study the properties of RRL stars as indicators of the mean physical parameters
of GCs in the Galaxy. But first, it is in order to provide an overview on how a star with initial mass M ≈ 2.0 M⊙
- 2.5 M⊙ is formed, and how it evolves into an RRL star with mass M ≈ 0.6 M⊙ - 0.8 M⊙ by the time it reaches
its pulsation evolutionary stage on the HB.
In order to propose a model that can approximately describe the initial state of a star, we first need to make
a number of reasonable assumptions that will be refined over time to produce an accurate model. The basic
assumptions are outlined next and were taken from Salaris et al. (2005):
• The stars are spherically symmetric and are made only of matter and radiation. Rotation and magnetic
fields are negligible.
• The evolution of the physical and chemical properties of a star has to be slow so we can describe them as
a series of states in hydrostatic equilibrium. This assumption means that the pressure increases toward the
center of the star. If the pressure increases then the density and temperature must increase as well.
• The matter in each stellar layer is very close to local thermodynamic equilibrium. This hypothesis implies
that the mean free path is much smaller than the radius of the star, and that the time elapsed between
collisions is much smaller than the timescale for the change of the microscopic properties of the gas.
This means that at each point within the star, radiation can be well described by the Planck function
corresponding to the unique temperature in common with the matter. This also means that each stellar
layer can be assumed to behave like a black body, with (almost) no net energy flux absorbed or emitted.
In reality there must be a small outgoing flux, otherwise stars would not shine.
• The effects of rotational mixing and atomic diffusion are negligible, so convection is the only mechanism
for chemical element transport within stars.
We now need to propose the minimum variables for which a star depends on for its evolution. The natural
variables that one can immediately think of are the mass m, the temperature T, the density ρ and the pressure
P, but that is just half of the problem. We also need to figure out how we can combine these variables in order
to come up with a first approximation and to see if there are any other unknown variables that we might have
ignored that could be playing a relevant role in the evolution of a star.
6 Chapter 2. Stellar structure and evolution of low-mass stars
Mass continuity
This equation comes from the mass-conservation law. It is clear that stars do not conserve their mass, but rather
they transform some of it into energy by means of nuclear fusion at their cores. The amount of mass transformed
into energy is small enough to be neglected as if there were no nuclear reactions taking place.
∂r
∂m
=
1
4πr2ρ
(2.1)
Hydrostatic equilibrium
For a star to exist, it first needs to be at equilibrium with itself. In this case, the forces acting on a shell inside
the star are the pressure and the force of gravity, so:
∂P
∂m
= −
Gm
4πr4
(2.2)
.
It is important to mention that, in general, the star is not in hydrostatic equilibrium, i. e., there is an imbalance
between the pressure and the force of gravity that leads to the shell to experience an acceleration causing the star
to pulsate, in which case, the last equation can be written as:
∂2r
∂t2
= −4πr2 ∂P
∂m
−
Gm
r2
(2.3)
Note that if the acceleration term is equal to zero, we can recover the original equation of hydrostatic equilibrium.
Energy conservation
Similar to the mass-conservation law, we can also make use of the energy-conservation law to describe how much
energy is being produced in a shell at a radius r inside a star. For this we use the parameter ϵ. This parameter
depends on three things: the density, the temperature and the chemical composition of the shell. The luminosity
produced at a radius r is related to ϵ as:
∂L
∂m
= ϵ − ϵν − ϵg (2.4)
where ϵ is the total energy produced due to nuclear reactions, ϵν is the energy lost due to neutrinos and ϵg is the
energy produced or lost due to gravitational expansion or contraction.
Energy transport
This equation describes how energy created due to nuclear reactions in the core of the star is transported from
its interior to its surface and then released into space as photons. In general, there are two mechanisms of energy
transport: radiation and convection. If the inner regions of a star are sufficiently transparent, the energy will
be transported by photons. In this case, the radiation mechanism is more efficient. On the other hand, if the
material is highly opaque to radiation, the radiation will get trapped momentarily causing the temperature and
Chapter 2. Stellar structure and evolution of low-mass stars 7
the pressure to rise locally; this has the effect of creating a convection current, so convective transport will be
more efficient in this case. The general form of the equation of energy transport is:
∂T
∂m
= −
GmT
4πr4P
∇ (2.5)
where the temperature gradient ∇ is defined as:
∇ ≡
∂lnT
∂lnP
. (2.6)
If the energy transport is radiative, then:
∇rad =
3
16πacG
κRLP
mT4
. (2.7)
where a= 7.5657×10−16 Jm−3K−4 is the radiation constant, c is the speed of light (2.9976×108 ms−1) and κR
is the Rosseland mean opacity estimated as:
1
κR
=
∫ ∞
0
1
κν
∂Bν
∂T dν
∫ ∞
0
∂Bν
∂T dν
(2.8)
where κν is the monochromatic opacity and Bν is the monochromatic Planck function. The equation 2.5 can also
be used to describe convective energy transport as long as ∇ has an appropriate value. The determination of this
value is extremely complicated since it depends on several approximations due to the fact that the flow of gas in
the convective region of a star is turbulent.
After the stellar structure equations have been mentioned, it is important to clarify what is considered to be a
low-mass star. Taking into account the aforementioned basic assumptions listed at the beginning of this chapter,
we also need to add that the stars have to be able to develop an electron-degenerate core after leaving the MS and
to ignite the triple α process in order to produce an active He-burning shell during their red giant phase. These
characteristics are only met if the mass of the star is at most M ≈ 2.0 M⊙ - 2.5 M⊙.
The birth and evolution of a low-mass star as outlined by Catelan (2007) will be described next and a schematic
is shown in Fig. 2.1:
1. A portion of a molecular cloud collapses due to the force of gravity once the Jeans mass has been exceeded.
This portion is still very extended and hence it is very cold (∼ 10K) and isothermal. Once the cloud has
started experiencing gravitational collapse, it will become optically thick, its temperature will rise and will
no longer be isothermal.
2. After the gravitational collapse, a small core forms and it starts accreting material causing an increase in
luminosity and temperature.
3. The accretion dissipates and the proto-star becomes visible. Its structure is fully convective and its lumi-
nosity decreases while the temperature remains almost constant (the Hayashi track). After the Hayashi
track, a radiative core is formed and so, the proto-star increases its temperature and its brightness again
while still going through gravitational collapse. The energy production by means of the proton-proton
chain starts converting hydrogen into helium in the core of the star which makes it settle on the ZAMS.
8 Chapter 2. Stellar structure and evolution of low-mass stars
4. When the star reaches the ZAMS, it stops contracting due to the pressure coming from the inside and
reaches hydrostatic equilibrium. The evolution of the star in this phase is very slow and its temperature
and luminosity increase very little.
5. At this stage, the hydrogen burning into helium at the core of the star ceases and moves to an external shell.
This marks the departure of the star from the MS. The helium core keeps growing due to hydrogen being
burned at the outer shell. Once the core has reached ∼ 10% of the total mass of the star (the Schönberg-
Chandrasekhar mass) the core collapses and becomes degenerate. This translates into an increase of heat
and to an energy release. The temperature at the core increases in such a way that now hydrogen is fused
into helium via the CNO cycle which becomes the primary source of energy of the star. Part of the energy
being produced is used to expand the envelope of the star which in consequence causes the star to cool
down and enters the sub-giant phase. The expansion of the envelope allows for convection to take place
as the star reaches the base of the RGB.
6. In the RGB phase, the convective envelope reaches deep inside the star and starts carrying material pro-
cessed at its core to the surface (first dredge-up).
7. As the hydrogen burning shell advances outwards, it will encounter a chemical composition discontinuity
created by the first dredge-up. This discontinuity provides extra fuel to the hydrogen-burning shell which
causes the star to momentarily revert the direction of its evolution before resuming its ascent on the RGB.
8. Once the star has reached the tip of the RGB, the temperature at its core is sufficiently high to ignite the
helium burning in a shell inside the core. Since the core is in a degenerate state the increase in temperature
does not translates into an expansion and cooling, so while the triple α reactions take place, this also
increases the temperature, causing the reactions to happen faster and so on. This gives place to a thermal
runaway process which will only stop after the degeneracy is lifted. In this stage these stars lose most of
their initial mass until they are left with M ≈ 0.6 M⊙ - 0.8 M⊙, which are typical for RRL stars. Once the
core is non-degenerate, it will start burning helium quiescently and the star will settle in the ZAHB.
9. The ZAHB phase is characterized by the quiescent helium-burning in the core of the star. Irrespective
of their mass, the stars have nearly the same luminosity in this stage. Depending on how much mass the
stars have lost during their RGB stage, the stars will position themselves on either the red side of the HB if
they have lost a small amount or on the blue side if they have lost a large amount. During this stage, some
stars will cross the IS, making them susceptible to pulsation. The stars affected by pulsation on the HB
are known as RRL stars.
10. The next phase corresponds to when the core of the star has used up all the helium and starts ascending
on the AGB increasing its luminosity. The core is now made out of carbon and oxygen and is inert, but
the helium burning continues in a shell and the hydrogen burning in a more exterior shell. After the mass
of the envelope has become very low, the star goes through a superwind phase, which means that the star
(now a post-AGB star) sheds its outermost layers leaving the core of the progenitor exposed which may or
may not lit a surrounding planetary nebula. After a quick evolution to the blue, the only remaining object
is the inert core of the star which after a decrease in luminosity finally ends its life as a white dwarf.
 
 
11.1 a
 
.
á
 
q
 
. 
Ñ 
e
 
cloud 
collapse 
or 
P 
 
 
! 
E 
G 
o 
  
log(L/L ¿) 
a
L
 
evolution 
of 
a 
low-mass 
star 
(schematic) 
  
  
 
 
6.0 
45 
40 
3.5 
3.0 
l
o
g
 
(
T
o
n
)
 
1.0
Chapter 2. Stellar structure and evolution of low-mass stars 9
Figure 2.1: Evolution of a low-mass star in the HR diagram. The numbers correspond to specific
stages in the evolution of the star described in Chapter 2 (Catelan, 2007).

11
Chapter 3
RR Lyrae stars
Figure 3.1: Schematic Hertzsprung-Russell diagram illustrating the loca-
tion of several classes of pulsating stars. The dashed line shows the zero-
age main sequence, the continuous curves are selected evolution tracks,
at masses 1, 2, 3, 4, 7, 12 and 20 M⊙, the dot-dashed line is the horizon-
tal branch and the dotted curve is the white-dwarf cooling curve. Picture
credit: Dr. Péter I. Pápics. Caption from Christensen-Dalsgaard (1998).
RRL stars are old, metal-poor low-mass
pulsating stars (M/M⊙ ≈ 0.6 - 0.8) with
periods between about 0.2 d and 1.2 d. In
the Hertzsprung-Russell diagram (HRD)
they are located on the HB where they
cross the IS (see Fig. 3.1).
This strip extends diagonally on the CMD
and most of the stars within it are pulsating
variables. In terms of surface temperature,
they are in the range of 6000 K - 7500 K
which makes them as of A or F spectral
types. As noted in Chapter 2, when these
stars reach the ZAHB, they will burn He in
their cores for about 108 years.
RRL stars can be found everywhere in our
Galaxy and also within the Local Group
galaxies, however, they are too faint to
be detectable beyond it. Given the time
it takes for a low-mass star to potentially
evolve into an RRL star (> 10 Gyrs),
their presence in a GC is also an indica-
tor of an old stellar population. RRL stars
are not expected to produce a significant
amount of heavy elements, this means that
the measured chemical abundances in their
atmospheres are more related to the in-
terstellar gas cloud from which they were
formed. This makes them excellent tracers
of the chemical composition history of the
Galaxy.
12 Chapter 3. RR Lyrae stars
The first person to detect RRL stars (and variable stars in general) in GCs was Solon I. Bailey at the Harvard
College Observatory in 1893 through photographic plates. After the independent discoveries of variable stars in
GCs such as 47 Tucanae and ω Centauri by E. C. Pickering and Williamina Flemming, Bailey conducted a study
of variable stars in photographs of 23 GCs between 1895 and 1898, discovering more than 500. According to
Pickering et al. (1901), Williamina Flemming discovered a seventh magnitude variable star in the constellation
of Lyra with a period of 0.56 d. He noted that the characteristics of its light curve were indistinguishable from the
cluster-type variables found by Bailey and it was labeled as RR Lyrae and to date, it is the brightest known member
of its class. Bailey noted that most of his discovered variables shared similar properties, such as their amplitudes
being of about one magnitude in the blue filter and their periods of pulsation of about a day. He divided them
into three separate categories: a, b and c, and since they were first discovered in GCs, they originally received
the name of cluster variables. As for the morphology of the light curves of a- and b-types, they show asymmetry
with roughly the same periods of pulsation (0.4 d - 1.0 d), the main difference being the amplitude since in the
a-type is larger (∼ 1.0 mag in the V filter) than in the b-type (∼ 0.6 mag in the V filter). For c-type stars, the
period of pulsation is shorter (0.2 d - 0.4 d) and the amplitude is smaller (∼ 0.5 mag in the V filter) compared to
those of the a- and b-types. The similarity between the a- and b-type stars prompted their merging into a single
category now known as RRab stars, while the c-type stars kept their original classification as RRc stars (see Fig.
3.2).
0.0 0.5 1.0
φ
14.5
15.0
15.5
16.0
V
V8
0.0 0.5 1.0
φ
14.6
14.8
15.0
15.2
V7
0.0 0.5 1.0
φ
14.7
14.8
14.9
15.0
V36
Figure 3.2: The different light curve shapes of RRL stars. From left to right: RRab (P =
0.559922 d), RRc (P = 0.312668 d) and RRd (P1 = 0.31596 d, P2 = 0.30424 d). The black
dots come from IAO and the blue dots from IAC. (Deras et al. 2018; Deras et al. 2019).
When plotting the amplitude of the RRL stars vs. their period of pulsation (Bailey Diagram, see Fig. 3.3), it is
apparent that the a-type and b-type stars shared a common region while the c-type stars were in another. One
important feature is that RRab stars pulsate in the fundamental radial mode while the RRc pulsate in the first-
overtone radial mode as first noted by Schwarzschild (1940). There is a more rare type of RRL star which has
been found to pulsate simultaneously in both the fundamental radial mode and the first-overtone radial mode.
These stars are called double-mode variables and satisfy a period ratio of P1/P0 = 0.745 (Alcock et al., 2000)
and have received the RRd designation (Nemec, 1985). These stars took particular relevance because due to their
double mode nature, they allow to determine their masses independently of the stellar evolution theory. A brief
description of radial and nonradial modes is provided in section 3.1.2. It is important to mention that during their
evolution along the HB, RRL stars may switch pulsation modes from fundamental to first overtone and back to
fundamental when they cross the IS.
Chapter 3. RR Lyrae stars 13
Figure 3.3: Period-amplitude diagram for RRL stars in the globular clus-
ter M3, based upon Cacciari et al. (2005). RRab stars known to show the
Blazhko effect have been excluded. The amplitudes of Blazhko variables
change during a secondary pulsation cycle (caption and figure taken from
Smith, H. A. et al. (2009)).
van Albada et al. (1973) proposed that
there exists a hysteresis zone on the HB
where the RRL cross the IS (see HF re-
gion in Fig. 3.4). RRL stars entering
this hysteresis zone would keep the pul-
sation mode that they had when they en-
tered, i.e., if the RRL were pulsating in
the fundamental radial mode (RRab), then
they would keep doing so while crossing
the either-or region to the blue side of the
IS and conversely, RRc stars would keep
pulsating in the first overtone while cross-
ing to the red side of the IS. This either-or
region is located near the center of the IS in
which both the fundamental and the first-
overtone modes can in principle be excited,
so it is shared by both RRab and RRc stars.
RRL stars follow a tight linear P-L rela-
tion in the K-band (Longmore et al., 1986)
although Catelan (2004) has derived P-
L relations in the UBVRI JHK Johnson-
Cousins-Glass system. This makes them
particularly useful as yardsticks to estimate
distances in the local universe.
Figure 3.4: Hysteresis of RRL stars on the IS. Filled circles represent RRab stars and empty
circles represent RRc stars. The H region is populated only by RRc stars, the F region only by
RRab stars and the HF region simultaneously by both RRab and RRc stars (Caputo et al., 1978).
14 Chapter 3. RR Lyrae stars
3.1 Driving mechanisms of pulsation
In order for a star to pulsate, it needs two fundamental drives: an energy source and a mechanism that excites
the pulsation and keeps it going. As for the energy source, the stars have an abundant production of it provided
by the nuclear engine at their cores. As noted by Eddington (1926), the energy needed to maintain the pulsation
is quite small, so the problem is not finding the energy source, but rather identifying the mechanisms that excite
and maintain the pulsations i.e., that transform the nuclear energy into mechanical oscillations. He proposed
that the mechanism of pulsation should resemble that of a thermodynamic heat engine: adding heat when the
temperature is high and releasing it when the temperature is low, during the pulsation cycle; a valve that regulates
the flow of energy. In the case of the RRL stars (and other well known radial pulsators such as Cepheids and δ
Scuti stars), their pulsations are driven by two different mechanisms namely, κ and γ that will be briefly described
in the following section.
3.1.1 The κ and γ mechanisms
The zone inside the star where the Eddington valve takes place is where the helium becomes doubly ionized as
explained by Zhevakin (1953) and Cox et al. (1958). This ionized zone is of particular interest since it creates
a region where the opacity plays an important role. The opacity can be characterized by the Rosseland mean
opacity as defined in Chapter 2. The way in which the Rosseland mean opacity depends on the temperature and
density can be described by the following relation:
κR = κ0ρnT−s (3.1)
where κ0 is a constant, ρ is the density and T is the temperature. Under ordinary circumstances, the opacity
would decrease as the temperature increases, but since ionization is taking place, the the opposite occurs: as the
temperature increases, the opacity increases as well. The increment in opacity is what traps the energy coming
from within the star. This mechanism is called the κ mechanism (Baker et al., 1962). A second mechanism
called the γ mechanism (Cox et al., 1966) also takes place and depends on the adiabatic exponent Γ3 which
controls how much heat a layer can gain during compression. When the star compresses, it absorbs heat and
instead of raising the temperature of the gas, it goes into ionization trapping the energy coming from the core.
At the same time, recombination occurs and the opacity decreases and in consequence all the energy that was
initially accumulated, is released over a short period of time in the subsequent expansion. It is important to note
that the κ and γ mechanisms can work together.
There is a third mechanism, the ϵ mechanism, that takes place in the regions of the star where the nuclear reactions
are being produced. This modulates the nuclear reaction rate in the core, because if the star is compressed, the
temperature increases, thus the production of energy increases as well and vice-versa. This behavior allows for
the pulsational instability to take place. But Catelan et al. (2015), concluded that temperature fluctuations at the
hydrogen-burning shell are of about ∼ 1 K while at the Helium-burning core are ∼ 0.5 K. In the case of the RRL
stars, their temperature fluctuations are of about 1000 K - 1500 K according to Kolenberg (2002) and of about
2000 K - 3000 K according to de Boer et al. (2010). This means that the ϵ mechanism does not play a relevant
role in the pulsation of RRL stars.
3.1. Driving mechanisms of pulsation 15
3.1.2 Radial and nonradial modes
Since stars are spherical three-dimensional objects, it is expected that they are prone to experience oscillations in
three perpendicular directions. These directions are best described by spherical coordinates i.e., (r, θ, ϕ), where
r is the radius of the star, θ is the co-latitude and ϕ is the longitude. Assuming a spherically symmetric star, the
displacements in each direction are given by:
ξr(r, θ, ϕ, t) = a(r)Ym
l (θ, ϕ)exp(−2iπνt), (3.2)
ξrθ(r, θ, ϕ, t) = b(r)
∂Ym
l (θ, ϕ)
∂θ
exp(−2iπνt), (3.3)
ξϕ(r, θ, ϕ, t) =
b(r)
sin θ
∂Ym
l (θ, ϕ)
∂θ
exp(−2iπνt), (3.4)
where a(r) and b(r) are the amplitudes, ν is the oscillation frequency and Ym
l are the spherical harmonics defined
as:
Ym
l (θ, ϕ) = (−1)m
√
2l + 1
4π
(l − m)!
(l + m)!
Pm
l (cos θ)exp(imϕ) (3.5)
and Pm
l are the Legendre polynomials given by:
Pm
l (cos θ) =
1
2l l!
(1 − cos2 θ)m/2 dl+m
d cosl+m θ
(cos2 − 1)l (3.6)
There are three numbers that specify the modes of pulsation on a star, namely n, l and m. The number n represents
the number of radial nodes; l is the angular degree of the mode and represents the number of surface nodes, and
m is the azimuthal order of the mode. In particular, |m| is how many surface nodes are lines of longitude. In
order to determine the number of nodal lines present on the surface of a star, it is sufficient to calculate the value
of l − |m|. The simplest pulsation mode that we can think of is when the star pulsates in the fundamental radial
mode. This condition is met when l = 0. In this case, the star expands and contracts spherically with the core
staying at rest (a node) while the surface experiences a displacement either inwards or outwards (an antinode).
In the case of the first overtone radial mode, there exists a radial node between the core of the star and its surface.
This radial node takes place in a concentric shell inside the star and does not move. The antiphase motions occur
above and below this shell. As previously mentioned, RRL stars can either pulsate in the fundamental radial
mode (RRab), in the first radial overtone (RRc), or simultaneously in both (RRd).
The simplest case for a nonradial pulsation mode is when l = 1 and m = 0. This produces a dipole, i.e., a nodal
line around the equator of the star, making one hemisphere expand while the other one contracts and vice versa.
For l = 2 it will produce a quadrupole and for l = 3, an octupole. In order for nonradial modes to occur, n ≥ 1.
This implies that there is at least one radial node within the star. Currently we can only resolve the surface of the
Sun, but not other stellar surfaces.
16 Chapter 3. RR Lyrae stars
3.2 The Blazhko effect
This effect first described by Sergey Blazhko in 1907 (Blazhko, 1907) is a long-term modulation of the amplitude,
phase and shape of the light and radial velocity curves. It has been detected in Cepheids, RRL stars, and δ Scu
stars (Breger, 2010). It spans a range between 50-500 times the period of pulsation. This effect is intermittent. To
try to explain this effect is beyond the scope and goals of this work since to date, there is no consensus regarding
the physical mechanism that drives it. Some hypotheses/ideas include:
• Magnetic oblique rotator/pulsator (Shibahashi, 2000).
• Nonresonant radial/nonradial double-mode pulsator (Cox, 2013).
• Resonant nonradial rotator/pulsator (Nowakowski et al. 2001, Nowakowski et al. 2003).
• Radial mode resonance of 9:2 (Buchler et al., 2011).
• Magnetic dynamo-driven convection (Stothers, 2011).
• Periodic energy dissipation driven by shock wave dynamics (Gillet, 2013).
In Fig. 3.5 we show two light curves of RRab stars in the GC NGC 6171 who display a noticeable Blazhko effect
in amplitude and phase.
0.0 0.5 1.0
φ
15.0
15.5
16.0
V
V16
0.0 0.5 1.0
φ
15.2
15.4
15.6
15.8
V5
Figure 3.5: Blazhko effect in two RRab stars in NGC 6171. Note that the light curves in each
panel are plotted with their respective period and phase but observed at different epochs (blue and
black dots). The left panel shows an amplitude modulation and the right panel a phase modulation,
both features of this effect. The black dots come from IAO and the blue dots from IAC (Deras
et al., 2018).
17
Chapter 4
Globular clusters
GCs are the largest and oldest stellar structures in the Galaxy with ages between 10.5 Gyrs - 13.5 Gyrs (Jimenez,
1998), placing their formation around the time of reionization (Forbes et al., 2015) and around the time of for-
mation of the Galaxy itself.
Figure 4.1: Globular cluster M13. Thousands of stars can be spatially
resolved as well as their individual brightness and color. Image credit:
Canada-France-Hawaii Telescope / Coelum. 1
These clusters contain from a few hundred
thousands to a million stars tightly bound
by the force of gravity and characterized
by high central densities. The majority of
them are located in the halo around the
Galactic core while a small amount are lo-
cated near the bulge. They usually occupy
a spherical volume of around 100 pc3 (see
Fig. 4.1).
To get an idea about their size, a vol-
ume with the size of the Solar System can
contain around 15,000 stars. According
to Lynn (1886), it was Alexander Ihle, a
German amateur astronomer, the first to
discover a GC in 1665, now known as
M22. Due to low-quality instrumentation
available at the time, he described it as a
cloud-like spot. It was not until the late
1700’s that William Herschel was able to
resolve individual stars since he had ac-
cess to the largest telescopes of his time.
It was Herschel himself who coined the
term Globular Cluster (Herschel, 1789).
There are currently roughly 150 GCs in
the MW although there are estimations that
this number can go up to 180 ± 20 (Ash-
man et al., 1992) since it is possible that
many of them are hidden behind the cloud
1http://www.cfht.hawaii.edu/HawaiianStarlight/HawaiianStarlight.html
18 Chapter 4. Globular clusters
of dust and gas in the disc. Almost every galaxy studied to date has their group of GCs associated to them (Har-
ris, 1991). The GCs can contain a wide variety of stars, similar to the ones found in the bulge of the Galaxy.
An assortment of variable stars can also be found in GCs of the likes of quiescent Low-Mass X-Ray Binaries
(LMXBS), cataclysmic variables (CVs), millisecond radio pulsars, W Vir stars, Blue Stragglers, SX Phe, and
chromospherically active binaries, to name a few (Heinke, 2010), but their signature stars are the RRL type,
which were discussed in Chapter 3.
4.1 On the origin of GCs
As for the origin of GCs themselves, the Λ Cold Dark Matter model (ΛCDM) predicts the formation of structures
from the bottom to the top, i.e., small structures merge together to form the larger structures (galaxies) we see
today. The MW itself is a clear example of this mechanism as Ibata et al. (1994) demonstrated with the dwarf
spheroidal galaxy Sagittarius going through a process of disruption, also halo stellar streams have been observed
to cross the solar neighborhood (Helmi et al., 1999), and the discovery of stellar debris from the dwarf galaxy
Gaia-Enceladus merge with our Galaxy (Helmi et al., 2018). Such events have naturally prompted the accretion
of GCs by the MW as it was in the case of this last event where at least 8 GCs were accreted according to Myeong
et al. (2018). What role do galaxy mergers play regarding the properties of GCs is still not clear. On one hand
they could aid in the assembly of giant molecular clouds while on the other hand they could also increase the
destruction rate of stars clusters.
Although the origin of GCs in the Galaxy is still not yet understood, knowing the age and metallicity of GCs
can help set rigorous constraints on models of their formation. By knowing the age and redshift can also help us
understand better the link between them and the formation of the Galaxy. Searle et al. (1978) were the first to
undertake the study of the origin of the 150 GCs in the Galaxy. The estimation of relative ages by Marín-Franch
et al. (2009) and homogeneous measurements of the metallicity have led to the discovery that the age-metallicity
relation for Galactic GCs is bifurcated, implying that young GCs located in the metal-poor branch have halo-like
kinematics, hence they are more likely to have been accreted whereas the young GCs located in the metal-rich
branch with disc-like kinematics are more likely to have formed in situ. This was also confirmed by Tonini
(2013) who performed Monte Carlo simulations of the assembly history of galaxies and concluded that the
metallicity distribution of the GCs is in general, bimodal, differentiating between metal-rich GCs formed in situ
and metal-poor GCs accreted from satellite galaxies. Out of the 150 GCs in the Galaxy, about 40% likely formed
in situ; 35% might have a known merger origin distributed between the Gaia-Enceladus (19%), the Sagittarius
dwarf galaxy (5%), the Helmi streams (6%), and the Sequoia galaxy (5%). 16% of the remaining GCs are
possibly associated to a group with high bound orbits while the rest possess low bound orbits, suggesting a more
heterogeneous origin (Massari et al., 2019). According to the aforementioned study, NGC 6205 formed as a
result of the merging of the dwarf galaxy Gaia-Enceladus while NGC 6712 and NGC 6171 were formed in situ.
There are currently a couple of possible scenarios on which the origin of the GCs could be explained in terms
of galaxy formation. The first scenario explains GCs formation as the result of special conditions in low-mass
dark matter halos at/or during the reonization epoch (Peebles et al. 1968; Trenti et al. 2015). For example, the
formation of compact stellar clusters due to H2 cooling, as suggested by Padoan et al. (1997) and formation
triggered by shocks induced by a hydrogen ionization front (Cen, 2001), to name a couple. Their advantages are
that the models rely on fundamental physical processes that are well-established at the time of early Universe
(z > 6). On the downside, the models tend to be more on the qualitative side and the conditions on which the
GCs formed are no longer taking place at lower redshifts, which would pose a problem when trying to explain
the formation of younger GCs. The second scenario suggests that the origin of GCs is a natural consequence of
galactic star formation process (Swinbank et al., 2011, & references therein) mainly for two reasons: the first one
is that the high turbulent pressure in galaxies leads to a high fraction of star formation (Kruijssen, 2012) which
4.2. On the evidence of multiple stellar populations in GCs 19
is similar to what is observed in active star-forming galaxies in the local Universe (Adamo et al. 2015; Johnson
et al. 2016); the second one is that the formation of massive giant molecular clouds could be prompted by the
high gas pressures in the turbulent interstellar medium (Reina-Campos et al., 2017).
On a last note, there is some evidence that another more exotic type of GCs might exist. The so called Dark
Globular Clusters (Taylor et al., 2015). The authors measured the dynamical masses Mdyn and the corre-
sponding mass-to-light ratio 
dyn
V of 89 compact stellar systems in the nearby elliptical galaxy NGC 5128. They
found two distinct sequences on the 
dyn
V - Mdyn plane where one of them "appears to be populated by a distinct
group of objects that require significant dark gravitating components such as central massive black holes and/or
exotically concentrated dark matter distributions." They suggest the possibility of the presence of dark matter
mass components to try to explain the high value of the 
dyn
V , which could not be explained by the mere presence
of intermediate-mass black holes.
4.2 On the evidence of multiple stellar populations in GCs
It was initially thought that all the stars in the GCs were formed at the same time from a giant molecular cloud and
have been usually considered as the best example of a simple stellar population. The supporting evidence was the
existence of well-defined MS - TO - SGB - RGB sequences on the CMD showing no bifurcations and/or spread
other than the expected photometric errors and a homogeneity in the chemical composition of their component
stars. This notion was then put into question in light of new evidence provided by more accurate photometric
and spectroscopic observations, which showed that multiple stellar populations (MSP) are present in almost all
the Galactic GCs. One of the problems that led to this affirmation was that while GCs are homogeneous in their
iron content, they also show a large star-to-star variation in the abundance of some light elements like C, N, O,
Al, etc. Cohen (1978) and Osborn (1971) provided the first empirical spectroscopic evidence that GCs might
be composed by MSP by finding chemical inhomogeneities in bright RGB stars in the clusters M3, M13 for the
former and M5, M10 for the latter. On the photometric side, the evidence in favor of MSP in GCs came with the
availability of the data from the Hubble Space Telescope (HST). This allowed to plot deep CMDs of known
GCs such as ω Cen and NGC 2808, which clearly showed the presence of more than one TO point, MS branch
and RGB (Bedin et al. 2004; Piotto et al. 2007).
Monelli et al. (2013) carried out a study of 22 Galactic GCs and proposed the CU,B,I index as a new diagnostic
tool for MSP and defined it as CU,B,I = (U − B) − (B − I), where U, B and I are the standard Johnson
filters. The selection of this particular difference of colors is that remarkably separates the stars with different
helium abundances ((B − I) color, Marino et al. (2008); Sbordone et al. (2011)) and light-element abundances
((U − B) color, Piotto et al. (2007); di Criscienzo et al. (2011)) in a plot of Mv vs. CU,B,I , which they called
a pseudo − CMD. Note that the pseudo-CMD appears reversed in comparison to a typical CMD. The reason
is that the CU,B,I index gets less negative for MS stars with increasing brightness, while the SGB bends towards
more negative colors. Among the clusters studied, they included NGC 6205 and NGC 6712. For NGC 6205,
they found that its RGB split into three RGBs namely RGBa, RGBb and RGBc. The differences between them
being the abundances of Na and O. Stars belonging to the RGBa branch are Na-poor and O-rich, stars belonging
to the RGBc branch are Na-enhanced and O-depleted, and stars belonging to the RGBb branch have intermediate
properties (see Fig. 4.2). For NGC 6712, I filter data were unavailable at the time so they defined an alternate
index CU,B,V which was noted to have a smaller sensitivity than the CU,B,I . Nevertheless, it allowed them to
identify well-defined substructures forming at least two RGBs (see Fig. 4.3).
On the other hand, Milone et al. (2013) defined an index CF275W,F336W,F438W = (mF275W −mF336W)− (mF336W −
mF438W), based on the HST filter system which was later used by Piotto et al. (2015) to efficiently separate MSP
20 Chapter 4. Globular clusters
Figure 4.2: MSP in NGC 6205. First panel: Pseudo-CMD of NGC 6205. Second panel: Zoom
of the RGB. Third panel: Separation of three stellar subpopulations. Na-poor and O-rich stars in
green, Na-enhanced and O-depleted in cyan and stars with intermediate properties in magenta.
UBI are the Johnson standard filters (Monelli et al., 2013).
Figure 4.3: MSP in NGC 6712. First panel: Pseudo-CMD of NGC 6712 (a1). Second panel:
Zoom of the RGB. The red line is the RGB fiducial line and was drawn by hand (a2). Third panel:
Rectified RGB (a3) and a histogram showing the distribution of ∆cubv color differences (a4). The
bimodal distribution found is evidence of the presence of two different stellar populations in NGC
6712 (Monelli et al., 2013).
4.2. On the evidence of multiple stellar populations in GCs 21
Figure 4.4: MSP in NGC 6171. The use of the CF275W,F336W,F438W index clearly shows the
presence of two different stellar populations on its RGB. Magnitudes and colors are in the HST
filter system (Piotto et al., 2015).
throughout the CMD in a sample of 57 Galactic GCs. The reason behind this choice of filters is that first gen-
eration stars, i.e., stars which are oxygen- and carbon-rich and nitrogen-poor, are brighter in F336W than in
F275W and F438W. For second generation stars whose material has been processed through the CNO-cycle,
the opposite is true: they are oxygen- and carbon-poor but nitrogen-rich. In consequence, they are relatively
bright in F275W and F438W but faint in F336W. This means that first generation stars are bluer than second
generation stars in one color (F336WF438W), but redder in another (F275WF336W). One of the GCs studied
in the sample mentioned above was NGC 6171 in which its pseudo-CMD clearly shows a bifurcation of its RGB
indicating the presence of at least two different stellar populations (see Fig. 4.4).
22 Chapter 4. Globular clusters
4.3 The Oosterhoff dichotomy
It has been noted since early works of Pieter Oosterhoff (Oosterhoff 1939; Oosterhoff 1944) that the GCs can be
differentiated by the frequency distribution of their periods of pulsation of their RRab and RRc stars (see Fig.
4.5). He based this result on the study of five GCs namely ω Cen, M3, M5, M15 and M53, but as the number
of GCs discovered in the MW increased, it became evident that this period distribution was not the same for all
clusters.
Figure 4.5: Period-frequency distribution of RRL variables in clusters
of Oosterhoff type I (top panel) and Oosterhoff type II (bottom panel)
(Clement et al., 2001).
We now classify GCs into two main
groups, called the Oosterhoff groups: the
OoI in which
⟨
Pab
⟩
≈ 0.55 d and OoII in
which
⟨
Pab
⟩
≈ 0.65 d. When values of
[Fe/H] became available, it was also noted
that OoI type clusters were more metal-
rich than the OoII type.
There is a special case in which GCs oc-
cupy an odd position in the plot
⟨
Pab
⟩
vs. [Fe/H], such as NGC 6388 and NGC
6441. These type of clusters are some-
times referred to as OoIII type clusters
or just atypically metal-rich OoII clusters.
When measuring
⟨
Pab
⟩
and
⟨
[Fe/H]
⟩
of
GCs, one finds that the OoI and OoII clus-
ters present an evident segregation, while
leaving a gap in between. This gap is of-
ten referred to as Oosterhoff gap (Catelan
2004; Catelan 2009). After analyzing the
GCs in the MW satellite galaxies, it was
found that the GCs in them fill this gap
and are called Osterhoff-intermediate stel-
lar systems, although it is important to note that not all galaxies follow this trend (see Fig. 5 in Catelan (2009)).
4.4 The color-magnitude diagram
The CMD is the observational counterpart of the HRD and plots the color index of a population of stars (the
difference between any two photometric bands) in the x-axis vs. the magnitude of the stars in a given photometric
band in the y-axis. The underlying assumption is that stars radiate as a black body enabling the use of Wien’s
displacement law. CMDs of GCs are an essential tool to study the fundamental properties of their constituent
stars since they show well defined regions which are representative of their different evolutionary stages. Among
the characteristic regions in a CMD of a GC we can find the MS, the TO point, the RGB, the HB and the AGB
which will be outlined in the following subsections (see Fig. 4.6).
4.4.1 The main sequence
This is the place where stars will spend most of their lifetime as long as they keep burning hydrogen mainly
through the proton-proton chain at their cores and it spans a wide range of colors starting from the bluer (hotter)
stars to the redder (cooler) ones. The position of the star on the MS will depend mainly on their initial mass.
4.4. The color-magnitude diagram 23
At this stage, the stars are in hydrostatic equilibrium where the outward thermal pressure is balanced by the
gravitational pressure from the outer layers. More massive stars will spend less time (∼ 100 Myrs) on the MS as
they will burn the hydrogen at their cores more rapidly, while low-mass stars like the Sun will spend ∼ 10 Gyrs
on it.
Figure 4.6: Scheme of a CMD (Lamers et al., 2017).
The depth of the MS at fainter magnitudes
depends exclusively on the observations.
Associated to the MS, is the TO point,
which is an indicator (but not necessarily)
of the exhaustion of hydrogen at the core of
the stars. Its location on the CMD appears
always between 2.0-3.0 mag fainter than
the HB and this difference is a function of
the age of the GC. It has been noted that
the TO point in GCs in the MW with high
metallicity, will be redder and fainter and
conversely, the less metallic, the brighter
and bluer their TO point will be. This is an
important feature, since a faint TO point is
an indicator that GCs are old (Hoyle et al.,
1955).
4.4.2 The red giant branch
The main feature of this stage is that the hy-
drogen burning has stopped at the cores of
the stars, and has moved out to an adjacent
layer. On the CMD, the stars move towards
brighter magnitudes and redder colors un-
til they reach the tip of the RGB. This as-
cension is halted by the ignition of the de-
generate helium core that has been piling
up helium ashes from the hydrogen burning in the outer layers. GCs with high metallicity display RGBs that are
shallower and redder than the ones with a lower metallicity.
4.4.3 The horizontal branch
The HB is conformed by stars that are burning helium at their cores which come from the RGB and have gone
through a substantial mass loss. The stars on the HB are bluer than the RGB and brighter than the MS which
implies an ample range of colors at roughly the same brightness. A well known feature of the HB is the existence
of the so-called "RRL gap" somewhere in its middle and is a defining characteristic of Population II stars. As
mentioned in Chapter 3, this RRL gap on the HB is where RRL stars can be found on the CMD of a GC and is
where the RRL cross the either-or region. In terms of RRL evolution on the HB and the Oosterhoff dichotomy,
in OoI GCs most RRL stars evolve into the either-or region from the red side of the HB, populating the zone
with RRab stars (which pulsate in the fundamental mode). In OoII GCs, the RRL stars would enter the either-or
region from the blue side of the HB, populating the zone with RRc stars (which pulsate in the first overtone).
The HB plays an important role in understanding GCs. Where the stars fall on the HB will depend on how
much mass they have left after the RGB phase (mass-loss events via the He-flashes). A way of describing its
24 Chapter 4. Globular clusters
morphology is through the Lee parameter defined as L ≡ (B-R)/(B+V+R) (Lee, 1990) where B, V and R are
the number of stars to the blue, inside and to the red of the IS. The morphology of the HB will depend mainly
on the metallicity of the GC, as first suggested by Sandage et al. (1960). HBs with high metallicity will appear
redder on the CMD due to a higher opacity in the envelopes of their stars. Other physical effects can also play
a role in the morphology of the HB, since not all GCs with low metallicity display an extended HB whereas
metal-rich GCs display a short and red HB, almost indistinguishable from the red clump. This is known as the
"second parameter problem", since the metallicity alone does not define the color of the HB, therefore, there must
be one or more hidden variables affecting the color distribution along the HB (van den Bergh 1967; Sandage
et al. 1967). There has been a discussion over the last few years with respect to the physical nature of this or
these parameters. Gratton et al. (2010) firmly established that age is the second main parameter affecting the
morphology of the HB, by means of reanalyzing public extensive photometric databases. In spite of this, they
do acknowledge that there may be the need of a third or more parameters in order to explain the median of colors
of HBs and suggested that variations in the helium content in the stars might be one of them.
4.4.4 The asymptotic giant branch
Once the helium at the core of the star is exhausted, the core shuts down and the burning of helium moves
to a shell around the core. The increase in energy production makes the star shift its position upwards in the
CMD, creating a track very close to the RGB but not overlapping it. The star looses a significant amount of
mass due to stellar winds. As the star looses mass and the helium and hydrogen are exhausted in the shells, the
thermal pressure dissipates the outer envelope (the planetary nebula phase), leaving only the inert core, that will
eventually turn into a white dwarf.
For our intended purposes in this work, the main feature of the HB is the presence of RRL stars. During their
evolution, a star may or may not become susceptible to pulsations, depending on the physical conditions of its
interior and on whether the driving mechanism of pulsation gets excited or not. In the case of RRL stars, they
become unstable and start to pulsate when they cross the IS. Determining the metallicity of GCs through the
study of the pulsations of RRL stars on the HB is of fundamental importance and constitutes the cornerstone of
this work. The methodology used will be described in detail in the next chapter.
25
Chapter 5
Fourier light curve decomposition of RR
Lyrae stars and their physical parameters
5.1 Finding the best period of pulsation
When dealing with variable stars, the first and most evident property that we would like to estimate is their
period of pulsation. An accurate determination of this period, can help us discriminate between different types
of pulsating stars, eclipsing binaries, pulsating modes, or multi-mode cases, particularly among the RRL stars.
For this reason, it is of paramount importance to be able to derive an accurate value of their periods of pulsation.
This has been proven helpful, since periods can provide insights regarding the evolutionary stage such as in the
case of secular period changes which will be discussed in Section 5.3.
We used two methods to determine the period of pulsation of the RRL stars. The first one is based on the
string-length method (Burke et al. 1970, Dworetsky 1983).
The advantage of using the string-length method is that it is particularly useful when we only possess relatively
few randomly spaced observations of high accuracy (Nobs < 20) over a long span of time. Essentially, what this
method does is to estimate the length of a given light curve as a function of the period in the phase space (mi,
ϕi), where m is the magnitude and ϕ is the phase. The shortest length is the one that gives the best phasing and
therefore, the best estimation of the period, i.e., when
SQ =
n−1
∑
i=1
[(mi − mi−1)
2 + (ϕi − ϕi−1)
2]1/2 + [(m1 − mn)
2 + (ϕ1 − ϕn)
2]1/2 (5.1)
is a minimum, with n being the number of observations. The number SQ is known as the string-length parameter.
The second method is based on performing Discrete Fourier Transformation to extract the fundamental pulsating
frequency from the light curves of pulsating stars. This was achieved by pre-whitening steps which successively
remove the previous most intense signals in order to tease out other potential oscillations from the residuals. For
this method we employed the software Period04 (Lenz et al., 2004).
26 Chapter 5. Fourier light curve decomposition of RR Lyrae stars and their physical parameters
5.2 Semi-empirical calibrations for metallicity, absolute magnitude and tem-
perature
One of the early attempts to try to link the light curves of RRL stars with their metallicity was carried out by
van Herk (1971). In a sample of 20 RRab stars, he was able to find indications of a correlation between the
metallicity parameter ∆S (Preston, 1959) and the height of a secondary bump in the phased light curve.
This approach was reassumed by Norman Simon while trying to provide a quantitative description of the so
called Hertzsprung progression (Hertzsprung, 1926) in terms of the Fourier decomposition of the light curves
of classical Cepheids (Simon et al., 1981). It can be described as a "bump" in the light curves whose periods are
in the range of 6 d < P < 16 d. The bump appears on the descending branch for Cepheids with periods up to 9
days and shifts its position towards the maximum light for 9 d < P < 12 d and moves at earlier phases for longer
periods.
In later papers, he extended his Fourier analysis to the RRL stars (Simon et al. 1982; Simon 1987) in the hopes
of finding a similar regular progression in their light curves as it was in the case of the classical Cepheids and
asserting RRab and RRc variables as two different objects in terms of their Fourier coefficients.
It was not until 1988 that Simon revived the idea of using the light curve structure of RRab stars to try to
determine a correlation to the metallicity (Simon, 1988a). He found that the period of pulsation plays a role
in this correlation as previously surmised by Lub (1977b) and based on a previous study of over 100 RRL stars
(Lub, 1977a). Simon compiled a list of 61 RRab and 12 RRc stars with highly accurate light curves and excellent
phase coverage and tried to determine a relation between ∆S and the Fourier parameter ϕ21. He then showed that
a relation between the Fourier parameter ϕ21 and the metallicity index ∆S exists among RRL stars with periods
P ≤ 0.575 d (Simon, 1988b). A more refined and robust method used in this thesis to determine metallicities of
RRL stars from the structure of their light curves will be described in the following paragraphs.
As mentioned in the beginning of this chapter, the non-linear and periodic nature of the light curves of the RRL
stars allow us to use Fourier series to describe them. The representation of the light curves can be written as:
m(t) = A0 +
N
∑
k=1
Akcos(
2π
P
k(t − E0) + ϕk) (5.2)
where m(t) is the magnitude at time t, A0 is the intensity weighted mean, P is the period of pulsation, Ak and
ϕk are the amplitude and phase of harmonic k and E0 is the epoch at t = 0, generally assumed as a selected time
of maximum brightness. The coefficients yielded by the Fourier decomposition can be used to estimate their
iron abundances with fair accuracy. The Fourier parameters are defined from the amplitudes and phases of the
harmonics in Eq. 5.2 as Rij = Ai/Aj and ϕij = jϕi − iϕj. One can transform the coefficients from cosine series
phases into sine series using the following relation, if needed:
ϕ
(s)
jk = ϕ
(c)
jk − (j − k)
π
2
. (5.3)
Jurcsik et al. (1996) derived a robust calibration for the estimation of the [Fe/H] from the light curves of the
RRab stars. They used a sample of 81 Galactic-field RRL stars with reliable light curves and High Dispersion
Spectroscopy (HDS) abundances whose metallicity had been previously carefully measured by both Suntzeff
et al. (1994) and Layden (1994). This sample does not include RRL stars with Blazhko effect. They found that
5.2. Semi-empirical calibrations for metallicity, absolute magnitude and temperature 27
a linear relation between [Fe/H], P and ϕ31 reproduces the spectroscopic determinations of the iron abundance.
To find this relation, they performed a two-parameter linear fit (P and ϕ31) up to 3σ, which yielded:
[Fe/H]J = −5.038 − 5.394 P + 1.345 ϕ
(s)
31 (5.4)
The standard deviation of the estimated abundance is:
σ2
[Fe/H] = 1.809σ2
ϕ31
+ 2K12P + 2K13ϕ31 + 2K23Pϕ31 + K11 + K22P2 + K33ϕ2
31 (5.5)
where K11 = 0.08910, K22 = 0.02529, K33 = 0.00374, K12 = 0.00116, K13 = –0.01753 and K23 = –0.00289.
Since the metallicity is given in the Jurcsik-Kovács scale, we need to transform it to the standard Zinn-West
scale (Zinn et al., 1984) which can be achieved through the following equation: [Fe/H]J = 1.431[Fe/H]ZW +
0.88 (Jurcsik, 1995). It is important to mention that similar relations can be found using different phases and/or
using amplitudes instead, but the standard deviations from the fit will be larger. In order to apply Eq. 5.4 to
the light curve of a given RRab star, this must follow the systematics defined by the sample of the 81 calibrator
stars. In other words, the light curve under analysis must be compatible with the light curves of the calibrators
within certain limits. To account for this, the authors proposed an indicator that can help to determine when the
calibration is applicable. They defined a Deviation Parameter (DF) for measuring the relative accuracy of the
prediction, such that DF = |Fobs − Fcalc|/σF, where Fobs is the observed value of the given Fourier parameter,
Fcalc is the predicted value from the other observed parameters and σF is the respective standard deviation. The
values of Fcalc in terms of the Fourier parameters, can be found in the work of Kovacs et al. (1998). For easiness
of use, the authors also introduced a parameter Dm, which is the maximum of the deviation parameters DF.
If Dm < 3.0, the light curve satisfies the compatibility condition. Since the Dm changes when Blazhko-like
modulations are present in the light curve of RRab stars, it is a good indicator of its presence if the compatibility
condition is not met. High values of the Dm parameter, however, might be also due to poor quality of the data
rather than to the presence of the Blazhko effect itself. Finally, one must be careful when using this calibration
since there exists a systematic difference which will predict higher values at the low metallicity end.
For the estimation of the absolute magnitude for RRab stars, Kovács et al. (2001) derived an empirical relation
based on 383 RRab stars in twenty GCs for which in principle, their distances and luminosities are known. The
calibration is of the form:
MV = −1.876 log P − 1.158 A1 + 0.821 A3 + 0.41. (5.6)
A calibration for the estimation of the metallicity of RRc stars analogous to the one for the RRab was derived by
Morgan et al. (2007). The authors carried out an analysis of the Fourier coefficients of 106 stars in 12 Galactic
GCs with cluster mean metallicities ranging from –2.2 to –1.0 (Morgan, 2003), to determine the best relation
between the three parameters by exploring all possible permutations of an equation of the form:
[Fe/H] = aP2 + bP + cϕ2
31 + dϕ31 + ePϕ31 + f . (5.7)
Only those fits that included at least one term in P and one in ϕ31 in the relation were taken into account. To
determine the best fit, they used two criteria: the first one is related to the quality of the data and the second
is related to the simplicity of the relation. The relation to reproduce the Zinn-West metallicity scale (Zinn et
al., 1984) contains 6 parameters, while the relation to reproduce the Carretta-Gratton metallicity scale (Carretta
28 Chapter 5. Fourier light curve decomposition of RR Lyrae stars and their physical parameters
et al., 1997) contains only 3 parameters. We adopted the calibration in the metallicity scale of Zinn-West, for
consistency with the relation derived to estimate the metallicity of RRab stars, namely:
[Fe/H]ZW = 52.466P2 − 30.075P + 0.131ϕ
2(c)
31 − 0.982ϕ
(c)
31 − 4.198ϕ
(c)
31 P + 2.424. (5.8)
In the case of the absolute magnitude of the RRc stars, Kovacs (1998) derived an empirical relation using 93 RRc
stars distributed in four Galactic GCs, in three GCs from the Large Magellanic Cloud and in the dwarf galaxy
Sculptor, with the resulting calibration:
MV = 1.061 − 0.961P − 0.044ϕ
(s)
21 − 4.447A4
1. (5.9)
The authors acknowledge, however, that the significance of this relation is low due to the noise in the light curves
and the size of the sample. Also, they argue that nonlinearity does not play a significant role so in order to fit any
physical parameter, linear combinations of the Fourier parameters can be used. The method followed to derive
Eq. 5.6 and Eq. 5.9 was to use any least squares solution for {ck} that minimizes an equation of the form:
D[Fe/H] =
1
N − M − 1
N
∑
i=1
[[Fe/H]i − c0 −
M
∑
k=1
ckFk(i)]
2 (5.10)
and {mk} for
DMv =
1
N − Nc − M
Nc
∑
j=1
Nj
∑
i=1
[V
j
i − dj − m0 −
M
∑
k=1
mkF
j
k(i)]
2 (5.11)
respectively, where Fk(i) represent the Fourier components (amplitudes, phases and period) of the i-th star. M
is the number of Fourier parameters selected that minimizes D[Fe/H] and DMv. N is the total number of stars,
Nc is the number of clusters, Nj is the number of stars in the j-th cluster, V
j
i is the observed average magnitude
of the i-th star in the j-th cluster and {dj} is the relative distance moduli.
The calibrations for the metallicity and the absolute magnitude have an associated uncertainty of 0.14 dex and
0.04 mag, respectively.
The values of MV from both RRab and RRc stars derived from Eq. 5.6 and Eq. 5.9 are transformed into
luminosities using the following equation:
log(L/L⊙) = −0.4(MV − Mbol
⊙ + BC). (5.12)
To calculate the bolometric correction, we used the equation BC = 0.06[Fe/H]ZW + 0.06 as derived by Sandage
et al. (1990). We have adopted the value of Mbol
⊙ = 4.75 mag.
The effective temperature of the RRab stars was estimated with the calibration given by Jurcsik (1998):
log(Teff) = 3.9291 − 0.1112(V − K)0 − 0.0032[Fe/H], (5.13)
1The original zero-point of this calibration was 1.261, but according to Arellano Ferro et al. (2010), it had to be lowered in order to
make the luminosities of the RRc stars consistent with the distance modulus of 18.5 for the Large Magellanic Cloud.
5.2. Semi-empirical calibrations for metallicity, absolute magnitude and temperature 29
where
(V − K)0 = 1.585 + 1.257P − 0.273A1 − 0.234ϕ
(s)
31 + 0.062ϕ
(s)
41 . (5.14)
For the effective temperature in RRc stars, we used:
log(Teff) = 3.7746 − 0.1452log(P) + 0.0056ϕ
(c)
31 , (5.15)
as derived by Simon et al. (1993).
It has been noted by Cacciari et al. (2005), however, that Eq. 5.13 and 5.15 (which are based on the Fourier
parameters), produce effective temperature offsets of about 40 K and 250 K hotter for RRab and RRc stars,
respectively, when compared to the color-temperature relations predicted by the temperature scales of Sekiguchi
et al. (2000). To account for this offset in the temperature, Arellano Ferro et al. (2010) and Yepez et al. (2020)
converted the observed colors (V − I) into effective temperatures, via the polynomial calibration derived by
Nemec (2004). This calibration, in turn, uses the HB models of VandenBerg et al. (2006) with the color-log Teff
relations as described by VandenBerg et al. (2003). The polynomial fit is of the form:
y = A0 + A1x + A2x2 + A3x3 + A4x4 + A5x5 + A6x6 + A7x7 (5.16)
where y = log Teff, x = (V− I)0 and the coefficients A0 = +3.9867, A1 = –0.9506, A2 = +3.5541, A3 = –3.4537,
A4 = –26.4992, A5 = +90.9507, A6 = –109.6680 and A7 = +46.7704. Comparing the old temperatures with the
new ones, their differences are of about 40 ± 44 K for the RRab stars and 300 ± 367 K for the RRc stars, as
found by Arellano Ferro et al. (2010).
For the estimation of the stellar mass of the RRL stars, we used:
log(M/M⊙) = 16.907 − 1.47logPF + 1.24log(L/L⊙)− 5.12log(Teff) (5.17)
as given by van Albada et al. (1971) where logPF is the fundamental period, and the stellar radii with:
L = 4πR2σT4 (5.18)
Nemec et al. (2013) derived their own P - ϕs
31 - [Fe/H] calibrations based on the analysis of 41 RRL stars located
in the FoV of the Kepler satellite (Basri et al., 2005). These stars were observed during ∼ 970 days of quasi-
continuous high-precision Q0 - Q11 long- and short-cadence Kepler photometry. From this sample, 37 are
RRab stars and 4 are RRc stars with 16 showing Blazhko effect. The spectroscopic [Fe/H] values, which were
compared to those obtained from the following calibrations, are derived from high dispersion spectra, so the
derived metallicities will be expressed in the HDS (Carretta et al., 2009) scale.
For the RRab stars, the derived relation is as follows:
[Fe/H] = b0 + b1 P + b2ϕ
(s)
31 (Kp) + b3ϕ
(s)
31 (Kp) P + b4 (ϕ
(s)
31 (Kp))2 (5.19)
where ϕ
(s)
31 is the mean ϕ
(s)
31 (Kp) value and [Fe/H] is the versatile wavelength analysis (VWA) spectroscopic
value, with the coefficient values b0 = −8.65± 4.64, b1 = −40.12± 5.18, b2 = 5.95± 1.72, b3 = 6.27± 0.96
30 Chapter 5. Fourier light curve decomposition of RR Lyrae stars and their physical parameters
and b4 = −0.72 ± 0.17, and a rms error of 0.084 dex. The Fourier parameter ϕ
(s)
31 (Kp), is calculated from
the light curve in the Kepler photometric system, thus, our Fourier parameters from the V light curve were
transformed into the Kepler system via the relation: ϕ
(s)
31 (Kp) = ϕ
(s)
31 + 0.151 (Nemec et al., 2011) before
applying the calibration. In general, Eq. 5.4 and Eq. 5.19 agree when [Fe/H] < -1.0, but the discrepancies grow
larger as [Fe/H] decreases. This model appears to correct this effect in part due to the inclusion of a broader
sample of stars and in part to the use of a non-linear model instead of a linear one.
In the case of RRc stars, the authors decided to include the 4 RRc stars in their Kepler sample to the database
from Morgan et al. (2007) and derived a new non-linear relation from them as well. This inclusion of the Kepler
targets had the effect of expanding the range of the relation to considerably higher metallicities. The relation
derived is as follows:
[Fe/H] = b0 + b1 P + b2ϕ
(c)
31 + b3ϕ
(c)
31 P + b4P2 + b5 (ϕ
(c)
31 )
2, (5.20)
with the coefficient values b0 = 1.70 ± 0.82, b1 = −15.67 ± 5.38, b2 = 0.20 ± 0.21, b3 = −2.41 ± 0.62,
b4 = 18 ± 8.70, and b5 = 0.17 ± 0.04, and a rms error of 0.13 dex. Finally, they concluded that Eq. 5.19 and
Eq. 5.20 can be used to determine metallicities of RRL stars with Blazhko effect as long as the modulation is
not too extreme and that enough Blazhko cycles are used when estimating average ϕ31.
All the aforementioned equations in this section constitute our primary methodology in estimating the physical
parameters of the RRL stars in the GCs of our sample and are summarized in Table 5.1 and Table 5.2 for RRab
and RRc stars, respectively.
Table 5.1: Summary of calibrations used for [Fe/H], Mv and T for RRab stars.
Parameter Calibration Source
[Fe/H] −5.038 − 5.394 P + 1.345 ϕ
(s)
31 Jurcsik et al. (1996)
[Fe/H] b0 + b1 P + b2ϕ
(s)
31 (Kp) + b3ϕ
(s)
31 (Kp) P + b4 (ϕ
(s)
31 (Kp))2 Nemec et al. (2013)
Mv −1.876 log P − 1.158 A1 + 0.821 A3 + 0.41 Kovacs (1998)
log(Teff) 3.9291 − 0.1112(V − K)0 − 0.0032[Fe/H] Jurcsik (1998)
Table 5.2: Summary of calibrations used for [Fe/H], Mv and T for RRc stars.
Parameter Calibration Source
[Fe/H] 52.466P2 − 30.075P + 0.131ϕ
2(c)
31 − 0.982ϕ
(c)
31 − 4.198ϕ
(c)
31 P + 2.424 Morgan et al. (2007)
[Fe/H] b0 + b1 P + b2ϕ
(c)
31 + b3ϕ
(c)
31 P + b4P2 + b5 (ϕ
(c)
31 )
2 Nemec et al. (2013)
Mv 1.061 − 0.961P − 0.044ϕ
(s)
21 − 4.447A4 Kovacs (1998)
log(Teff) 3.7746 − 0.1452log(P) + 0.0056ϕ
(c)
31 Simon et al. (1993)
All the estimated averages of the physical parameters obtained from these calibrations, were weighted by their
respective uncertainties using the standard equation wi = 1/σ2
i .
5.3. Period changes in RR Lyrae stars 31
5.3 Period changes in RR Lyrae stars
The study of the evolution of stars from their formation to their final stages is unattainable from a temporal
perspective. Human lifespans pale in comparison to that of stars. Is in this sense that the study of secular period-
changes in RRL stars, which are a consequence of stellar evolution, is of interest. To study them can help to
constrain and test evolutionary models in and around of the HB stages. Nevertheless, these period changes are
difficult to measure since we need accurate observations over a long time base. Theory predicts that, as the RRL
stars on the HB slowly evolve either towards its blue or red side, their period-change rates will be very close to
zero except towards the end of the HB evolution, i.e. when the stars start burning He in a shell around the core;
that is right before the AGB phase, when evolution happens fast. At this stage, the period change rates, β = Ṗ,
can be between +0.1 and +0.15 day/Myr (Lee, 1991). Period changes are usually the result of stellar evolution
along the HB, particularly might be due to a fast crossing of pre-ZAHB stars through the IS towards the blue
(Silva Aguirre et al., 2008), although it is also important to note that period changes may also very well be the
result of non-evolutionary effects, such as stochastic processes related to mixing events in the core of the star
(Balázs-Detre et al. 1965; Sweigart et al. 1979).
In particular, RRL stars have been observed in GCs such as M5 (Arellano Ferro et al., 2016), NGC 6171 (Deras
et al., 2018) and NGC 4147 (Luna et al., 2019) for some 50 - 80 years. Gathering more and more data over
this time base enabled us to detect and measure these period changes. From an evolutionary standpoint, period
changes are expected since the basic pulsation equation P
√
⟨
ρ
⟩
= Q depends on the average density of the star
which significantly varies during their lifetimes. The O − C method (Observed time o f maximum light -
Calculated time o f maximum light) is a diagnostic tool which helps studying the possible differences between
measured features in the light curve of a star and its predicted values. It is the standard method used to find errors
in the estimation of the period of pulsation in stars or to find authentic secular variations. To make use of it, we
need to select an easily identifiable feature of the light curve of an RRL star that can be measured with a high
degree of accuracy. Assuming that the light curves are well covered over a pulsation cycle, we can use the time
of maximum brightness of an RRab star. This is ideal since it is unequivocally well defined. In the case of an
RRc star, the bisector of its light curve at its time of maximum can also provide a good proxy of the moment
of maximum brightness. The basic assumption in this method is the existence of an accurate and stable clock,
for which we use the period of pulsation of the RRL stars. Also, a long time baseline relative to the period of
pulsation is needed. In order to construct an O − C diagram we first need to accurately determine the period of
pulsation of the RRL star. A lengthy discussion of this method and its equations is described in Sterken (2005)
and Arellano Ferro et al. (2016) and is as follows:
First, we need to adopt an initial ephemeris E0 which corresponds to an epoch of maximum brightness. P0 is the
estimated period at time E0 and N is the number of cycles that have taken place between the observed time of
maximum and E0. So,
C = E0 + P0N (5.21)
where N is defined as:
N =
⌊O − E0
P0
⌋
. (5.22)
The incomplete brackets imply rounding down to the nearest integer.
As a way of studying the period changes in RRL stars, let us propose an initial quadratic model of the form:
O − C = A0 + A1N + A2N2. (5.23)
32 Chapter 5. Fourier light curve decomposition of RR Lyrae stars and their physical parameters
Now, substituting Eq. 5.21 in 5.23 we obtain:
O − [E0 + P0N] = A0 + A1N + A2N2. (5.24)
Rearranging terms we have:
O = (A0 + E0) + (A1 + P0)N + A2N2. (5.25)
Now, the last equation depends only on N, but we want to know how the observed value changes when the
number of cycles changes, and so:
P(N) =
dO
dN
= (A1 + P0) + 2A2N. (5.26)
Let us now define the rate of period change as β such that:
β ≡ Ṗ =
dP
dt
= 2A2
dN
dt
(5.27)
Since the time elapsed to a given epoch is simply the period times the number of cycles, t = PN, then:
dN
dt
=
1
P
(5.28)
and so, we can rewrite Eq. 5.27 as:
β =
2A2
P
. (5.29)
Finally, if N = 0 and P = P0 then:
β = β0 =
2A2
P0
. (5.30)
The period-change parameter β can yield three possible outcomes:
• β > 0 implies that the period of pulsation of the star is increasing over time. If the star is located on the HB,
it means that it is evolving towards the red side. This also means that a plot of O − C versus the number
of cycles N shows a parabola with a positive coefficient of the quadratic term A2, i.e., a positive parabola.
• β = 0 implies that the period of pulsation of the star remains constant over time, so its plot will be a constant
function over N number of cycles.
• β < 0 implies that the period of pulsation of the star is decreasing over time. If the star is located on the
HB, it means that it is evolving towards the blue side. This also means that a plot of O − C versus the
number of cycles N shows a parabola with a negative coefficient of the quadratic term A2, i.e., a negative
parabola.
We must note that sometimes, if the time of the initial epoch E0 and the observed time of maximum are much
longer than the period of pulsation of a star, the O − C diagram can exceed several cycles which have to be
accounted for, specially if there are significant gaps in their time series. As it was mentioned at the beginning
5.3. Period changes in RR Lyrae stars 33
of this chapter, period changes in RRL stars can have an origin that it is not compatible just with their stellar
evolution exclusively. If these are indeed the cases, the period changes will produce irregular and complicated
shapes of the O − C diagrams, very different from the parabolic cases.
From a GC perspective, evolutionary models predict that the mean values of period changes in the RRL stars on
a very red HB will be close to zero (see Fig. 15 in Catelan (2009)). According to Fig. 5.1 one would expect that
more luminous RRL stars go through large period changes. In fact, it has been found that some RRL stars with
high positive values of β (e.g., V77 in M5 Arellano Ferro et al. (2016)) and V17 in Deras et al. (2018)), where
this is the case. This suggests that they are in a more advanced evolutionary state, on their way to the AGB while
also finding RRL stars with negative β which might indicate that they are at the pre-ZAHB stage.
To close up this chapter, it is relevant to highlight that measuring period changes in RRL stars can provide us
with information on their evolutionary state. Also, their study constitutes a proof of stellar evolution in "real
time" and is the only feature that we can actually measure within a single human lifetime.
34 Chapter 5. Fourier light curve decomposition of RR Lyrae stars and their physical parameters
Figure 5.1: Distribution of the RRLs of NGC 6171 on the HB. Continuous lines are the evolu-
tionary tracks for [Fe/H]=–1.48, and M = 0.60M⊙, 0.62M⊙, 0.64M⊙ and 0.66 M⊙ from Dorman
(1992). The numbers along the tracks indicate the corresponding period-change rates calculated
by Jurcsik et al. (2001) expressed in d/Myr. Green and blue symbols represent RRc and RRab
stars respectively. Tagged RRab stars with a line segment attached are the ones with remarkable
period changes. The direction of the line segment represents the the direction of the period change:
positive to the right (V17) and negative to the left (V10, V12 and V16).
35
Chapter 6
Physical parameters of RR Lyrae stars in
NGC 6171 (M107)
6.1 Overview
NGC 6712 (M107) is a sparse GC located in the constellation of Ophiuchus (α = 16h32′31.86′′, δ = −13◦03′13.6′′,
J2000) and very close to the Galactic bulge (l = 3.37◦ , b = 23.01◦). It was first discovered by Pierre Méchain in
1782 and rediscovered by William Herschel in 1793. It is the last GC added to the Messier catalog. The distance
to the cluster and its reddening are given by Harris (1996) (2010 edition) as 6.4 kpc and E(B − V) = 0.33,
respectively. The CVSGC lists 26 variables 23 of which are RRL, one Mira, one Lb and one SX Phe. In this
article, we present the results of the first ever determination of physical parameters of RRL stars in NGC 6171
by means of Fourier decomposition of their light curves.
6.2 Observations and transformation to the standard system
The observations were carried out at two locations: The IAO using the 2 m telescope between 2015-2016 and at
OAN-SPM in 2017 using the 0.84 m telescope. The number of analyzed images in the V filter was 467 and 479
in the I filter.
For the processing of the images and the photometry we made use of the Difference Imaging Analysis technique
with its pipeline implementation DanDIA (Bramich 2008; Bramich et al. 2013) which is outlined in Appendix B.
This allowed us to recover 3827 stars in the FoV of our images. We then transformed the instrumental magnitudes
of the stars into the Johnson-Kron-Cousins standard system (Landolt, 1992). We used 79 standard stars for the
images from the IAO and 73 for the images from OAN-SPM in both V and I filters. The values for the standard
magnitudes were taken from the catalog of Photometric Standard Fields compiled by Stetson (2000). All the
data of the variable star population in NGC 6171 are reported in Table 6.1.
6.3 On the reddening of NGC 6171
To estimate an accurate value of the reddening in the direction of the cluster is fundamental in the determination
of its distance. The estimation of the reddening of this cluster has yielded a wide range of values, from 0.25 to
0.46 (see Smith, H. A. et al. (1986) for a summary). Therefore, we decided to estimate the reddening using an
independent method for the RRab stars in our FoV. The method, proposed by Sturch (1966), is based on the fact
that the intrinsic color (B − V)0 of these stars is constant near minimum light. Also, we used the calibration
derived by Guldenschuh et al. (2005) where (V − I)0,min = 0.58 ± 0.02 which allowed us to estimate E(V − I).
36 Chapter 6. Physical parameters of RR Lyrae stars in NGC 6171 (M107)
Table 6.1: Data of Variable stars in NGC 6171 in the FoV of our images with the exception of
V23 whose light curve in the V filter was taken from Clement et al. (1997) (Deras et al., 2018).
Star ID Type
⟨
V
⟩
(mag)
⟨
I
⟩
(mag) AV (mag) AI (mag) P [days] α (J2000.0) δ (J2000.0)
V4 RRc 15.493 14.667 0.462 0.343 0.282132 16:32:25.28 -13:05:55.4
V5 RRab Bl - - 0.4831 0.3301 0.702375 16:32:47.90 -13:05:57.3
V6 RRc Bl 15.614 14.766 0.523 0.424 0.259627 16:32:31.32 -13:04:25.6
V7 RRab Bl - - 1.1391 0.7461 0.497474 16:32:35.00 -13:04:18.9
V8 RRab 15.578 14.523 1.094 0.668 0.559922 16:32:32.82 -13:03:59.9
V9 RRc 15.632 14.674 0.496 0.340 0.320601 16:32:30.18 -13:03:37.9
V10 RRab Bl 15.782 14.811 0.995 0.680 0.415559 16:32:28.08 -13:03:10.3
V11 RRab 15.673 14.543 0.680 0.444 0.592809 16:32:32.60 -13:02:44.9
V12 RRab Bl 15.698 14.694 1.0341 0.6781 0.472833 16:32:35.98 -13:02:16.6
V13 RRab 15.834 14.781 1.193 0.812 0.466797 16:32:30.10 -13:02:06.1
V14 RRab - - - - 0.481620 16:32:33.13 -13:01:55.5
V15 RRc 15.622 14.724 0.483 0.378 0.288590 16:32:33.20 -13:01:17.6
V16 RRab Bl 15.675 14.617 0.8011 0.4851 0.522796 16:32:27.33 -13:01:25.0
V17 RRab 15.593 14.538 1.058 0.681 0.561168 16:32:25.16 -13:02:07.2
V18 RRab 15.739 - 0.811 - 0.561404 16:32:37.13 -12:59:41.9
V19 RRc Bl 15.697 14.952 0.503 0.322 0.278762 16:32:47.79 -13:00:32.7
V20 RRab 15.708 14.610 0.789 0.537 0.578107 16:32:34.08 -13:02:26.8
V212 RRc 16.519 15.660 0.566 0.407 0.258715 16:32:37.61 -13:05:42.5
V23 RRc 15.626 - 0.439 - 0.323344 16:32:13.96 -13:03:01.5
V24 RRab 15.600 14.530 0.882 0.657 0.523949 16:32:31.98 -13:03:09.7
V26 SX Phe 17.616 16.377 - - 0.055275 16:32:32.66 -13:03:00.4
Bl: RRL stars with Blazhko effect.
1 Maximum observed amplitude.
2 Not a cluster member
6.4. Physical parameters of RRL stars in NGC 6171 37
Table 6.2: Physical parameters of the RRab and RRc stars in NGC 6171. The numbers in paren-
theses indicate the uncertainty on the last decimal place (Deras et al., 2018).
RRab
ID [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V81 -1.40(13) -1.30(13) 0.566(1) 3.811(7) 1.673(1) 0.69(6) 5.50(1)
V11 -1.40(38) -1.30(38) 0.623(4) 3.804(1) 1.651(2) 0.65(1) 5.54(1)
V13 -1.37(15) -1.26(15) 0.661(2) 3.821(7) 1.635(1) 0.72(6) 5.03(1)
V17 -1.31(13) -1.20(12) 0.581(3) 3.812(8) 1.668(1) 0.67(6) 5.44(1)
V18 -1.35(70) -1.24(68) 0.623(7) 3.810(2) 1.651(3) 0.66(1) 5.40(2)
V20 -1.27(23) -1.16(22) 0.614(2) 3.809(8) 1.655(1) 0.64(6) 5.44(1)
V24 -1.58(29) -1.50(33) 0.690(3) 3.810(1) 1.624(1) 0.67(4) 5.21(7)
Weighted mean -1.33(1) -1.22(1) 0.596(1) 3.813(4) 1.661(1) 0.68(3) 5.34(1)
σ ± 0.12 ± 0.12 ± 0.051 ± 0.002 ± 0.006 ± 0.01 ± 0.07
RRc
ID [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V4 -0.81(17) -0.75(10) 0.577(9) 3.875(1) 1.669(4) 0.57(1) 4.07(1)
V9 -1.07(10) -0.96(8) 0.562(5) 3.870(1) 1.675(2) 0.51(1) 4.21(1)
V15 -1.02(15) -0.91(11) 0.582(9) 3.873(1) 1.667(4) 0.56(1) 4.11(2)
V212 -0.89 (1.43) -0.81(92) 0.624(5) 3.877(5) 1.650(2) 0.60(5) 3.95(9)
V23 -1.26(29) -1.14(26) 0.569(1) 3.868(1) 1.672(4) 0.51(1) 4.23(2)
Weighted mean -1.02(7) -0.90(5) 0.569(3) 3.871(1) 1.672(1) 0.60(1) 4.13(1)
σ ± 0.19 ± 0.19 ± 0.010 ± 0.002 ± 0.002 ± 0.02 ± 0.04
1 Not included in the average of [Fe/H].
2 Not a cluster member, hence not included in the calculation of mean physical parameters.
We then transformed to E(B−V) by using the relation E(V − I)/E(B−V) = 1.259 from Schlegel et al. (1998).
We used all of the RRab stars with the exception of V18, since its light curve displays a prominent gap near its
minimum. We estimated an average reddening E(B−V) = 0.45± 0.03 which is the value we adopted throughout
this work.
6.4 Physical parameters of RRL stars in NGC 6171
As with the reddening, there are many values reported in the literature for the metallicity of this cluster. Most of
the determinations come from the metallicity index ∆S defined by Preston (1959). Following the procedure to fit
the light curves of the RRL stars described in Chapter 5, we estimated the values for the metallicity as [Fe/H]ZW
= –1.33 ± 0.12 for the RRab stars and as [Fe/H]ZW = – 1.02 ± 0.19 for the RRc stars. The light curves of the
RRL stars can be seen in Fig. 6.1 and their physical parameters are reported in Table 6.2.
Using the reddening derived in Section 6.3, we estimated the distance to NGC 6171 using Eq. 5.6 and Eq. 5.9
for the RRab and RRc stars, respectively. The estimated distance was 5.3 ± 0.3 kpc for the RRab stars and 5.3
± 0.2 kpc for the RRc stars. We also used the P-L relation in the I filter derived by Catelan et al. (2004) from
which we found a distance of 5.6 ± 0.2 kpc and 5.4 ± 0.3 kpc assuming the metallicity found for the RRab and
RRc stars respectively. Finally, we used the P-L relation for the SX Phe stars proposed by Cohen et al. (2012)
38 Chapter 6. Physical parameters of RR Lyrae stars in NGC 6171 (M107)
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V4
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V5
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V6
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V7
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V8
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V9
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V10
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V11
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V12
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V13
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V14
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V15
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V16
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V17
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V18
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V19
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V20
0.0 0.5 1.0
φ
15.0
15.5
16.0
16.5
V
,
I
V21
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
V23
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,
I
V24
Figure 6.1: Light curves in V and I filters of RRL stars in NGC 6171. The light curves are a
combination of the data obtained at Hanle (black symbols) and at San Pedro Mártir (blue symbols).
The continuous red line represents the Fourier fit (Deras et al., 2018).
6.5. The color-magnitude diagram of NGC 6171 and its HB 39
and found a distance of 5.2 ± 0.4 kpc. All the above determinations of the distance are strictly independent and
agree satisfactorily within the uncertainties as shown in Table 6.3.
Table 6.3: Distance comparison to NGC 6171 from the different methods used in this work (Deras
et al., 2018)
Method Distance [kpc]
RRab Fourier decomposition 5.3 ± 0.3
RRc Fourier decomposition 5.3 ± 0.2
SX P-L 5.2 ± 0.4
RRab / RRc I-band P-L 5.6 ± 0.2 / 5.4 ± 0.3
6.5 The color-magnitude diagram of NGC 6171 and its HB
Using the magnitude weighted means of V and V − I of the 3827 stars in our FoV, we were able to build the
CMD of NGC 6171 shown in Fig. 6.2. The intensity-weighted means
⟨
V
⟩
and
⟨
V
⟩
-
⟨
I
⟩
were used in the case
of all of our variables. These were calculated via the Fourier fits of their light curves in Fig. 6.1 and are listed in
Table 6.1. We explored the light curves of the stars at the tip of the RGB looking for variability and were able to
identify two new SR-type stars (V27 and V28).
Our derived basic parameters of distance, reddening and age are consistent with the CMD as shown by the over-
laying of two theoretical isochrones and a ZAHB taken from the Victoria-Regina stellar models of VandenBerg
et al., 2014. The age of the cluster 10.98 Gyrs, was derived by De Angeli et al. (2005) from a differential age esti-
mation approach. For the isochrones, we adopted the metallicity values yielded by the RRab and RRc stars, i.e.,
[Fe/H]ZW = -1.33 (blue) and -1.02 (green), respectively. The metallicity adopted for the ZAHB was [Fe/H]ZW =
–1.31. These models were reddened and shifted by our estimated values of reddening (E(B − V) = 0.45) and
distance (5.4 ± 0.1 kpc), which are consistent with the observed CMD.
The average of the pulsation period of the RRab stars is 0.53 d which indicates NGC 6171 as an OoI-type cluster.
The RRab and RRc stars are well separated by the FORE, with the exception of V9. This splitting of the pulsation
modes occurs in all OoII-type clusters but only in some of the OoI-type. NGC 6171 is a good example of a GC
with a very red HB morphology, as indicated by the value of the Lee index (Lee, 1990), L = (B-R)/(B+V+R) =
–0.74.
6.6 Contributions made by the student to this article
• I cleaned, reduced and analyzed over 800 science images of the GC.
• I obtained the instrumental magnitudes and transformed them to the standard system to obtain the light
curves of the star present in the GC.
• I estimated the reddening of the GC.
• I estimated the Fourier coefficients and the physical parameters related to them.
• I estimated the distance to the cluster using different types of variable stars.
• I made Figures. 1, 2, 3, 4, 5 and A1.
• I made Tables 1, 2, 3, 4, 5, 6 and A1.
40 Chapter 6. Physical parameters of RR Lyrae stars in NGC 6171 (M107)
0.5 1.0 1.5 2.0 2.5
V − I
13
14
15
16
17
18
19
20
V V 21
V 9
NGC 6171 (M107) V 25
V 26
V 27
V 28
Figure 6.2: CMD of NGC 6171. Blue circles correspond to RRab stars, green circles to RRc
stars and the magenta circle to the SX Phe. The open green circle corresponds to an RRc star
(V21) that is not a member of the cluster. V27 and V28 are variables newly found in the present
work. The two isochrones are from VandenBerg et al., 2014, with Y = 0.25 and [α/Fe] = +0.4,
corresponding to an age of 11 Gyrs and to a [Fe/H]RRab = -1.33 (blue) and to a [Fe/H]RRc = -1.02
(green). The ZAHB for [Fe/H] = -1.31 and isochrones have been reddened by E(B − V) = 0.45
and shifted to the distance of the RRab and RRc stars, i.e. 5.3 kpc. The continuous and dashed
vertical lines represent the red edge of the first overtone IS reddened by E(B − V) 0.33 and 0.45
respectively (Deras et al., 2018).
• I wrote Sections 2, 2.1, 2.2, 3, 4.1, 6, 6.1 and 6.2.
• I co-wrote Sections 1, 4.2, 5, 7 and 8.
• I conducted a search for variable stars.
Received: 7 May 2018 Accepted: 4 October 2018
DOI: 10.1002/asna.201813489
O R I G I N A L A R T I C L E
Physical parameters of RR Lyrae stars in NGC 6171
D. Deras1,* A. Arellano Ferro1 S. Muneer2 S. Giridhar2 R. Michel3
1Instituto de Astronomía, Universidad Nacional
Autónoma de México, Ciudad de México, Mexico
2Indian Institute of Astrophysics, Bangalore, India
3Observatorio Astronómico Nacional, Instituto de
Astronomía, Universidad Nacional Autónoma de
México, Ensenada, Mexico
*Correspondence
D. Deras, Instituto de Astronomía, Universidad
Nacional Autónoma de México, Ciudad de México
CP 04510, Mexico.
Email: dderas@astro.unam.mx
Funding Information
DGAPA-UNAM, IN105115), IN106615-17.
We present an analysis of VI CCD time series photometry of the globular clus-
ter NGC 6171. The main goal is to determine individual physical parameters for
single-mode RR Lyrae stars present in the field of view of our images, by means of
light curve Fourier decomposition and well-established calibrations and zero points.
This leads to the estimation of the mean values of the metallicity and distance for the
parental cluster. From the RRab stars, we find [Fe/H]ZW =−1.33± 0.12 and distance
d = 5.3 ± 0.3 kpc, and from the RRc stars we find [Fe/H]ZW =−1.02± 0.19 and dis-
tance d = 5.3 ± 0.2 kpc. Independent methods such as the P–L (period–luminosity)
relations for RRab and SX Phe stars enable the estimation of a weighted average
distance to the cluster of 5.4 ± 0.1 kpc. We confirm the amplitude modulations of
the Blazhko type of five RRab and the non-membership of V21 to the cluster. The
color–magnitude diagram is consistent with an age of 11 Gyrs. The distribution of
RRab and RRc stars seems well segregated around the first overtone red edge of
the instability strip. This positions NGC 6171 among OoI-type clusters where the
pulsating modes are neatly separated in the horizontal branch. We report two new
irregular variables of the Lb type.
KEYWORDS
fundamental parameters – globular clusters: Individual (NGC 6171) – horizontal
branch – RR Lyrae stars
1 INTRODUCTION
NGC 6171 (M107) is a very disperse globular cluster (appar-
ent diameter ∼13′) located in the constellation of Ophiuchus
(𝛼 = 16h32′31.86′′, 𝛿 =−13◦03′13.6′′, J2000), and very close
to the Galactic bulge (l = 3.37◦, b = 23.01◦). In the compi-
lation of Harris (1996) (2010 edition), the distance from the
Sun and the reddening for the cluster are given as 6.4 kpc
and E(B − V)= 0.33, respectively. The Catalogue of Vari-
able Stars in Globular Clusters (CVSGC; Clement et al. 2001;
2015 edition) lists 26 variables of which 23 are RR Lyrae, one
Mira (V1), one Lb variable (V25), and one SX Phe (V26). The
first 24 variable stars were discovered by Oosterhoff (1938),
while V25 was discovered by Lloyd Evans & Menzies (1973)
and V26 by McCombs et al. (2012). The cluster member-
ship of the RR Lyrae stars in NGC 6171 was determined by
Cudworth et al. (1992).
The historical data available for this cluster spans 82 years,
which enabled Arellano Ferro et al. (2018) to perform an
analysis of secular changes in the periods of the RR Lyrae
stars present in the cluster. It was found that 82% of the sam-
ple had stable periods, while significant period variations
were found only in four stars, three of which had decreas-
ing periods; if interpreted as of evolutionary origin, it implies
evolution toward the blue side of the horizontal branch (HB).
However, no further analysis of the light curves has ever been
performed.
In this paper, we report the analysis of time series photom-
etry of NGC 6171 in the VI bands. We make use of Fourier
decomposition of the RR Lyrae stars light curves to find indi-
vidual values of [Fe/H], distance, mass, effective temperature,
and radii, and hence the average metallicity and distance to
the parental cluster. We also briefly discuss the morphol-
ogy of the HB and the age of the cluster. The rest of the
paper is organized as follows: In Section 2, we describe the
observations and the transformation to the standard system. In
Section 3, we discuss the reddening of the cluster. In Section
4, we perform the Fourier light curve decomposition, present
Astron. Nachr. / AN. 2018;339:603–614 www.an-journal.org © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 603
604 DERAS ET AL.
TABLE 1 Observations log of NGC 6171. Data are from two sites,
Hanle (Han) and San Pedro Mártir (SPM). Columns NV and NI give the
number of images taken with the V and I filters, respectively. Columns
tV and tI provide the exposure time, or range of exposure times. In the
last column the average seeing is listed
Date Site NV tV (s) NI tI (s) Seeing (′′)
20150626 Han 20 30–40 20 8 2.0
20150627 Han 22 30 22 8 2.1
20160518 Han 60 30 60 10 2.2
20160519 Han 43 30 43 10 1.9
20170628 SPM 86 40–50 90 25–30 5.1
20170629 SPM 16 40 17 25 2.3
20170630 SPM 39 40 41 25 1.8
20170701 SPM 40 40 40 25 1.8
20170702 SPM 38 40 39 25 2.2
20170703 SPM 35 40 36 25 1.9
20170704 SPM 37 40 38 25 2.0
20170705 SPM 31 40 33 25 2.2
Total — 467 — 479 — —
the physical parameters, and discuss the Oosterhoff type and
the period–amplitude diagram. In Section 5, we evaluate the
distance to NGC 6171 derived from an assortment of meth-
ods along with the implications of the adopted reddening. In
Section 6, we frame the resulting iron abundance [Fe/H] in the
perspective of the numerous previous estimates. In Section 7,
we highlight the RR Lyrae distribution on the HB along with
overall consistency of the color–magnitude diagram (CMD)
and the age of the cluster. In Section 8 we summarize our con-
clusions. Finally in Appendix A we offer detailed comments
on specific stars.
2 OBSERVATIONS
The observations used for this work were performed at two
different locations. The first set of data was obtained using
the 2-m telescope of the Indian Astronomical Observatory
in Hanle, India, on two epochs, each consisting of two consec-
utive nights. The first epoch spans the nights of June 26—27,
2015, and the second epoch spans the nights of May 18–19,
2016. The detector used was a SITe002 CCD of 2048× 2048
pixels with a scale of 0.296 arcsec/pix and a field of view
(FoV)≈ 10.1 arcmin2. The second set of data was obtained
with the 0.84-m Ritchey–Chrétien telescope of the San Pedro
Mártir Observatory (SPM), México, during eight nights
between June 28 and July 5, 2017. The detector used was
a 2048× 2048 pixel ESOPO CCD (e2v CCD42-90) with
1.7 e−/ADU (analog-to-digital unit) gain, a readout noise of
3.8e−, a 0.44361 arcsec/pix scale, and an FoV ≈ 7.4 arcmin2,
along with the Mexman filter wheel. The log of the observa-
tions is given in Table 1.
The observations described above were published in elec-
tronic format by Arellano Ferro et al. (2018). In that paper,
only the V data of the RR Lyrae stars were employed to
discuss the secular period variations. In the present paper,
we aim to calculate the physical parameters of RR Lyrae via
the Fourier light curve decomposition and extend our analy-
sis to the V and I bands for all variables present in the FoV
of our images. In the following subsections, we describe the
transformation to the standard system.
2.1 Difference image analysis
We have employed difference imaging analysis (DIA) with its
pipeline implementation DanDIA (Bramich 2008; Bramich
et al. 2013) to obtain high-precision photometry for all the
point sources in the FoV of the CCD for the reduction of our
data. DanDIA creates a reference image by stacking the best
quality images in each filter, and then it subtracts them from
the rest of the images in the collection. Each star’s differential
flux is then determined by means of the point spread function
(PSF) calculated by DanDIA from a large number of isolated
stars in the FoV, allowing for spatial variation on the chip.
This enabled us to construct an instrumental light curve for
each star in our FoV.
2.2 Transformation to the standard system
Having obtained the instrumental light curves, we applied
the methodology described by Bramich & Freudling (2012)
to correct for possible systematic photometric errors, which
tend to be significant for the brightest stars. The method
solves for magnitude offsets that should be applied to each
photometric measurement from the given image. In our case,
these offsets were of the order of 5 mag in the bright end and
negligible for fainter stars.
The calibrated light curves of all the point sources in our
FoV were transformed to the Johnson–Kron–Cousins stan-
dard magnitude system (Landolt 1992). We did this by using
standard stars in our FoV identified in the catalog of Pho-
tometric Standard Fields (Stetson 2000). The transformation
equations for each dataset are shown in Figure 1, where the
mild color dependence of the standard minus the instrumental
magnitudes can be seen.
3 ON THE REDDENING OF NGC 6171
Earlier estimates of E(B−V) in NGC 6171 span a large range,
from 0.25 to 0.46, which were obtained by an assortment of
methods (see Table 2 of Smith & Hesser 1986 for a summary).
For example, the use of a variety of reddening-independent
photometric colors favored values between 0.31 and 0.38
(Zinn 1980; Zinn 1985). The adopted value of E(B − V) in
the catalog of Harris (1996) is 0.33. The largest value of
0.46 has been estimated by van den Bergh (1967). Incom-
patibilities of the low reddening value with 100-𝜇m dust
emission data (Dutra & Bica 2000) and between photomet-
ric and spectroscopic Teff for cluster giants have been recently
DERAS ET AL. 605
1.0 1.5 2.0 2.5 3.0 3.5
1.0 1.5 2.0 2.5 3.0 3.5
−0.8
−1.0
−1.2
V
−
v
N = 79
V-v = 0.057(±0.019)(v-i)-1.051(±0.025)
v−i
v−i
−1.0
−1.2
−1.4
I−
i
I-i = 0.111(±0.029)(v-i) - 1.260(±0.037)
N = 79
0.6 0.8 1.0 1.2 1.4 1.6 1.8
0.6 0.8 1.0 1.2 1.4 1.6 1.8
−2.0
−2.1
−2.2
V
−
v N = 73
V-v = -0.111(±0.016)(v-i)-2.021(±0.014)
−2.6
−2.7
I−
i
I-i = 0.037(±0.019)(v-i) - 2.644(±0.016)
N = 73
FIGURE 1 Transformation equations in V and I filters between the
instrumental and the standard photometric systems. These equations were
calculated using a set of standard stars (N = 79 for Hanle and N = 73 for San
Pedro Mártir [SPM]) in the field of NGC 6171. The top panel corresponds
to Hanle observations and the bottom panel to SPM
noted (O’Connell et al. 2011). These authors point out that
these inconsistencies can be alleviated if E(B − V) is 0.45 and
0.46, respectively.
In an attempt to clarify the above incompatibilities, we esti-
mated the reddening for the RRab stars individually using
the method explained by Sturch (1966), according to which
these stars have a constant intrinsic color (B − V)o near min-
imum light, between phases 0.5 and 0.8. This, coupled with
the calibration of (V − I)o in this range of phases by Gulden-
schuh et al. (2005) as ̄(V − I)o,min = 0.58±0.02 mag, allowed
us to estimate the individual values of E(V − I) for a sample
of RRab stars. These were converted to E(B − V) using the
ratio E(V − I)/E(B − V)= 1.259 derived by Schlegel et al.
(1998). The light curves were phased with the periods listed
in Table 2 and were adopted from the analysis of Arellano
Ferro et al. (2018). The resulting values for the reddening are
listed in the second-to-last column of Table 3. For V17, we
have used minimum values of V and I to calculate (V − I)min,
as we were constrained by the data gap in 0.5–0.8 phases.
Ignoring the value for V18, which has a prominent gap near
minimum light, we derived an average E(B − V) estimate
of 0.45± 0.03. This value is in good agreement with those
derived using reddening maps and the calibration of Schlegel
et al. (1998), namely 0.46± 0.01. Although one could expect
differential reddening in this cluster given its location near the
Galactic bulge, there is no evidence of it other than the large
scatter in the values of the reddening found by the variety of
approaches and samples employed by many authors. In what
follows, we will adopt the value of E(B − V)= 0.45 and dis-
cuss the consequences of adopting lower values on the cluster
distance, the age, and the CMD.
4 THE RR LYRAE STARS
4.1 Fourier decomposition and physical parameters
of RR Lyrae stars
Since RR Lyrae stars are the most relevant stars in globular
clusters and are good proxies of the stellar evolution on the
HB, it is useful to determine their physical parameters such
as [Fe/H], log(L/L⊙), log Teff, mass, and radius. This can be
achieved by means of the Fourier decomposition of their light
curves in V into harmonics and by using semiempirical cal-
ibrations that can correlate the Fourier parameters with the
physical quantities. The V and I light curves of the RR Lyrae
stars are shown in Figure 2.
A given light curve of a RR Lyrae star can be represented
by the following equation:
m(t) = A0 +
N
∑
k=1
Ak cos
(
2𝜋
P
k (t − E) + 𝜙k
)
, (1)
where m(t) is the magnitude at time t, P is the period of pul-
sation, and E is the epoch. In order to calculate the Fourier
parameters, a least-squares approach is used to estimate the
best fit for the amplitudes Ak and phases 𝜙k of the sinusoidal
components. The phases and amplitudes of the harmonics
in Equation 1, i.e., the Fourier parameters, are defined as
𝜙ij = j𝜙i − i𝜙j, and Rij =Ai/Aj.
Well-tested calibrations and zero points have been used sys-
tematically by our group over the last few years (Arellano
Ferro et al. 2017). Since other calibrations have been used by
other authors, below we list the specific equations used in this
work. For metallicity and absolute magnitude of the RRab
stars, the calibrations of Jurcsik & Kovács (1996) and Kovács
& Walker (2001) were employed, respectively:
[Fe∕H]J = −5.038 − 5.394 P + 1.345 𝜙
(s)
31
, (2)
MV = −1.876 log P − 1.158 A1 + 0.821 A3 + K. (3)
Note that the calibration for the metallicity is given in
the Jurcsik–Kovács scale but can be transformed to the
Zinn–West scale (Zinn & West 1984) with the equation:
[Fe/H]J = 1.431[Fe/H]ZW + 0.88 (Jurcsik 1995).
For the RRc stars, we employed the calibrations given
by Morgan et al. (2007) and Kovács & Kanbur (1998),
respectively:
[Fe∕H]𝑍𝑊 = 52.466 P2 − 30.075 P + 0.131 𝜙
(c) 2
31
− 0.982 𝜙
(c)
31
− 4.198 𝜙
(c)
31
P + 2.424, (4)
MV = 1.061 − 0.961 P − 0.044 𝜙
(s)
21
− 4.447 A4. (5)
606 DERAS ET AL.
TABLE 2 Data of variable stars in NGC 6171 in the FoV of our images with the exception of V23 whose light curve in the V filter was taken from
Clement & Shelton (1997)
Star ID Type <V> (mag) <I> (mag) AV (mag) AI (mag) P (days) 𝜶 (J2000.0) 𝜹 (J2000.0)
V4 RRc 15.493 14.667 0.462 0.343 0.282132 16:32:25.28 −13:05:55.4
V5 RRab Bl — — 0.483a 0.330a 0.702375 16:32:47.90 −13:05:57.3
V6 RRc Bl 15.614 14.766 0.523 0.424 0.259627 16:32:31.32 −13:04:25.6
V7 RRab Bl — — 1.139a 0.746a 0.497474 16:32:35.00 −13:04:18.9
V8 RRab 15.578 14.523 1.094 0.668 0.559922 16:32:32.82 −13:03:59.9
V9 RRc 15.632 14.674 0.496 0.340 0.320601 16:32:30.18 −13:03:37.9
V10 RRab Bl 15.782 14.811 0.995 0.680 0.415559 16:32:28.08 −13:03:10.3
V11 RRab 15.673 14.543 0.680 0.444 0.592809 16:32:32.60 −13:02:44.9
V12 RRab Bl 15.698 14.694 1.034a 0.678a 0.472833 16:32:35.98 −13:02:16.6
V13 RRab 15.834 14.781 1.193 0.812 0.466797 16:32:30.10 −13:02:06.1
V14 RRab — — — — 0.481620 16:32:33.13 −13:01:55.5
V15 RRc 15.622 14.724 0.483 0.378 0.288590 16:32:33.20 −13:01:17.6
V16 RRab Bl 15.675 14.617 0.801a 0.485a 0.522796 16:32:27.33 −13:01:25.0
V17 RRab 15.593 14.538 1.058 0.681 0.561168 16:32:25.16 −13:02:07.2
V18 RRab 15.739 — 0.811 — 0.561404 16:32:37.13 −12:59:41.9
V19 RRc Bl 15.697 14.952 0.503 0.322 0.278762 16:32:47.79 −13:00:32.7
V20 RRab 15.708 14.610 0.789 0.537 0.578107 16:32:34.08 −13:02:26.8
V21b RRc 16.519 15.660 0.566 0.407 0.258715 16:32:37.61 −13:05:42.5
V23 RRc 15.626 — 0.439 — 0.323344 16:32:13.96 −13:03:01.5
V24 RRab 15.600 14.530 0.882 0.657 0.523949 16:32:31.98 −13:03:09.7
V26 SX Phe 17.616 16.377 — — 0.055275 16:32:32.66 −13:03:00.4
Bl, RR Lyrae with Blazhko effect.
aMaximum observed amplitude.
bNot a cluster member.
The coefficients can be transformed from cosine series
phases into sine series via the relation 𝜙
(s)
𝑗𝑘
= 𝜙
(c)
𝑗𝑘
− (j − k)
𝜋
2
,
when necessary.
In Table 3, we list the Fourier parameters of the RR Lyrae
identified in NGC 6171, and in Table 4 the physical param-
eters derived from the Fourier decomposition. It is worth
recalling that the determination of the iron abundance [Fe/H]
of the RRab stars is valid only if they comply with the devi-
ation parameter (Dm) (Jurcsik & Kovács 1996) not exceed-
ing the value of 3.0. The Dm value for each RRab star is
listed in Table 3 in column 11. Since a few stars have only
marginally larger values of Dm, we relaxed the criterion in
order to include these stars and have a larger sample. For
V8, with Dm = 6.8, its Fourier parameters are reported but
they were not taken into account when calculating the aver-
age of [Fe/H]. Also not included were the variables V5, V6,
V7, V12, V16, and V19 since they show Blazhko-like ampli-
tude modulations, and V14 that has an incomplete light curve.
Despite V21 being a nonmember, we calculated its physical
parameters.
The iron abundance in the scale of Zinn & West (1984)
can be converted into the scale of Carretta et al. (2009)
via the equation [Fe/H]UVES =−0.413+ 0.130[Fe/H]ZW −
0.356[𝐹𝑒∕H]2
ZW
, which are also listed in Table 4.
With the period, luminosity, and temperature known for
each RR Lyrae star, its mass and radius can be estimated
from the equations log M/M⊙ = 16.907–1.47 log PF + 1.24
log (L/L⊙)− 5.12 log Teff (van Albada & Baker 1971) and
L = 4𝜋R2𝜎T4, respectively.
4.2 Bailey diagram and Oosterhoff type
The period–amplitude plane for RR Lyrae stars, also known
as the Bailey diagram, is shown in Figure 3 for the VI band
passes. The periods and amplitudes are listed in Table 2. In
most cases, we took the amplitudes corresponding to the best
fit provided by the Fourier decomposition of the light curves
unless there were glaring gaps such as in the case of V14,
or where amplitude and phase modulations were significant,
like in V5 and V7. In these cases, the star was ignored, or,
in other cases, the maximum amplitude was measured and
the star was plotted with a triangle in Figure 3. With the
exception of these Blazhko variables, the distribution of RRab
stars in both VI band passes shows a mild scatter around the
OoI locus found by Cacciari et al. (2005) for M3 and by
Kunder et al. (2013a) for NGC 2808. This confirms the OoI
type of NGC 6171. While averaging the periods of the RRab
stars, we found <Pab>= 0.53 d, which also defines NGC
6171 as of the OoI type. None of the RRab stars falls near
the evolved sequences, i.e., the segmented loci, thus none of
them seems to be clearly ahead in its evolution towards the
asymptotic giant branch. V17 was found to have a significant
positive secular period change (Arellano Ferro et al. 2018), as
expected in stars advanced in their evolution. However, while
DERAS ET AL. 607
TABLE 3 Fourier coefficients of RRab and RRc stars in NGC 6171 (all in V mag). The numbers in parentheses indicate the uncertainty on the last
decimal place. Also listed are the reddening values EBV (E[B−V]) for each individual RRab star calculated from the method used by Sturch (1966)
and the deviation parameter Dm
ID A0 A1 A2 A3 A4 𝝓21 𝝓31 𝝓41 EBV Dm
RRab
V8 15.578 (1) 0.357 (1) 0.199 (1) 0.118 (1) 0.085 (1) 4.032 (10) 8.295 (14) 6.467 (20) 0.45 6.8
V10 15.782 (1) 0.338 (13) 0.156 (13) 0.137 (12) 0.075 (11) 3.530 (109) 7.654 (138) 5.679 (211) 0.37 3.7
V11 15.673 (2) 0.229 (3) 0.120 (2) 0.064 (2) 0.037 (2) 4.010 (40) 8.426 (40) 6.750 (98) 0.50 3.4
V12 15.698 (1) 0.429 (2) 0.169 (1) 0.064 (1) 0.030 (1) 3.922 (11) 8.066 (19) 5.748 (45) 0.43 3.3
V13 15.834 (1) 0.415 (2) 0.194 (1) 0.135 (1) 0.081 (1) 3.816 (11) 7.954 (16) 5.834 (26) 0.46 1.5
V16 15.675 (2) 0.292 (2) 0.153 (2) 0.073 (2) 0.028 (2) 3.896 (21) 8.143 (31) 6.264 (75) 0.47 2.9
V17 15.593 (1) 0.340 (2) 0.188 (1) 0.115 (2) 0.075 (1) 4.024 (14) 8.396 (14) 6.554 (28) 0.43 1.4
V18 15.739 (3) 0.282 (5) 0.143 (5) 0.084 (5) 0.046 (4) 4.162 (50) 8.355 (74) 6.494 (117) — 2.2
V20 15.708 (1) 0.268 (2) 0.137 (1) 0.082 (1) 0.042 (1) 4.097 (20) 8.504 (25) 6.697 (40) 0.47 1.2
V24 15.600 (16) 0.287 (21) 0.144 (21) 0.105 (20) 0.071 (17) 3.760 (218) 7.964 (306) 5.850 (442) 0.47 2.3
RRc
V4 15.493 (1) 0.228 (2) 0.032 (2) 0.020 (2) 0.014 (2) 4.996 (58) 3.743 (94) 2.482 (140)
V6 15.614 (1) 0.250 (1) 0.049 (1) 0.013 (1) 0.023 (1) 4.429 (25) 3.817 (102) 1.383 (67)
V9 15.632 (1) 0.233 (1) 0.018 (1) 0.019 (1) 0.010 (1) 4.890 (48) 4.167 (50) 2.554 (90)
V15 15.622 (1) 0.240 (2) 0.032 (2) 0.019 (2) 0.013 (2) 4.838 (49) 3.596 (82) 2.114 (125)
V19 15.697 (1) 0.215 (1) 0.042 (1) 0.013 (1) 0.012 (1) 4.137 (31) 2.255 (97) 2.142 (109)
V21 16.519 (8) 0.230 (11) 0.047 (11) 0.012 (11) 0.012 (11) 4.638 (248) 3.129 (855) 1.099 (920)
V23 15.626 (1) 0.207 (2) 0.019 (2) 0.013 (2) 0.009 (2) 4.783 (101) 3.947 (146) 2.505 (198)
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V4
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V5
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V6
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V7
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V8
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V9
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V10
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V11
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V12
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V13
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V14
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V15
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V16
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V17
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V18
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V19
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V20
0.0 0.5 1.0
φ
15.0
15.5
16.0
16.5
V
,I
V21
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
V23
0.0 0.5 1.0
φ
14.0
14.5
15.0
15.5
16.0
V
,I
V24
FIGURE 2 Light curves in V and I filters of RR Lyrae stars in NGC 6171. The light curves are a combination of the data obtained at Hanle (black symbols)
and at San Pedro Mártir (blue symbols). The continuous red line represents the Fourier fit. The scale of the V-axis is the same for all plots with the exception
of fainter V21, which is not a cluster member. In the case of V23, the V light curve was taken from Clement & Shelton (1997). The data for all this light
curves have been published in electronic format by Arellano Ferro et al. (2018)
in the V-band the scatter is larger and it may not preclude
V17 from being inconsistent with the evolved stars locus, in
the I-band all stable RRab stars are rather consistent with the
non-evolved sequence.
5 THE [Fe/H] VALUE OF NGC 6171
NGC 6171 is a moderately metal-rich cluster of the OoI type.
In Table 5 we list the various metallicity values found in
608 DERAS ET AL.
TABLE 4 Physical parameters of the RRab and RRc stars in NGC 6171. The numbers in parentheses indicate the
uncertainty on the last decimal place
ID [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
RRab
V8a −1.40 (13) −1.30 (13) 0.566 (1) 3.811 (7) 1.673 (1) 0.69 (6) 5.50 (1)
V11 −1.40 (38) −1.30 (38) 0.623 (4) 3.804 (13) 1.651 (2) 0.65 (10) 5.54 (1)
V13 −1.37 (15) −1.26 (15) 0.661 (2) 3.821 (7) 1.635 (1) 0.72 (6) 5.03 (1)
V17 −1.31 (13) −1.20 (12) 0.581 (3) 3.812 (8) 1.668 (1) 0.67 (6) 5.44 (1)
V18 −1.35 (70) −1.24 (68) 0.623 (7) 3.810 (15) 1.651 (3) 0.66 (12) 5.40 (2)
V20 −1.27 (23) −1.16 (22) 0.614 (2) 3.809 (8) 1.655 (1) 0.64 (6) 5.44 (1)
V24 −1.58 (288) −1.50 (325) 0.690 (29) 3.810 (50) 1.624 (12) 0.67 (40) 5.21 (7)
Weighted mean −1.33 (1) −1.22 (1) 0.596 (1) 3.813 (4) 1.661 (1) 0.68 (3) 5.34 (1)
𝜎 ±0.12 ±0.12 ±0.051 ±0.002 ±0.006 ±0.01 ±0.07
RRc
V4 −0.81 (17) −0.75 (10) 0.577 (9) 3.875 (1) 1.669 (4) 0.57 (1) 4.07 (1)
V9 −1.07 (10) −0.96 (8) 0.562 (5) 3.870 (1) 1.675 (2) 0.51 (1) 4.21 (1)
V15 −1.02 (15) −0.91 (11) 0.582 (9) 3.873 (1) 1.667 (4) 0.56 (1) 4.11 (2)
V21b −0.89 (1.43) −0.81 (92) 0.624 (50) 3.877 (5) 1.650 (20) 0.60 (5) 3.95 (9)
V23 −1.26 (29) −1.14 (26) 0.569 (10) 3.868 (1) 1.672 (4) 0.51 (1) 4.23 (2)
Weighted mean −1.02 (7) −0.90 (5) 0.569 (3) 3.871 (1) 1.672 (1) 0.60 (1) 4.13 (1)
𝜎 ±0.19 ±0.19 ±0.010 ±0.002 ±0.002 ±0.02 ±0.04
aNot included in the average of [Fe/H].
bNot a cluster member, hence not included in the calculation of mean physical parameters.
the literature and the methods used to determine them. We
include the results from Fourier decomposition found in this
work for comparison. We can see that many of the results
come from the spectroscopic metallicity index ΔS defined
by Preston (1959). These values agree well, within uncer-
tainty, with the value derived for the RRc stars in the present
work. The result for the RRab stars, [Fe/H]=−1.33± 0.12,
although in reasonable agreement with the other estimates,
is the lowest value. Nevertheless, it is in excellent agree-
ment with the trend of the Mv versus[Fe/H]ZW relation (see
Figure 6) found by Arellano Ferro et al. (2017) for the RRab
stars from a sample of 24 globular clusters.
As a further test, we applied Equations 2 and 4 to the
Fourier parameters published by Clement & Shelton (1997)
for the stable members of NGC 6171 and found [Fe/H]ZW
−1.31± 0.02 and −1.03± 0.04 for the RRab and RRc stars,
respectively, which are very similar to the results from the
decomposition of our own light curves in the present work.
We note that the value from the RRc stars is similar to the
spectroscopic results.
6 THE DISTANCE TO NGC 6171
The estimation of the distance to globular clusters relies heav-
ily upon the accurate determination of the reddening. For
NGC 6171, the values of distance and reddening that were
adopted by Harris (1996) are 6.4 kpc and E(B − V)= 0.33,
respectively. However, as discussed in Section 3, the value of
E(B − V) from a large number of independent estimates has
been found to have a large scatter. From the average reddening
calculated from the minimum color in RRab stars, we have
adopted E(B − V)= 0.45.
To estimate the distance to NGC 6171, we made use of
four independent methods, all of which yield consistent val-
ues within their respective uncertainties. We describe these
methods in the following sections.
6.1 From the RR Lyrae stars
For the RR Lyrae stars, the values of the absolute magnitude
MV of each star were employed to calculate the individual dis-
tances. Since two independent calibrations for RRab and RRc
stars for MV were used, two equally independent values of the
distance to NGC 6171 were found: 5.3± 0.3 and 5.3± 0.2 kpc
for the RRab and RRc stars, respectively. We stress at this
point that the individual values of E(B − V) listed in Table 3
for the RRab stars were employed in the calculations. For the
RRc stars, since we do not know the individual reddenings, we
adopted the mean from the RRab stars, i.e., E(B − V)= 0.45.
An alternative approach is via the period–luminosity (P–L)
relation for RR Lyrae stars in the filter (Catelan et al.
2004) MI = 0.471–1.132 log P + 0.205 log Z, with log
Z = [M/H]− 1.765; [M/H]= [Fe/H]− log(0.638 f + 0.362)
and log f = [𝛼/Fe] (Salaris et al. 1993). Thus, adopting a
value for [Fe/H] and the reddening, one can obtain the cor-
responding average distance. Applying this relation to the
RRab and RRc stars, each with their [Fe/H]ZW value given
in Table 4, and adopting [𝛼/Fe]=+0.3, we found the aver-
age distance <d>= 5.6 ± 0.2 kpc and <d>= 5.4 ± 0.3 kpc
respectively, which are consistent with the distances obtained
DERAS ET AL. 609
−0.6 −0.4 −0.2
−0.6 −0.4 −0.2
0.0
0.5
1.0
1.5
A
V
V5
V 7
V10
V12
V16
V17
logP
0.0
0.5
1.0
A
I
V5
V 7
V10 V12
V16
V17
FIGURE 3 Bailey diagram for NGC 6171. Filled and open symbols
represent RRab and RRc stars, respectively. Triangles correspond to stars
with Blazhko modulations. The continuous and segmented lines in the top
panel are the loci for unevolved and evolved stars in M3 according to Cacciari
et al. (2005). The black parabola was obtained by Kunder et al. (2013a) for
RRc stars in 14 OoII clusters. The red parabolas were calculated by Arellano
Ferro et al. (2015) from a sample of RRc stars in five OoI clusters and
avoiding Blazhko variables. In the bottom panel, the black segmented locus
was found by Arellano Ferro et al. (2011) and Arellano Ferro et al. (2013)
for the OoII clusters NGC 5024 and NGC 6333, respectively. The blue solid
and segmented loci for unevolved and evolved stars, respectively, are from
Kunder et al. (2013b). For a more detailed discussion, see Section 4.2
from the Fourier approach described above. RR Lyrae stars
with Blazhko amplitude modulations (V5, V6, V7, V12, V16,
V19), incomplete light curve (V14, V18), noIdata available
(V23), or not a member of the cluster (V21) were ignored.
6.2 From the SX Phe star
There is only one known SX Phe star in NGC 6171, namely
V26. The period and the intensity-weighted <V> and <I>,
as well as other relevant data for this star, are listed in
Table 2. In order to calculate the distance to NGC 6171 from
this SX Phe, we adopted the P–L relation for SX Phe stars
from Cohen & Sarajedini (2012) for the fundamental mode
Mv = (−1.64± 0.11)− (3.39± 0.09) log Pf . As was done with
the RR Lyrae stars, we also adopted E(B − V)= 0.45.
0.0 0.5 1.0
φ
17.4
17.5
17.6
17.7
17.8
V
FIGURE 4 Light curve of V26. The black and blue dots correspond to the
data from Hanle and San Pedro Mártir, respectively
TABLE 5 Historical values of [Fe/H] for NGC 6171
Method [Fe/H]
Spectroscopic metallicity index ΔS −0.82a
High-resolution spectroscopy −1.00b
Spectroscopic metallicity index ΔS −0.83, −0.94c
Spectroscopic metallicity index ΔS −0.84d
Mean from previous ΔS determinations −0.90e
Webbink’s empirical relation −1.09f
Revised ΔS method −0.99g
Spectroscopic analysis −0.93h
RRab Fourier decomposition −1.33i
RRc Fourier decomposition −1.02i
aZinn (1980).
bPilachowski et al. (1981).
cSmith & Perkins (1982).
dSmith & Manduca (1983).
eDa Costa et al. (1984).
fCudworth et al. (1992).
gClementini et al. (2005).
hO’Connell et al. (2011).
iThis work.
The resulting distance to V26 is d = 5.2 ± 0.4 kpc, which
is in agreement with the distance obtained above from the
RR Lyrae stars. This confirms that V26 is a fundamental pul-
sator and a member of the cluster. The position of V26 in the
CMD diagram is shown in Figure 5. Since there is only one
SX Phe, the quoted uncertainty corresponds to the uncertainty
associated with the P–L relation.
It is worth noting that the light curve of the SX Phe is very
disperse (Figure 4), which is due to the fact that the star is
faint and is located near the central region of the cluster. We
found no traces of a secondary pulsating mode.
The weighted mean of the distance rendered by the above
four approaches is 5.4 ± 0.1 kpc. In Table 6 we compare this
result with some values from the literature.
6.3 The tip of the red giant branch as a distance
indicator
The luminosity of the true tip of the RGB (TRGB) can in prin-
ciple be used to estimate the distance to globular clusters.
It has been argued on theoretical grounds that the neutrino
magnetic dipole moment enhances the plasma decay process,
610 DERAS ET AL.
0.5 1.0 1.5 2.0 2.5
V−I
13
14
15
16
17
18
19
20
V V 21
V 9
NGC 6171 (M107) V 25
V 26
V 27
V 28
FIGURE 5 Color versus magnitude diagram of NGC 6171. Blue circles
correspond to RRab stars, green circles to RRc stars, and the magenta circle
to the SX Phe. The open green circle corresponds to an RRc star (V21) that
is not a member of the cluster. V27 and V28 are variables newly found in
the present work. The two isochrones are from VandenBerg et al. (2014),
with Y = 0.25 and [𝛼/H]= 0.4, corresponding to an age of 11 Gyr and to a
[Fe/H]RRab =−1.33 (blue) and to a [Fe/H]RRc =−1.02 (green). The zero-age
horizontal branch for [Fe/H]=−1.31, and isochrones have been reddened
by E[B−V]= 0.45 and shifted to the distance from the RRab and RRc
stars, i.e., 5.3 kpc. The continuous and dashed vertical lines represent the
red edge of the first overtone instability strip reddened by E(B−V) 0.33 and
0.45, respectively. See Section 7 for a discussion
TABLE 6 Other values of the distance to NGC 6171
Method Distance (kpc)
Assuming Mv =+0.5 for the HB. 7.2a
Absolute and apparent magnitude of the HB 6.2± 0.9b
From bolometrically corrected magnitudes 5.6c
Fourier decomposition; P-L RRL, SX Phe 5.4± 0.1d
aPeterson & King (1975).
bCudworth et al. (1992).
cShetrone et al. (2009).
dThis work.
postpones helium ignition in low-mass stars, and extends
the red giant branch, and hence the brightest stars in a given
cluster may lie below the true TRGB. Viaux et al. (2013)
found that the difference for the case of M5 is between 0.04
and 0.16 mag. By a comparison of their noncanonical mod-
els with the observed TRGB in 25 clusters, Arceo-Díaz et al.
(2015) concluded that, on average, the theoretical TRGB is
some 0.26 bolometric magnitudes brighter than the observed
one with a standard deviation (SD) of 0.24 (see their Table 5).
For the case of NGC 6171, we note from the CMD in
Figure 5 that the three brightest stars are about 0.98 mag
fainter than the theoretical TRGB as calibrated by Salaris &
Cassisi (1997):
M
𝑡𝑖𝑝
𝑏𝑜𝑙
= −3.949 − 0.178[M∕H] + 0.008[M∕H]2. (6)
However, it is likely that our photometry for this bright
star is still affected by systematic errors that could not be
fully removed by the post calibration described in Section
2.2, and hence the TRGB bends toward lower luminosities,
as suggested by the comparison with the isochrones (see
Section 7). Therefore, this approach does not lead to a reliable,
independent estimate of the cluster distance.
7 THE CMD AND THE AGE OF NGC 6171
The CMD of NGC 6171 in Figure 5 was built using the
magnitude-weighted means of V and V − I of the 3827 mea-
sured stars in the FoV of our images. For all variables, the
intensity-weighted means <V> and <V>−<I>were used.
These were calculated from the Fourier fits of light curves in
Figure 2 and are listed in Table 2.
In order to confirm that the CMD is consistent with
the basic parameters found in this work, i.e., reddening,
metallicity, and distance, we overlaid theoretical isochrones
and a zero-age horizontal branch (ZAHB) taken from the
Victoria-Regina stellar models of VandenBerg et al. (2014),
which were kindly made available by the authors. For the age
of the cluster, we refer to the differential age estimation by
De Angeli et al. (2005) from which an age of 10.98 Gyrs can
be inferred for NGC 6171. We adopted the isochrones cor-
responding to 11.00 Gyrs and for [Fe/H]=−1.33 (blue) and
−1.02 (green), as suggested from the RRab and RRc stars,
respectively. The ZAHB for [Fe/H]=−1.31 is also shown.
These models were reddened by E(B − V)= 0.45 and shifted
to the distance of 5.3 kpc, producing a satisfactory match with
the observed CMD.
We must stress at this point that, if the alternate values of
E(B − V)= 0.33 and distance 6.4 kpc were used, both the
isochrones and the ZAHB would be way off to the red of the
observed stellar distributions.
The distribution of RRab and RRc in the instability strip on
the HB should be noted. Except for V9, the two modes seem
to be neatly separated, as it occurs in all OoII-type clusters but
only in some OoI clusters (Arellano Ferro et al. 2018). The
split border between the modes was calibrated by Arellano
Ferro et al. (2016) at (V − I)0 = 0.45–0.46 and was inter-
preted as the red edge of the first overtone (RFO) instability
strip. Hence no RRc stars should be found to the red side of
this border. In Figure 5, this border, reddened by 0.33 and
0.46, is shown as continuous and segmented vertical lines,
respectively. In order to place RFO in the observed separa-
tion of modes, a reddening of between 0.35 and 0.40 would
be required. We believe that all the disagreements in the value
of reddening, and those commented in Section 3, may suggest
the presence, to some extent, of differential reddening, which,
given the Galactic position of the cluster, should not come as
a surprise.
Let us turn our discussion toward the RRc variable V9,
located beyond the RFO of the instability strip. Its position
DERAS ET AL. 611
in the CMD is firm given the high photometric quality of the
V and I light curves. One possible explanation can be based
upon higher reddening than the average value. If we consider
that the star should be situated somewhere between two RFOs
in Figure 5, a reddening value higher by 0.08–0.12 than the
average reddening would be required to reconcile its location
in the CMD. This would imply that the star in the present posi-
tion is dimmer by 0.25–0.35 magnitude. While advocating the
possibility of the presence of differential reddening in NGC
6171, we should refrain from further speculative discussion
on this star.
In spite of the above reddening demeanors, we consider
NGC 6171 as one more example of a very red HB morphol-
ogy ((B – R)/(B + V + R)=−0.74) OoI cluster, with the two
pulsation modes well split.
8 DISCUSSION
In this paper, we presented the results of the first ever deter-
mination of physical parameters of RR Lyrae stars in NGC
6171 by means of Fourier decomposition of their light curves.
By performing high-precision photometry, we were able
to extract the light curves of 3827 stars in our reference image,
20 of which are RR Lyrae, 1 SX Phe, and 1 Lb variables.
Of the three Lb variables, the variability of two is newly
announced in this work (V27 and V28).
An estimation of the reddening via the minimum color of
RRab stars suggests a mean value of E(B − V)= 0.45. How-
ever, historical determinations of E(B − V) display a large
scatter between 0.25 and 0.48. This fact may seem to sug-
gest that some differential reddening may be present. Taking
individual reddening values for the RRab stars and an aver-
age of 0.45 for the RRc stars, and considering the results
from the P–L relations for the RR Lyrae and SX Phe stars,
we found an overall average distance of 5.4 ± 0.1 kpc. For
RRab and RRc, the metallicities are [Fe/H]ZW − 1.33± 0.12
and− 1.02± 0.19, respectively. It is worth noticing that the
results from the RRc stars are closer to the spectroscopic
results, which are indisputably the most accurate and are not
affected by the reddening.
The above distance and iron values tend to be at the lower
end of the range of previous estimates, as shown in Table 6.
Particularly, [Fe/H]ZW of −1.33 from the RRab seems a bit
too low. We should highlight, however, that the isochrone for
this metallicity fits the observations very well. If the isochrone
for [Fe/H]ZW =−1.02 from the RRc stars is shifted upward
to match the data, it would imply an even shorter distance of
5.0 kpc.
Let us refer to the Mv versus [Fe/H]ZW relation for RR
Lyrae. From the homogeneous Fourier decomposition of RR
Lyrae stars in a sample of 24 clusters of both OoI and OoII
types, Arellano Ferro et al. (2017) found that the slope of the
relationship is different when calculated from RRab and from
RRc stars. In Figure 6, the two trends for the RRab and RRc
−2.0 −1.5 −1.0
[Fe/H]
ZW
0.4
0.5
0.6
0.7
M
v
Trend from RRab
Trend from RRc
6171(RRab )
6171(RRc )
FIGURE 6 Position of NGC 6171 on the [Fe/H]–MV plane according to
the Fourier decomposition results of RRab and RRc stars. The lines are the
trends found by Arellano Ferro et al. (2017) for a family of 24 globular
clusters
stars are shown and the corresponding position of NGC 6171
given the values in Table 4. This shows that despite the differ-
ence in the metallicity from the two stellar groups, the present
results are consistent with the homogeneous Fourier analysis
in a larger sample of globular clusters.
ACKNOWLEDGMENTS
We acknowledge the financial support from DGAPA-UNAM
(grants IN106615-17, IN105115) and from CONACyT (Méx-
ico). D.D. is grateful to CONACyT, México, for a PhD schol-
arship. We have made extensive use of the SIMBAD and ADS
services, for which we are grateful.
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How to cite this article: Deras D, Arel-
lano Ferro A, Muneer S, Giridhar S, Michel R.
Physical parameters of RR Lyrae stars in
NGC 6171. Astron. Nachr. 2018;339:603–614.
https://doi.org/10.1002/asna.201813489
APPENDIX
A. COMMENTS ON THE INDIVIDUAL
VARIABLES
A.1. V5, V6, V7, V12, V16, AND V19
All these stars show clear signs of amplitude and/or phase
modulations.
A.2. V10
Although the general distribution of RR Lyrae stars in the
period–amplitude plane follows that of an OoI cluster, V10
shows a peculiar position despite having a well-defined light
curve as noted in Section 4.2. Clement & Shelton (1997)
found a mild irregularity in the light curve of V10, claim-
ing that its shape is not consistent from cycle to cycle. They
attribute this behavior to a probable Blazhko effect, which
would explain the off-trend position of this star and also
would make it consistent with the position of variables V12
and V16 in the Bailey diagram. A close inspection of the
light curve of V10 published by Arellano Ferro et al. (2018)
indeed seems to indicate a mild amplitude modulation. Arel-
lano Ferro et al. (2018) report that this star is going through a
secular period decrease.
A.3. V19
Kovács et al. (1986) noted this star as a double-mode pul-
sator, but the first overtone to fundamental mode period ratio
P1/P0 was not found to be the canonical for radial pulsations.
Given that the light curve presented in this work is very well
defined over the time span of 2 years, we performed a fre-
quency analysis but did not detect a significant secondary
frequency.
A.4. V21
This RRc star is, in fact, farther than the cluster, as it falls well
below the HB in the CMD of Figure 5. Its distance, from the
calibration described in Section 4.1 is 7.8 kpc. The nonmem-
ber status of this star was first noted by Dickens (1970) and
later confirmed by Cudworth et al. (1992).
A.5. V25, V27, and V28
V25: The variability of this star was reported by Lloyd Evans
& Menzies (1973, 1977). They acknowledge that their classi-
fication as variable depends only on one V and one I plates.
The star was classified as Lb by Clement et al. (2001) in
their CVSGC. On the basis of our present data, we confirm
the irregular, long-term variable nature of this star, which is
shown in Figure A1.
V27, V28: We report here the variability of these two stars.
Like V25, they present long-term irregular variations, and
hence we classify them as Lb variables. Their light curves are
displayed in Figure A1. The mean magnitudes and equato-
rial coordinates of V25, V27, and V28 are given in Table A1.
Their position near the TRGB can be seen in the CMD of
Figure 5.
DERAS ET AL. 613
0.15 0.20 0.25
HJD + 2457200
12.98
13.00
13.02
13.04
13.06
13.08
13.10
V
0.20 0.25 0.30
HJD + 2457201
12.98
13.00
13.02
13.04
13.06
13.08
13.10
V
0.30 0.35 0.40
HJD + 2457527
12.98
13.00
13.02
13.04
13.06
13.08
13.10
V
0.30 0.35 0.40
HJD + 2457528
12.98
13.00
13.02
13.04
13.06
13.08
13.10
V
0.65 0.70 0.75
HJD + 2457932
12.98
13.00
13.02
13.04
13.06
13.08
13.10
V
0.15 0.20 0.25
HJD + 2457200
13.10
13.12
13.14
13.16
13.18
13.20
13.22
13.24
V
0.20 0.25 0.30
HJD + 2457201
13.10
13.12
13.14
13.16
13.18
13.20
13.22
13.24
V
0.30 0.35 0.40
HJD + 2457527
13.10
13.12
13.14
13.16
13.18
13.20
13.22
13.24
V
0.30 0.35 0.40
HJD + 2457528
13.10
13.12
13.14
13.16
13.18
13.20
13.22
13.24
V
0.65 0.70 0.75
HJD + 2457932
13.10
13.12
13.14
13.16
13.18
13.20
13.22
13.24
V
0.15 0.20 0.25
HJD + 2457200
13.19
13.20
13.21
13.22
13.23
13.24
13.25
13.26
V
0.20 0.25 0.30
HJD + 2457201
13.19
13.20
13.21
13.22
13.23
13.24
13.25
13.26
V
0.30 0.35 0.40
HJD + 2457527
13.19
13.20
13.21
13.22
13.23
13.24
13.25
13.26
V
0.30 0.35 0.40
HJD + 2457528
13.19
13.20
13.21
13.22
13.23
13.24
13.25
13.26
V
0.65 0.70 0.75
HJD + 2457932
13.19
13.20
13.21
13.22
13.23
13.24
13.25
13.26
V
V25
V27
V28
FIGURE A1 Long-term variability in three Lb type stars. V27 and V28, which are newly announced in the present paper
FIGURE A2 Finding chart built from the V reference image. The FoV is approximately 4.8× 4.8 arcmin2. Note that in the chart North is down and East is to
the left
614 DERAS ET AL.
TABLE A1 Irregular variable giants Lb in NGC 6171
ID <V> (mag) <I> (mag) 𝜶 (J2000.0) 𝜹 (J2000.0)
V25 13.048 10.691 16:32:31.41 −13:02:59.1
V27 13.137 11.084 16:32:24.00 −13:02:10.4
V28 13.220 11.166 16:32:39.00 −13:02:50.5
53
Chapter 7
Period changes in the RR Lyrae stars of NGC
6171 (M107)
7.1 Overview
In this article we studied the secular period variations in 22 RRL stars of NGC 6171, based on photometric data
with a time-base of 80 years, obtained between 1937 and 2017. We employed the time of maximum light drifting
approach by means of the O − C diagrams.
7.2 Observations
We have recently acquired observations from two sites: the 2 m telescope at the IAO, Hanle, India during 2015
and 2016 and from the 0.84 m telescope at SPM, during 2017. These data were transformed from the instrumental
magnitude to the standard photometric system of Johnson-Kron-Cousins (Landolt, 1992) using the standard stars
in our FoV provided by Stetson (2000). The data from 1935-1994 were kindly made available by Prof. Christine
Clement of the University of Toronto. All the data at our disposal span about 82 years. A summary of the data
available is presented in Table 7.1.
The method employed to study the secular period changes in the RRL stars in NGC 6171 is described in Section
5.3.
Table 7.1: Sources of photometric data of the RRL stars in NGC 6171 (Arellano Ferro et al.,
2018).
Authors Years Band
Oosterhoff (1938) 1935 pg
Coutts et al. (1971) 1946-1970 pg
Kukarkin (1961) 1959-1960 pg
Mannino (1961) 1959-1960 pg
Dickens (1970) 1966-1967 Bpg and Vpg
This article 1972-1991 Bpg
Clement et al. (1997) 1993-1994 CCD V
This article 2015-2017 CCD V
54 Chapter 7. Period changes in the RR Lyrae stars of NGC 6171 (M107)
7.3 Times of maximum brightness and the O − C diagrams
Our approach was to estimate as many times of maximum brightness as possible in the available data. In those
stars where the data sets covered several years, we were able to recover numerous times of maximum light and
produced the dense O − C diagrams in Fig. 7.1. The O − C residuals were calculated by identifying an epoch
and estimating the period at that particular epoch. The period and epoch (P0 and E0) in Table 7.2 were used as
initial values. As seen in Section 5.3, in general there are only two point distributions in a O −C diagram: linear
and parabolic. If the distribution of points is linear it means that the period of the star is not changing in time
and if the distribution is parabolic it means that the period is increasing (if the parabola is positive) or decreasing
(if the parabola is negative). One drawback of the O − C method is that it is very sensitive to the counting of
cycles, especially when dealing with long time bases and short periods of pulsation. This can cause the value
of the estimated period to drift because we either assumed a wrong initial value or because it is authentically
changing. Not considering this effect can lead the O − C diagram to show irregular shapes, suggesting false
period variations.
7.4 Period changes and the evolution on the HB
It has been noted that positive period change rates due to evolutionary origin tend to occur at a higher frequency
in GCs that show blue HB structures (L=(B-R)/(B+V+R) ≈ 1) as confirmed by models of the HB calculated
by Lee (1991) and Catelan (2004). Catelan (2009) noted the behavior of β as a function of L and shows that
in clusters with a red HB, the average value of β should be about 0 days/Myr. This is the case for NGC 6171
since it possesses a notably red HB (L =-0.74). From Table 7.2, we see that the overall average of β values is
-0.086 ± 0.346 days/Myr, which given the uncertainties, is not significantly different from zero. For the RRL
stars in NGC 6171, we found large secular period change rates in four stars namely: V10, V12, V16 and V17.
V10, V12 and V16 have large negative values of β, which implies evolution to the blue. The only star showing
a positive increase in its period is V17 with β =+0.624 ± 0.021 day/Myr. Conversely, by finding that only four
RRL stars are experiencing period changes, this means that the rest (82%) have kept a stable period of pulsation
in a 80-year span, which is quite remarkable.
7.5 Contributions made by the student to this article
• I refined the periods of the RRL stars to better phase their light curves with an accuracy of up to ∼ 10−6 d.
This resulted in the periods being updated and allowed a better cycle counting when searching for and/or
estimating period changes, which was the main goal of this article.
• I contributed to the recovery of historical times of maximum.
• I participated with the rest of the team in the discussions and decisions towards the final structure of the
paper.
• I gained quite a bit of experience regarding the issues of generating a scientific text based on our own data.
By being involved in it I learned about the evolutionary relevance of secular period changes.
7.5. Contributions made by the student to this article 55
Table 7.2: New periods and period change rates for RRLs in NGC 6171. The largest values found
of β0 are in boldface. The uncertainties in β0 correspond to the uncertainty in the coefficient A2
in Eq. 5.30 (Arellano Ferro et al., 2018).
Variable Variable P0 E0 E0 + A0 P0 + A1 β0
Star ID Type (+2 400 000) (+2 400 000) (O − C)
(days) (HJD) (HJD) (days) (d Myr−1)
V2 RRab 0.5710 42538.841 42538.840 0.571021 −0.069 ± 0.062
V3 RRab 0.5663 41844.561 41844.570 0.566344 +0.055 ± 0.095
V4 RRc 0.282130 57528.3538 57528.3633 0.282132 +0.005 ± 0.008
V5 RRab 0.7024 57200.193 57200.3342 0.702376 −0.062 ± 0.058
V6 RRc 0.259635 57527.2486 57527.2307 0.259627 +0.058 ± 0.029
V7 RRab 0.4975 44018.667 44018.635 0.497474 −0.155 ± 0.117
V8 RRab 0.5599 44371.599 44371.588 0.559922 +0.091 ± 0.034
V9 RRc 0.3206 57200.2279 57200.2369 0.320601 +0.012 ± 0.014
V10 RRab 0.415506 57528.4208 57528.4707 0.4155586 −0.714 ± 0.039
V11 RRab 0.5928 57528.3744 57528.3821 0.592809 +0.096 ± 0.022
V12 RRab 0.472833 49125.719 57527.331 0.472830 −0.756 ± 0.168
V13 RRab 0.4668 44371.865 44371.873 0.466797 +0.070 ± 0.230
V14 RRab 0.4816 43275.6195 43275.6356 0.48162 +0.067 ± 0.010
V15 RRc 0.288589 57200.1676 57200.0880 0.288590 +0.001 ± 0.066
V16 RRab 0.522798 57200.1894 57200.1503 0.522796 −1.088 ± 0.045
V17 RRab 0.561168 41860.578 41860.593 0.5611675 +0.624 ± 0.021
V18 RRab 0.561404 57528.4361 57528.4361 0.561404 +0.040 ± 0.128
V19 RRc 0.278766 57528.3812 57528.3666 0.278762 +0.005 ± 0.009
V20 RRab 0.5781 41863.748 41863.765 0.578107 −0.003 ± 0.018
V21 RRc 0.258724 57201.2263 57201.203 0.258715 −0.004 ± 0.053
V23 RRc 0.323343 49477.617 49477.6189 0.323344 +0.076 ± 0.056
V24 RRab 0.523977 49123.856 49123.821 0.523947 −0.261 ± 0.128
56 Chapter 7. Period changes in the RR Lyrae stars of NGC 6171 (M107)
Figure 7.1: O − C residuals as a function of number cycles of RRL stars in NGC 6171 (Arellano
Ferro et al., 2018).
Received: 30 October 2017 Revised: 12 January 2018 Published on: 15 April 2018
DOI: 10.1002/asna.201813409
O R I G I N A L A R T I C L E
Period changes in the RR Lyrae stars of NGC 6171 (M107)
A. Arellano Ferro1,* P. Rosenzweig2 A. Luna1 D. Deras1 S. Muneer3 S. Giridhar3
R. Michel4
1Instituto de Astronomía, Universidad Nacional
Autónoma de México, Ciudad Universitaria,
Mexico City, Mexico
2Grupo de Astrofísica Teórica, Facultad de
Ciencias, Universidad de Los Andes, Venezuela
3Indian Institute of Astrophysics, Bangalore, India
4Observatorio Astronómico Nacional, Instituto de
Astronomía Universidad Nacional Autónoma de
México, Ensenada, Mexico
*Correspondence
A. Arellano Ferro, Instituto de Astronomía,
Universidad Nacional Autónoma de México,
Ciudad Universitaria, CP 04510, Mexico.
Email: armando@astro.unam.mx
Funding Information
CDCHTA—Universidad de Los Andes (ULA),
C-1992-17-05-B. Instituto de Astronomía
(UNAM). National University of México.
CONACyT. DGAPA-UNAM, IN105115,
IN106615-17.
Based on photometric data obtained between 1935 and 2017, O−C diagrams were
built for 22 RR Lyrae stars in the globular cluster NGC 6171, leading to the discov-
ery of secular period changes in four variables for which we have calculated period
change rates 𝛽. In contrast, we find that 82% of the sample stars have stable peri-
ods over the last 82 years. For the stable-period stars, the whole database has been
employed to refine their periods. Among the period-changing stars, three (V10, V12,
and V16) have decreasing periods that are larger than expected from stellar evo-
lution. Despite these individual cases of significant period change rate, the global
average of the measured period changes in the cluster is basically zero, in conso-
nance with theoretical predictions for clusters with redder horizontal branches. The
hitherto unpublished observations, now brought into public domain, are employed
to calculate a set of times of maximum light, which is used in the present analysis.
KEYWORDS
globular clusters: individual (NGC 6171) – stars: variables: RR Lyrae
1 INTRODUCTION
The study of secular period changes of RR Lyrae stars (RRLs)
in globular clusters may play a decisive role in testing hor-
izontal branch (HB) evolution models. However, measuring
secular period changes from observations is difficult as accu-
rate observations over a very long time base are required. Only
a few clusters have been studied from data covering more than
60 years, for example, M3 (Corwin & Carney 2001; Jurcsik
et al. 2012), M5 (Arellano Ferro et al. 2016; Szeidl et al.
2011), NGC 6934 (Stagg & Wehlau 1980), M14 (Wehlau &
Froelich 1994), M15 (Silbermann & Smith 1995), NGC 7006
(Wehlau et al. 1999), and 𝜔 Cen (Jurcsik et al. 2001). Theory
predicts that blueward or redward evolution near the zero-age
HB (ZAHB) is slow and produces very small period change
rates, except toward the end of the HB evolution, when the
values of 𝛽 = Ṗ can be between +0.1 and +0.15 day Myr−1
(Lee 1991). However, in several of the above studies, stars
with significantly large values of 𝛽, both positive and nega-
tive, have been reported. These high values may be the result
of nonevolutionary effects, like stochastic processes related to
mixing events in the core of the star (Balazs-Detre & Detre
1965; Sweigart & Renzini 1979) or to the fast crossing of the
instability strip (IS) of pre-ZAHB stars on their evolution to
the blue, with 𝛽 ≤−0.3 day Myr−1 (Silva-Aguirre et al. 2008).
In the present investigation, we have focused our attention
on the globular cluster NGC 6171 (M107 or C1629-129 in
the IAU nomenclature, 𝛼 = 16h32m31.86s, 𝛿 =+13◦03′13.6′′
J2000, galactic coordinates l= 3◦.37, b=+23◦.01).
A previous study of secular period changes in NGC 6171
was published by Coutts & Sawyer Hogg (1971), who
included photographic data of 22 RRLs taken between 1935
and 1970; however, the times of maximum light were rather
scanty. A commentary of the period change rates based on
statistical theoretical grounds was published by Gryzunova
(1979a, 1979b). Supplemented with data from Las Cam-
panas 1972–1991 (now published in the present work), Las
Campanas 1993–1994 (Clement & Shelton 1997), and our
Hanle 2015–2016 and San Pedro Mártir 2017 CCD obser-
vations, the time base extends to ∼82 years, constitutes a
significant improvement, and encourages a new approach to
the study of the secular period changes of the RRLs of this
cluster.
The photometric data were used to calculate as many times
of maximum light as possible, and these were employed to
investigate the secular period behavior of 22 RRLs in NGC
Astron. Nachr. / AN. 2018;339:158–167 www.an-journal.org © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 158
ARELLANO FERRO ET AL. 159
TABLE 1 Sources of photometric data of the RRLs in NGC 6171
Authors Years Band
Oosterhoff (1938) 1935 pg
Coutts & Sawyer Hogg (1971) 1946–1970 pg
Kukarkin (1961) 1959–1960 pg
Mannino (1961) 1959–1960 pg
Dickens (1970) 1966–1967 Bpg and Vpg
Table 3 (this article) 1972–1991 Bpg
Clement & Shelton (1997) 1993–1994 CCD V
Table 2 (this article) 2015–2017 CCD V
6171. The sources and temporal distribution of the data are
indicated in Table 1.
The present article is organized as follows: in section 2, we
briefly describe our 2015–2017 observations; in section 2.1,
we summarize the 1935–1994 data taken from the literature;
in section 3, the O−C method is described. In section 4, the
approach used to estimate the times of maximum brightness
is explained, and it contains the individual O−C diagrams
and the resulting refined periods and period change rates.
In section 5, we discuss our results in the context of stel-
lar evolution, and finally, in section 6, our conclusions are
summarized.
2 OBSERVATIONS
The most recent V CCD time series observations used in
this article substantially extend the time baseline, in many
cases, to ∼ 82 years. These observations were performed on
four nights, between June 26, 2015 and May 5, 2016, with
the 2.0 m telescope at the Indian Astronomical Observatory
(IAO), Hanle, India. For seven nights between June 29 and
July 5, 2017, data were obtained with the 0.84 m telescope
at the San Pedro Mártir Observatory (SPM), México. A total
of 292 images obtained in the Johnson–Kron–Cousins V fil-
ter are used for the purpose of the present analysis. I images
were also obtained. The instrumental system was converted
into the Landolt–Johnson/Kron–Cousins standard system via
standard stars in the field of the cluster provided by Stetson
(2000). The standard system V–I magnitudes and their uncer-
tainties for the RRLs are published in electronic format, and
we present a small portion of it in Table 2.
2.1 The 1935–1994 data of NGC 6171
The data obtained between 1935 and 1994 have been system-
atically assembled by Prof. Christine M. Clement, who kindly
made them available to us. The original sources are summa-
rized in Table 1. The published data have been used as given
in the original articles without any further manipulation. The
observations from the years 1972 to 1991 were obtained by
Clement with the University of Toronto 61-cm telescope at
the Las Campanas Observatory of the Carnegie Institution of
TABLE 2 2015–2017 V I photometry data of the RRLs in NGC
6171. A full version of this table is available in electronic format
(see Table S1, Supporting Information)
Variable
Star ID Filter
HJD
(days)
Mstd
(mag)
𝝈m
(mag)
V4 V 2,457,200.14416 15.673 0.006
V4 V 2,457,200.14769 15.667 0.006
⋮ ⋮ ⋮ ⋮ ⋮
V4 I 2,457,200.13190 14.797 0.009
V4 I 2,457,200.13824 14.787 0.009
⋮ ⋮ ⋮ ⋮ ⋮
V5 V 2,457,200.14416 15.287 0.004
V5 V 2,457,200.14769 15.286 0.004
⋮ ⋮ ⋮ ⋮ ⋮
V5 I 2,457,200.13190 14.327 0.007
V5 I 2,457,200.13824 14.302 0.007
⋮ ⋮ ⋮ ⋮ ⋮
Washington. A total of 420 photographs were taken on plates
with 103aO emulsion, exposed through a GG385 filter. The
plates were measured on a Cuffey iris photometer. Some of
these data were used, but not published, in the study of the
Fourier parameter 𝜑31 by Clement et al. (1992). We are now
publishing these data in an electronic format, and a small
fraction of the table is included in the printed version of this
article as Table 3. These data were taken in an almost yearly
basis, and the light curves of most variables are well covered,
allowing good estimation of the time of maximum light nearly
every year. Thus, this dataset is crucial for the interpretation
of the O−C diagrams.
2.2 The 2015–2016 data of NGC 6171
In Figure 1, the V light curves tabulated in Table 2 and
obtained in 2015–2016 (black symbols) and 2017 (blue sym-
bols) are displayed. They have provided a few recent occur-
rences of maximum light that extend the time base to 82 years.
They have been phased with the corrected period estimated
below and the seasonal time of maximum. Note the consis-
tency of the 2015–2016 and the 2017 data and the evident
variation in amplitude in some stars, likely due to the Blazhko
effect. The employment of these light curves as indicators
of the mean metallicity and distance of the parental cluster,
via the Fourier decomposition, will be reported elsewhere.
Comments on individual stars can be found in section 4.2.
3 THE O−C APPROACH TO THE SECULAR
PERIOD CHANGES
The observed minus the calculated (O−C) residuals of a
given feature in the light curve, as an indication of miscalcula-
tions or authentic variations of the pulsation or orbital period,
using a single given phase of the light curve as a reference, is
a standard approach that has been in use for many decades, for
160 ARELLANO FERRO ET AL.
TABLE 3 Las Campanas photographic data (B mag) taken between 1972 and 1991 by C. Clement and previously unpublished. This is an extract
from the full table, which is only available with the electronic version of the article (see Supporting Information)
HJD −2,400,000 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12
41,446.819 16.86 16.44 15.56 15.91 16.02 15.33 16.44 15.77 16.84 16.07 16.89
41,447.640 – – – – – – – – – – –
41,447.676 15.84 15.89 15.56 16.29 15.60 16.29 15.25 – 16.86 16.61 16.61
…
HJD −2,400,000 V13 V14 V15 V16 V17 V18 V19 V20 V21 V23 V24
41,446.819 16.79 16.89 15.66 16.84 16.66 15.24 15.91 16.70 16.95 16.15 –
41,447.640 – – 15.77 16.05 15.77 15.99 16.44 – – 15.89 –
41,447.676 16.98 16.73 15.79 16.42 15.93 16.02 15.89 16.02 17.16 15.73 16.29
…
FIGURE 1 Light curves of the 2015–2017 data, except V23 for which data from Clement & Shelton (1997) were plotted as this star is not in the field of the
2015–2017 data. Black symbols are used for the 2015–2016 data from Hanle. Blue symbols are for the 2017 data from SPM. The light curves were phased
with the new periods listed in Table 5 (except for V5) and the local epochs of the maximum light for a given dataset as listed in Table 4, see section 4.2 for a
discussion on individual stars
example, in Cepheids, RRLs, and contact binary stars (e.g.,
Arellano Ferro et al. 2016; Arellano Ferro & Rosenzweig
1997; Coutts & Sawyer Hogg 1969). Then, it is convenient
to select a feature that facilitates the accurate determination
of the phase. For RRLs, the maximum brightness is a good
choice as it is well constrained, particularly for the RRab type,
as opposed to the longer-lasting time of minimum. To predict
the time of maximum, we adopt an ephemeris of the form
C = E0 + P0N, (1)
ARELLANO FERRO ET AL. 161
where E0 is an adopted origin or epoch of reference, P0 is the
period at E0, and N is the number of cycles elapsed between
E0 and C. An initial estimate of the number of cycles between
the observed time of maximum O and the reference E0 is
simply
N =
⌊
O − E0
P0
⌋
, (2)
where the incomplete brackets indicate the rounding down to
the nearest integer. However, it must be noted that if the time
between E0 and the observed time of maximum O is much
larger than the period, and the period change rate is large
enough, the O−C difference can exceed one or more cycles,
and there must be a correction for these extra cycles to obtain
a correct O−C diagram. This exercise may prove to be dif-
ficult if there are large gaps in the time series, but it is rather
straightforward otherwise.
A plot of the number of cycles of N versus O−C is usu-
ally referred as the O−C diagram, and its appearance can
make evident secular variations of the period or the fact that
the period P0 used in the ephemerides is wrong, in which
case the distribution of the O−C residuals is linear and tilted.
For NGC 6171, the distribution of the observations over the
last 80 years enabled us to accurately estimate the number of
cycles required in order to produce coherent O−C diagrams.
Let us assume, as an initial model, a quadratic distribution
of the O−C residuals as a function of time, represented by
the number of cycles N elapsed since the initial epoch E0. The
linear and quadratic cases are then particular solutions of the
representation:
O − C = A0 + A1N + A2N2, (3)
or
O = (E0 + A0) + (P0 + A1)N + A2N2. (4)
Considering the derivative, the period at any given N is
P(N) =
𝑑𝑂
𝑑𝑁
= (P0 + A1) + 2A2N. (5)
From the above equations, it is straightforward to demon-
strate that the period change rate 𝛽 = Ṗ at N = 0 and P=P0
is given by
𝛽 = 𝛽0 =
2A2
P0
, (6)
and that if the O−C distribution is linear, that is, A2 = 0, then
the correct epoch and period are given by E0 +A0 and P0 +A1,
respectively. A detailed derivation of the above equations can
be found in Arellano Ferro et al. (2016).
4 TIMES OF MAXIMUM BRIGHTNESS
AND THE O−C DIAGRAMS
We have estimated as many times of maximum brightness
as possible with the available data. When a light curve is
covered near the maximum, estimating the time of the maxi-
mum brightness is fairly straightforward, and the uncertainty
is small. Error bars would be similar in size to the sym-
bols in the O−C diagrams. Interpolating between competing
maxima was necessary only in a few cases. For the datasets
covering several years (e.g., Coutts & Sawyer Hogg 1971,
for 1946–1970 or data in Table 3 for the years 1972–1991),
we searched for clear times of maximum through the whole
collections and were able to recover numerous maxima, pro-
ducing the highest density of data in the O−C diagrams in
Figure 2.
The complete collection of times of maximum light is given
in Table 4. To calculate the O−C residuals, we proceeded as
follows: first, an epoch with a well-covered light curve was
identified, and the period at that epoch was estimated. This
period and epoch were adopted as initial values P0 and E0.
Generally, the data from 2015 to 2016 were proper for this
aim, except for a few incomplete light curves, particularly near
the time of maximum. In those cases, the data from Clement
& Shelton (1997) or from the previously unpublished data
from Clement (Table 3) were used. For some cases with a
linear distribution of O−C values, we used the period from
the catalogue of variable stars in globular clusters (CVSGC)
(Clement et al. 2001; 2015 edition). While these periods are
quoted to only four digits, and true periods may be slightly
different, the approach to the period correction is not sen-
sitive to the selection of P0 as a different P0 will produce
a different slope A1, but the corrected period P0 +A1 (see
Equation 5) will be the same. The adopted initial ephemerides
are summarized in Table 4. The resulting O−C diagrams are
shown in Figure 2 for every variable included in the present
work.
It is obvious from these diagrams that there are only two
types of O−C distributions: the linear distribution, which
is obtained in the time of maximum predicting ephemerides
(given in column 5 of Table 5), and once the correct period
is employed, it produces a horizontal distribution and implies
a nonchanging period, and the parabolic distribution, which
implies a secular period variation whose rate can be calculated
with Equation 6. It is worth noting at this point that the O−C
approach to the secular period changes is particularly sensi-
tive to the counting of cycles, which is very easy to lose when
dealing with a long time base and short-period stars, as is the
present case. As the O−C difference drifts, either because the
assumed initial period is wrong or because it is authentically
changing, the calculated maximum may skip a few cycles rel-
ative to the corresponding observed one. If this is not properly
considered, the O−C diagram may show intriguing shapes,
which could be misinterpreted as irregular period variations.
This has already been stressed by Arellano Ferro et al. (2016)
for the case of the RRLs in M5. In the present case of NGC
6171, we do not see any irregular variations but only con-
stant periods (linear) or a few parabolic secular variations
(quadratic). It is pointed out, however, that there are cases
where the horizontal O−C distributions display a consider-
able scatter (e.g., V6, V7, V18, and V21). This may be the
consequence of stochastic fluctuations of the period and/or of
162 ARELLANO FERRO ET AL.
FIGURE 2 O−C residuals as a function of number of cycles (filled circles). For stars with linear distributions, the O−C residuals were calculated using the
refined ephemerides (columns 5 and 6 in Table 5). The O−C residuals plotted with a triangle, filled and empty when two solutions were tried, stand out from
the rest of the distribution and were not considered in the adopted solutions. Solid lines represent the finally adopted period change solution for cases with 𝛽
different from zero. Dashed horizontal lines are plotted at (O−C)= 0 as a reference. In the case of V5, we explicitly show the linear distribution showing the
constancy of its period. An apparent abrupt period change in V14 is illustrated by the empty circles and two linear solutions. These and a few cases with
peculiarities are discussed in more detail in section 4.2
uncertainties in the estimations of the times of maximum light
due to the limited quality of the observations.
Table 5 summarizes the initial assumed ephemerides, the
corrected periods and epochs, and—in the corresponding
cases—the calculated period change rate 𝛽.
4.1 Consistency test for the refined periods
and period change rates
Once a period change rate has been calculated (quadratic
case), or the period has been duly corrected (linear case), a
natural test is to phase the light curve with the new ephemeris.
In Figure 1, the light curves from 2015 to 2016 are phased
either with corrected period (linear cases) or with P(N) for
the corresponding N = 0 in Equation 5 (parabolic cases).
In fact, for a secularly changing period, Equation 5 allows
the calculation the period at any given number of cycles
N elapsed from the origin E0. Each corresponding period
should properly phase the data taken at that value of N. As an
example, in Figure 3, we phase the light curve of star V12 over
the last 82 years using the “local” periods and epochs as pre-
dicted by the parabola in Figure 2 and Equation 5 and listed
ARELLANO FERRO ET AL. 163
TABLE 4 Observed times of maximum light, O, for the RRLs in
NGC 6171 and their corresponding O−C residuals calculated with
the given ephemerides for each variable. The sources of either the
times of maximum light or the data employed to calculate them are
coded in column 4 as follows: Oo (Oosterhoff 1938), CouSH
(Coutts & Sawyer Hogg 1971), Man (Mannino 1961), Kuk
(Kukarkin 1961), Di (Dickens 1970), Clem (Las Campanas, this
article), ClSh (Clement & Shelton 1997), and Tab3 (Hanle and
SPM, this article). This is an extract from the full table, which is
available in the electronic version of the article (see Table S2,
Supporting Information)
Variable P0 (days) E0 (HJD)
V2 0.571021 2,442,538.8397
O (HJD) O – C (days) No. of cycles Source
2,427,930.951 −0.0346 −25582 Oo
2,432,004.682 0.0340 −18448 CouSH
2,437,052.486 0.0142 −9608 Kuk
2,437,100.448 0.0104 −9524 Man
2,439,258.334 0.0088 −5745 Di
2,440,693.874 0.0025 −3231 CouSH
2,441,448.747 −0.0140 −1909 Clem
2,442,135.701 0.0020 −706 Clem
2,442,537.670 −0.0276 −2 Clem
2,442,538.841 0.0013 0 Clem
2,442,935.727 0.0279 695 Clem
2,443,273.736 −0.0075 1287 Clem
2,443,632.899 −0.0165 1916 Clem
2,444,020.635 −0.0037 2595 Clem
2,444,371.805 −0.0115 3210 Clem
2,444,759.535 −0.0046 3889 Clem
2,445,083.888 0.0086 4457 Clem
2,445,850.780 0.0197 5800 Clem
2,446,562.828 0.0047 7047 Clem
2,447,298.862 −0.0071 8336 Clem
in Table 6. In all cases, the light curve and time of maximum
are consistent with the secular period change. The appearance
of the curve is only limited by the quality of the photometry
of a given dataset.
4.2 Comments on individual stars
While the O−C diagrams in Figure 2 generally show a clear
period behavior, in a few cases, the O−C distribution may
admit alternative solutions, as we describe in the following
paragraphs. Since the previous study of the period changes in
NGC 6171 by Coutts & Sawyer Hogg (1971), good-quality
data have been obtained, and hence, richer O−C diagrams
can be produced, and a detailed comparison with the results
of these authors is probably inadequate. Some comments on
specific stars might, however, be in order.
V2 and V3. These RRLs are not included in the field of view
of our images, and the historical data are scarce. However, the
data in Table 3 enable the estimation of numerous times of
maximum light and hence the analysis of the secular behavior
of the period.
V5. The O − C residuals in Figure 2 display a clear lin-
ear distribution, leading to a refined period of 0.702376 day.
A negative period change rate and an abrupt negative period
change have been reported for this star by Coutts & Sawyer
Hogg (1971) and Gryzunova (1979a), respectively. Our solu-
tion displays a rather constant period. The estimated max-
imum from 2015 data shows a significant and unexpected
phase displacement and was not considered in the adopted
solution. We noted, however, that the predicted period in
Table 5 (0.702376 day) fails to properly phase the light curves
from 2015 to 2017, for which a shorter period, 0.695248 day,
had to be invoked to better scale the curves. We do not have a
clear explanation for this behavior and speculate that the star
might have undergone a stochastic variation of its period. The
amplitude variation between 2015–2016 and 2017 should be
noted.
V6. Although the O − C distributions of this star suggest
a linear distribution, the scatter is significant and is probably
due to the bump near maximum, which makes the estimation
of the time of maximum brightness inaccurate.
V7. In our data, the light curve of this star displays a large
difference between 2015–2016 and 2017 and suggests a large
amplitude modulation as observed in stars with the Blazhko
effect. Clement & Shelton (1997) noticed the cycle-to-cycle
variations near minimum light and the peculiar harmonics
amplitude rations relative to other RRab stars in the cluster.
Stars with the Blazhko effect often display not only amplitude
but also phase modulations, which likely explains the large
scatter in the O − C diagram for this star.
V10. A period increase was reported by Coutts & Sawyer
Hogg (1971) who calculated 𝛽 = 1.1 day Myr−1. The diagram
in Figure 2 shows a decreasing nature of the period at a rate
𝛽 =−0.714± 0.039 day Myr−1. The discrepancy between the
two investigations is caused by an error in the period due to an
uncertainty in the number of cycle counts in the CSH study.
The richer O − C diagram in our current study has resolved
this ambiguity.
V12. Our analysis of this star shows a decreasing period
with 𝛽 =−0.756± 0.168 day Myr−1. The O − C diagram,
however, displays a significant dispersion. We note that
Clement & Shelton (1997) found the Fourier parameters, par-
ticularly 𝜑3 and 𝜑4, to be peculiar among those in other
RRab stars. A close inspection of Figure 3 shows a distinc-
tive slope change on the rising branch in 1993, which is not
apparent in the other light curves. This suggests that the light
curve shape might undergo secular variations and probably
stochastic oscillations of the time of maximum. The ampli-
tude variation between 2015–2016 and 2017 data should be
noted, confirming the amplitude modulations reported by
Clement & Shelton (1997).
V14. The O − C residuals for this star show a pecu-
liar change in slope at about HJD 2443275.6 days or May
1977 if a period of 0.4816 day is used. Although this slight
change of slope could also be fitted by a parabola, imply-
ing 𝛽 =+0.32± 0.07 day Myr−1, we rather prefer the slight
164 ARELLANO FERRO ET AL.
TABLE 5 New periods and period change rates for RRLs in NGC 6171. The uncertainties in 𝛽0 correspond to the uncertainty in the coefficient A2
in Equation 6
Variable star ID Variable type P0 (days) E0 (+2,400,000) (HJD) E0 +A0 (+2,400,000) (HJD) P0 +A1 (days) 𝜷0 (O−C) (day Myr−1)
V2 RRab 0.5710 42,538.841 42,538.840 0.571021 −0.069 ± 0.062
V3 RRab 0.5663 41,844.561 41,844.570 0.566344 +0.055 ± 0.095
V4 RRc 0.282130 57,528.3538 57,528.3633 0.282132 +0.005 ± 0.008
V5 RRab 0.7024 57,200.193 57,200.3342 0.702376 −0.062 ± 0.058
V6 RRc 0.259635 57,527.2486 57,527.2307 0.259627 +0.058 ± 0.029
V7 RRab 0.4975 44,018.667 44,018.635 0.497474 −0.155 ± 0.117
V8 RRab 0.5599 44,371.599 44,371.588 0.559922 +0.091 ± 0.034
V9 RRc 0.3206 57,200.2279 57,200.2369 0.320601 +0.012 ± 0.014
V10 RRab 0.415506 57,528.4208 57,528.4707 0.4155586 −0.714 ± 0.039
V11 RRab 0.5928 57,528.3744 57,528.3821 0.592809 +0.096 ± 0.022
V12 RRab 0.472833 49,125.719 57,527.331 0.472830 −0.756 ± 0.168
V13 RRab 0.4668 44,371.865 44,371.873 0.466797 +0.070 ± 0.230
V14 RRab 0.4816 43,275.6195 43,275.6356 0.48162 +0.067 ± 0.010
V15 RRc 0.288589 57,200.1676 57,200.0880 0.288590 +0.001 ± 0.066
V16 RRab 0.522798 57,200.1894 57,200.1503 0.522796 −1.088 ± 0.045
V17 RRab 0.561168 41,860.578 41,860.593 0.5611675 +0.624 ± 0.021
V18 RRab 0.561404 57,528.4361 57,528.4361 0.561404 +0.040 ± 0.128
V19 RRc 0.278766 57,528.3812 57,528.3666 0.278762 +0.005 ± 0.009
V20 RRab 0.5781 41,863.748 41,863.765 0.578107 −0.003 ± 0.018
V21 RRc 0.258724 57,201.2263 57,201.203 0.258715 −0.004 ± 0.053
V23 RRc 0.323343 49,477.617 49,477.6189 0.323344 +0.076 ± 0.056
V24 RRab 0.523977 49,123.856 49,123.821 0.523947 −0.261 ± 0.128
period change and hence the two slopes depicted. The two
slopes would imply corrected periods, P1= 0.481617 day and
P1= 0.481624 day, which are identical to the fifth digit. We
failed to observe the star near maximum brightness between
2015 and 2017.
V15. A slightly tilted O − C diagram is found with two
discrepant values in 2015 and 2017, which are otherwise
consistent between them. This may suggest an abrupt period
change, which is to be confirmed in the future if new times of
maximum light become available.
V16. This is a clear and strong period-decreasing star
with 𝛽 =−1.088± 0.045 day Myr−1. Coutts & Sawyer Hogg
(1971) also found a similar result and calculated a period
decrease rate of −1.6 day Myr−1.
V17. The O − C diagram in Figure 2 shows a positive
parabola and a corresponding 𝛽 =+0.748± 0.021. This is the
only star in the sample for which we have found a positive
period increase that, if ascribed to evolution, may indicate a
redward evolution.
V21. In spite of the fact that this is a field star (Cudworth
et al. 1992; Dickens 1970) projected on the cluster field, it
was included in the present study due to the fact that, similar
to the cluster members, photometric coverage was available
for 82 years. Hence, although the scatter in the O − C dia-
gram is substantial, its period has been refined and reported
in Table 5.
V22. This star is not included in the present study. V22 is
not a cluster member (Sawyer Hogg 1973) and out of the field
of most studies, except that of Oosterhoff (1938). Hence, the
historical data are very scarce.
V23. The O − C diagram shows a linear distribution with
a small tilt, which implies a tiny correction to the period. The
1935 light curve from Oosterhoff (1938) is very scattered,
and we could not estimate a reliable time of maximum. We
note that the two oldest maxima, corresponding to data from
Coutts & Sawyer Hogg (1971) from July 1946 and July 1955,
are not fitted by the more recent linear distribution; however,
these are two bona fide maxima from a rather scattered light
curve. We note that, in the article by Dickens (1970), the star
labeled as V1 in fact corresponds to V23.
5 PERIOD CHANGES AND EVOLUTION IN
THE HB
As stars evolve across the IS, their pulsation period should
either increase or decrease if evolution is toward the red or the
blue, respectively. However, other nonevolutionary reasons
for period changes have been suggested, such as stochastic
variations (Balazs-Detre & Detre 1965) or mixing events in
the core of a star at the HB that may alter the hydrostatic
structure and pulsation period (Sweigart & Renzini 1979). In
addition, irregular and complicated secular period variations
have been claimed for some RRLs (e.g., Jurcsik et al. 2001;
Szeidl et al. 2011), which indeed would be difficult to recon-
cile with stellar evolution exclusively. However, for the RRLs
ARELLANO FERRO ET AL. 165
FIGURE 3 A selection of light curves of V12 over the past 82 years, phased with the corresponding period and epoch as predicted by the parabola in
Figure 1 and Equation 5. Note the variations in amplitude. These ephemerides are listed in Table 6
TABLE 6 A selection of epochs and periods of star V12 over
80 years. The phased light curves are shown in Figure 2
E0 P (days) Year
2,427,931.957 0.47289860 1935
2,436,728.475 0.47287822 1959
2,437,050.549 0.47287747 1960
2,440,747.669 0.47286889 1970
2,441,454.580 0.4728665 1972
2,443,281.546 0.4728627 1977
2,446,185.774 0.4728567 1985
2,448,011.631 0.4728529 1990
2,449,125.719 0.47284945 1993
2,449,482.737 0.47284862 1994
2,457,527.331 0.47282997 2015–2016
2,457,939.681 0.47283244 2017
in M5, it has been argued that there is no need to claim for
irregular period variations as an improper counting of cycles,
particularly in long-time baseline sets of times of maximum
light, may be responsible for apparent irregularities (Arellano
Ferro et al. 2016).
At present, there is much evidence that there is no prefer-
ential positive or negative values of 𝛽 in the RRLs in a given
cluster, and for a summary, the reader is referred to the discus-
sion of Arellano Ferro et al. (2016). Perhaps the exceptions are
𝜔Cen, for which an average of 𝛽 =+0.170± 0.561 day Myr−1
can be calculated from Table 6 of Jurcsik et al. (2001) and
IC 4499 with 𝛽 =+0.29± 0.60 day Myr−1 from Kunder et al.
(2011) (their Table 1 without three extreme cases). In addi-
tion, in the extensive investigations on secular period vari-
ations in globular clusters (e.g., Arellano Ferro et al. 2016;
Corwin & Carney 2001; Silbermann & Smith 1995), no sig-
nificant differences have been found for the average values of
𝛽 for the populations of RRab and RRc stars.
Models of the HB calculated by Lee (1991) and Catelan
et al. (2004) confirm that positive period change rates of evo-
lutionary origin occur mostly in globular clusters with blue
HB structures, that is, with large values of the HB structure
parameter = (B−R)/(B+V +R), where B, V, and R are the
number of stars to the blue, inside, and to the red of the
IS, respectively. Figure 15 of Catelan (2009) displays such
behavior of 𝛽 as a function of  and shows that, in red HB
clusters, the average value of 𝛽 should be about zero. NGC
166 ARELLANO FERRO ET AL.
6171 has a very red HB, with =−0.74; hence, the above
models predict an average 𝛽 ∼ 0 day Myr−1. In fact, the overall
average of 𝛽 values in Table 5 is −0.086± 0.346 day Myr−1,
which is not significantly different from zero given the typical
uncertainties of 𝛽.
Even in clusters like M3 and M5, where the overall period
change rates average nearly zero, as predicted by canonical
models, it has been common to isolate individual cases with
significantly large values of 𝛽. In the case of NGC 6171, we
found large secular period change rates in four stars in a sam-
ple of 22. Three of these have large negative values of 𝛽, which
implies evolution to the blue. The only period-increasing case
is V17 with 𝛽 =+0.624± 0.021 day Myr−1. This period rate
is comparable to the rate found in some RRLs in other clus-
ters; in M5 (=+0.31), for example, arguments have been
made for V8 (+0.474 day Myr−1), V7 (+0.474 day Myr−1),
and V25 (+0.933 day Myr−1) or the more moderate V77
(+0.340 day Myr−1), V87 (+0.369 day Myr−1), and V90
(+0.114 day Myr−1) in favor of them being stars in a
truly advanced evolution (Arellano Ferro et al. 2016); in
M3 (=+ 0.18), we have V10 (0.385 day Myr−1), V47
(+0.393 day Myr−1), V69 (+0.414 day Myr−1), and V83
(+0.345 day Myr−1) (Corwin & Carney 2001). Jurcsik et al.
(2001) calculated period change rates across the IS between
−0.026 and +0.745 day Myr−1 based on post-HB evolution-
ary tracks of Dorman (1992) for [Fe/H]=− 1.48 and masses
of 0.60–0.66 M⊙. Thus, all the above quoted positive period
changes may be consistent with an evolutionary origin, as well
as for our present case of V17.
On the other hand, as discussed by Silva-Aguirre et al.
(2008), pre-ZAHB stars crossing the IS at high evolv-
ing rates may have values of 𝛽 ∼−0.3 day Myr−1 but can
reach values <−0.8 day Myr−1. From the calculations of
Jurcsik et al. (2001), based on Dorman’s (1992) post-ZAHB
models, the fastest blueward evolution reaches the rate of
−0.7× 10−10 day day−1 or about −0.026 day Myr−1. Thus, it
is tempting to suggest that variables V10, V12, and V16 in
NGC 6171 are examples of pre-ZAHB stars. However, we
should consider that, according to the statistics produced by
the simulations of Silva-Aguirre et al. (2008) for the case of
M3, only one pre-ZAHB is expected every 60 bona fide HB
stars, and only 22% of them would fall in the IS, where they
can be disguised as RRLs. Assuming that these statistics hold
for NGC 6171, with about 110 stars in the HB, it implies that
not even one pre-ZAHB RR Lyrae-like pulsator should be
found. Nevertheless, RR Lyrae with large negative values of
𝛽 is a rather common feature in several clusters. In M3 itself,
there are five stars with 𝛽 <−0.4 day Myr−1 (Corwin & Car-
ney 2001); in M5, there are five stars with 𝛽 <−0.3 day Myr−1
(Arellano Ferro et al. 2016) and four in NGC 6934 (Stagg
& Wehlau 1980). Thus, it is probably not unlikely that
V10, V12, and V16 in NGC 6171 are indeed pre-ZAHB
stars.
Perhaps the most remarkable result in the present article is
the high percentage of stars with a stable period. It should
be noticed that the four stars with changing period in NGC
6171 are RRab stars and that 18 of the 22 stars studied, that is,
82%, have retained a constant period for at least 80 years. In
M5, 34% of its RRLs were found to have unchanging periods
over a 100-year interval (Arellano Ferro et al. 2016). Nat-
urally, the question of whether this result would be subject
to change after one or two decades of accurate estimation of
times of maximum may be posed. Note that the uncertainties
in the beta values in Table 5 are generally of a few hundreds
of day Myr−1 and that, if the linear distributions in Figure 2
are forcibly fitted with a parabola, the quadratic term leads to
very small and nonsignificant values of 𝛽. We conclude then
that with the data on hand, we are unable to detect period vari-
ations, positive or negative, below these limits. Thus, period
changes for stars evolving very near the HB at very low rates
may pass undetected.
6 SUMMARY
Pulsation period changes have been analyzed via the times
of maximum light for 22 RRLs in NGC 6171. Archival data
collected from the literature, previously unpublished data
spanning 19 years, and recent CCD observations enable a
span of up to 82 years for most of the sample stars, which
makes this work the first significant study of period changes
in NGC 6171. Secular period variations were found for four
stars, three with significant decreasing periods and one (V17)
with increasing period. No signs of irregular period varia-
tions were found in the RRLs of this cluster, but instead,
they all have either a remarkably stable period or a secular
period change that can be represented by a parabolic O−C
diagram.
The overall average of the period change rates found in
NGC 6171 is not significantly different from zero, as expected
from the canonical evolutionary models of the HB for a cluster
with a red HB. Notwithstanding this fact, individual stars with
large positive and negative period changes have been found,
a trend also observed in M3 and M5.
In NGC 6171, we have found a single case with posi-
tive 𝛽 (V17) that seems to be consistent with the period
change rate expected in a truly advanced stage of evolu-
tion toward the AGB. On the contrary, a few cases emerged
with values of 𝛽 significantly negative, which cannot be
reconciled with post-HB evolutionary predictions and may
be examples of pre-core-helium-burning stars on their con-
traction towards the ZAHB. The majority of the RRLs in
this cluster, both RRab and RRc, display a stable period
for at least 82 years, well within the uncertainties of the
O−C approach. Under the paradigm that period changes
are a consequence of stellar evolution, it must be con-
cluded that these stable stars are evolving very slowly and
their putative period changes are, given the data presently
available, undetectable by the approach described in this
work.
ARELLANO FERRO ET AL. 167
ACKNOWLEDGMENTS
We are grateful to Prof. Christine M. Clement for encour-
aging this work and for supplying us with her collection of
relevant historical data of NGC 6171 and her own exten-
sive unpublished observations taken between 1972 and 1991
and allowing us to publish them in the present article. Her
comments and suggestions to the manuscript are gratefully
appreciated. We are also indebted to the anonymous referee
for his or her useful suggestions and enlightening comments.
We acknowledge the financial support from DGAPA-UNAM,
México via grants IN106615-17 and IN105115 and from
CONACyT (México). P.R. is grateful for the financial sup-
port from the PREI program of the National University of
México and the hospitality of the Instituto de Astronomía
(UNAM). P.R. warmly acknowledges the financial support of
the CDCHTA—Universidad de Los Andes (ULA) through
project C-1992-17-05-B. We are thankful to Carlos Chavarría
for his help with some of the observations in SPM. We have
made extensive use of the SIMBAD and ADS services, for
which we are thankful.
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SUPPORTING INFORMATION
Additional supporting information may be found online in the
Supporting Information section at the end of the article.
How to cite this article: Arellano Ferro A, Rosen-
zweig P, Luna A, Essam A, Deras D, Muneer S,
Michel R. Period changes in the RR Lyrae stars of NGC
6171 (M107), Astron. Nachr./AN, 2018;339:158–167.
https://doi.org/10.1002/asna.201813409.
67
Chapter 8
A new study of the variable star population in
the Hercules globular cluster (M13; NGC
6205)
8.1 Overview
This is a very bright cluster (V ≈ 5.8 mag) in the direction of the Hercules constellation (α = 16h 41’ 41.24”,
δ = +36◦27′35.5′′, J2000), and located in the Galactic halo (l = 59.01◦, b = 40.91◦), at a distance of 7.1 kpc
(Harris, 1996) from the Galactic center. Its variable star population is comprised by RRL, SX Phe, W Vir, red
giants and pulsars. A notable feature of the CMD of this cluster is its prominent blue-tailed HB. In this article
we present the results of the analysis performed on the variable star population of the GC NGC 6205 (M13) and
implement the results from Gaia-DR2 to perform a star membership analysis.
8.2 Observations and transformation to the standard system
The data were acquired at two sites: At the IAO in Hanle, India using the 2 m telescope in two separate seasons
during 2014; and at IAC in Tenerife with the 0.80 m telescope in four separate seasons during 2016. A total of
953 images in the V filter and 922 in the I filter were obtained. For the reduction of our data we followed the
method described in Appendix B, with which we were able to recover the light curves of about 16 280 stars in
the V filter and 16 278 stars in the I filter in our FoV. We then transformed the acquired instrumental magnitudes
to the standard photometric system. For that, we made use of 368 standard stars in the FoV of the CCD at the
IAC and of 367 stars in the FoV of the CCD at the IAO. The data of the variable stars present in the cluster is
reported in Table 8.1.
8.3 The physical parameters of RRL stars in NGC 6205
We performed an analysis of the light curves of the RRL stars by using Fourier decomposition to obtain the
amplitudes and phase constants which are related to their physical parameters by the semi-empirical calibrations
mentioned in Chapter 5. From these calibrations we estimated the metallicity of the GC as [Fe/H]ZW=–1.603
± 0.002 from the only RRab and
⟨
[Fe/H]ZW
⟩
=–1.72 ± 0.12 from three RRc. The results of this analysis is
reported in Table 8.2. The light curves of the RRL stars are shown in Fig. 8.1.
Nemec et al. (2013) proposed an alternate set of equations also based in the Fourier coefficients to estimate the
metallicity of GCs. The values we obtained from these formulations are [Fe/H]ZW=–1.57 for the RRab stars
68 Chapter 8. A new study of the variable star population in the Hercules globular cluster (M13; NGC 6205)
Table 8.1: Data of variable stars in M13 in the FoV of our images (Deras et al., 2019).
Star ID Type
⟨
V
⟩ ⟨
I
⟩
AV AI P HJDmax α (J2000.0) δ (J2000.0) Gaia-DR2 Source
(mag) (mag) (mag) (mag) (days) + 2450000
V1 CW 14.04 13.49 1.04 0.71 1.4590 7573.7624 16:41:46.47 36:27:27.59 1328057184175359488
V2 CW 12.99 12.32 0.92 0.66 5.1108 7555.5896 16:41:35.89 36:27:48.27 1328057867076964864
V5 RRc Bl 14.79 14.37 0.53 0.34 0.381784 7555.5102 16:41:46.36 36:27:39.76 1328057179882276480
V6 CW 14.081 13.41 – – 2.1129 7554.6234 16:41:47.96 36:29:09.50 1328059413265281920
V7 RRc 14.90 14.53 0.34 0.26 0.312668 7569.5321 16:41:37.11 36:26:28.71 1328057768297931904
V8 RRab 14.84 14.25 0.85 0.55 0.750303 7568.5076 16:41:32.64 36:28:01.92 1328058077530125568
V9 RRc Bl 14.82 14.37 0.53 0.36 0.392724 7566.5022 16:41:46.25 36:27:37.75 1328057184175404544
V111 SR 11.86 10.34 >0.06 >0.05 92.04 – 16:41:36.63 36:26:35.51 1328057763997754112
V151 SR 12.09 10.67 >0.01 >0.01 30.04 – 16:41:47.02 36:25:57.22 1328057081103004032
V171 SR 11.93 10.46 >0.05 >0.02 43.04 – 16:41:50.91 36:28:54.18 1328058696010941568
V18 L 12.29 10.95 >0.12 >0.05 41.254 – 16:41:24.07 36:25:30.52 1328057523479525760
V191 SR 12.00 10.53 >0.02 >0.002 30.0 – 16:41:31.99 36:28:29.82 1328058077536075264
V20 SR 12.06 10.54 >0.11 >0.06 40.04 – 16:41:23.52 36:30:17.04 1328105150377778048
V241 SR 11.94 10.36 >0.05 >0.03 45.04 – 16:41:41.94 36:26:51.76 1328057149822883328
V25 RRc 14.62 14.23 0.50 0.30 0.429529 7566.5022 16:41:42.70 36:27:30.86 1328057905736994560
V31 RRd 14.39 13.81 0.034 0.018 0.32904 7567.4298 16:41:46.26 36:28:55.11 –
0.048 0.020 0.31936
V32 SR 14.14 13.26 >0.03 >0.02 33.0 – 16:41:23.03 36:28:05.10 1328104669341378432
V331 SR 11.98 10.49 >0.11 >0.07 33.0 – 16:41:50.26 36:24:15.37 1328056703145773696
V34 RRc Bl 14.81 14.34 0.40 0.32 0.389497 7568.4786 16:41:49.08 36:27:07.58 1328057111162818560
V35 RRc 14.81 14.54 0.22 0.18 0.319991 7566.4249 16:41:41.45 36:27:47.15 1328057905737064576
V36 RRd 14.82 14.51 0.047 0.034 0.31596 7566.4299 16:41:43.48 36:26:25.97 1328057149822551424
0.058 0.032 0.30424
V37 SX Phe 17.18 – 0.07 0.06 0.049411 – 16:41:42.45 36:28:21.50 –
V38 SR 12.13 10.66 >0.16 >0.12 32.04 – 16:41:38.67 36:25:37.66 1328057012383752704
V391 SR 11.92 10.40 – – 56.04 – 16:41:42.52 36:26:55.73 1328057145522507648
V401 SR 12.10 10.67 >0.01 >0.01 33.04 – 16:41:49.69 36:27:48.89 1328058657351049344
V41 SR 17.18 11.98 >0.13 >0.09 42.54 – 16:41:45.69 36:27:57.36 1328057935796514560
V421 SR 11.88 10.39 >0.02 >0.03 40.04 – 16:41:35.49 36:27:27.35 1328057871377540224
V43 L 12.46 11.14 >0.08 >0.04 – – 16:41:27.07 36:28:00.11 1328058107595085440
V44 L 12.15 10.72 >0.04 >0.03 – – 16:41:39.16 36:27:21.86 1328057802657985792
V45 L 12.64 11.41 >0.02 >0.03 – – 16:41:41.11 36:27:22.65 1328057905737161344
V46 SX Phe 16.04 15.41 0.15 0.14 0.052186 7574.5437 16:41:41.34 36:27:05.10 1328057905737081856
V47 SX Phe 16.88 16.55 0.29 0.31 0.065256 7554.5293 16:41:38.76 36:26:20.90 1328057802657690880
V50 SX Phe 16.94 16.55 0.33 0.31 0.061754 7568.5076 16:41:32.45 36:28:51.70 1328058146249921024
V542 RRc 14.90 14.59 0.16 0.13 0.295374 7568.4743 16:42:07.65 36:29:36.53 1328058970889601408
V552 SX Phe 17.60 17.40 0.23 0.37 0.040505 7554.5817 16:41:48.47 36:26:13.58 1328057081095055488
V562 SX Phe 17.21 16.88 0.18 0.28 0.024140 7572.5045 16:41:45.63 36:26:54.87 1328057149815076736
V572 W Uma 18.60 17.98 0.52 0.46 0.285416 7553.38383 16:41:21.37 36:28:17.92 1328104669336332544
V582 L 13.77 12.71 >0.07 >0.03 – – 16:41:40.70 36:27:16.16 1328057905737008768
V592 L 12.27 10.89 >0.07 >0.05 – – 16:41:33.92 36:30:05.51 1328058249334814208
V602 ? 14.47 13.61 >0.10 >0.10 0.494497 – 16:41:44.53 36:27:52.58 1328057940089940736
Bl: RRL stars with Blazhko effect.
1 Saturated in our IAC images.
2 New variable found in the present work.
3 Time of minimum.
4 Periods determined by Osborn et al. (2017).
8.4. Membership determination with Gaia 69
0.0 0.5 1.0
14.6
14.8
15.0
V
V5
0.0 0.5 1.0
φ
14.2
14.4
14.6
I
0.0 0.5 1.0
14.6
14.8
15.0
V
V7
0.0 0.5 1.0
φ
14.4
14.5
14.6
I
0.0 0.5 1.0
14.4
14.6
14.8
15.0
V V8
0.0 0.5 1.0
φ
14.1
14.3
14.5
I
0.0 0.5 1.0
14.6
14.8
15.0
V
V9
0.0 0.5 1.0
φ
14.1
14.3
14.5
I
0.0 0.5 1.0
14.4
14.6
14.8
V V25
0.0 0.5 1.0
φ
14.1
14.2
14.3
I
0.0 0.5 1.0
14.3
14.4
14.5
V
V31
0.0 0.5 1.0
φ
13.7
13.8
13.9
I
0.0 0.5 1.0
14.6
14.8
15.0
V
V34
0.0 0.5 1.0
φ
14.2
14.4
14.6
I
0.0 0.5 1.0
14.7
14.8
14.9
V
V35
0.0 0.5 1.0
φ
14.4
14.5
14.6
I
0.0 0.5 1.0
14.7
14.8
14.9
15.0
V
V36
0.0 0.5 1.0
φ
14.4
14.5
14.6
I
0.0 0.5 1.0
14.8
14.9
15.0
V
V54
0.0 0.5 1.0
φ
14.5
14.6
14.7
I
Figure 8.1: RRL in NGC6205 (Deras et al., 2019).
and [Fe/H]ZW=–1.74 for the RRc stars. These values are in good agreement with the ones derived using Eq.
5.4 and Eq. 5.8 derived by Jurcsik et al. (1996) and Morgan et al. (2007), respectively, within their respective
uncertainties.
In order to estimate the distance to NGC 6205 we used the P-L relations from the RRL stars, the SX Phe and the
W Vir, the level of the HB and the contact binary. These distances are reported in Table 8.3.
8.4 Membership determination with Gaia
We carried out an analysis of membership of the stars to NGC 6205 using the high-quality astrometric data by
Gaia-DR2 (Gaia Collaboration et al., 2018). The method used for this determination is based on the paper by
Bustos Fierro et al. (2019) which uses proper motions of the stars and is outlined in Appendix C. For this GC,
70 Chapter 8. A new study of the variable star population in the Hercules globular cluster (M13; NGC 6205)
Table 8.2: Physical parameters obtained from the Fourier fit for the RRab and RRc stars. The
numbers in parentheses indicate the uncertainty on the last decimal place (Deras et al., 2019).
RRab star
Star [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V81 -1.603 ± 0.002 -1.536 ± 0.002 0.378 ± 0.016 3.794 ± 0.008 1.749 ± 0.006 0.68 ± 0.07 6.50 ± 0.05
RRc stars
Star [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V7 -1.53(27) -1.44(30) 0.611(5) 3.866(1) 1.655(2) 0.524(6) 4.180(9)
V25 -1.81(15) -1.81(19) 0.542(6) 3.854(1) 1.683(3) 0.412(3) 4.571(13)
V35 -1.39(48) -1.28(48) 0.586(6) 3.867(1) 1.666(3) 0.517(9) 4.215(12)
V542 - - 0.652(6) 3.883(33) 1.639(2) 0.449(172) 3.804(10)
Weighted Mean -1.72 -1.66 0.586 3.856 1.665 0.45 4.28
σ ± 0.12 ± 0.15 ± 0.003 ± 0.001 ± 0.001 ± 0.01 ± 0.01
1 The uncertainties in the parameters come from the Fourier fit of the light curve, and are small due to its low scatter. 2 Not included in
the averages of the physical parameters.
Table 8.3: Distance comparison to NGC 6205 from the different methods used in this work (Deras
et al., 2019).
Method Distance [kpc]
RRab Fourier decomposition 7.61
RRc Fourier decomposition 6.8 ± 0.3
RRab / RRc I-band P-L 7.0 ± 0.2
SX Phe P-L 7.2 ± 0.7
Type II Cepheids P-L 7.1 ± 0.6
Level of the ZAHB 7.1 ± 0.1
Eclipsing Binary V57 6.9 ± 0.2
Weighted mean 7.1 ± 0.1
1 Based only on one RRab. Not included in the average.
Gaia detected 52 802 stars in a region of 40’ radius around the center of the cluster, for which 23 070 stars seem
to be members. The result of this determination is shown in Fig. 8.2.
8.5 The color-magnitude diagram of NGC 6205 and its HB
We used the likely member sources to do a cross-match with our own data, and in doing so, we found 7630
stars in common. The original and cleaned CMD of IAC data are shown in panels 1 and 2 of Fig. 8.3 and the
cleaned CMD from the Hanle data in panel 3. On the cleaned CMD, we overlaid a pair of isochrones built with the
theoretical models of VandenBerg et al. (2014) (light blue) and Marigo et al. (2017) (green) using a metallicity of
⟨
[Fe/H]
⟩
RR
= –1.58 in both cases. The isochrones and the ZAHB have been reddened by E(B −V) = 0.02 and
shifted to a distance of 7.1 kpc and are consistent with an age of 12.6 Gyrs. The Oosterhoff type of the cluster can
be assessed by different indicators, e.g. the only known RRab star (V8) whose period is 0.750303 d, indicates
that this GC is an OoII-type. The main feature of the CMD of this cluster is its extreme blue HB. The RRL
population on the HB is dominated by RRc-type stars. The value of the Lee index (Lee, 1990) which describes
the morphology of the HB is L = 0.95. The distribution of RRL stars on the HB shows a clear segregation around
the FORE, which seems to be common in all OoII-type clusters. Two double-mode or RRd-type stars are also
present. We also discovered seven new variables: one RRc, two SX Phe stars, three SR and one contact binary.
8.5. The color-magnitude diagram of NGC 6205 and its HB 71
Figure 8.2: Left panel: cluster field of NGC 6205; central panel: VPD of proper motions of stars
in NGC 6205; right panel: CMD of NGC 6205 (Gaia photometric system). The red dots corre-
spond to member stars and the grey dots to field stars according to the astrometric membership
determination (Deras et al., 2019).
−0.4 0.0 0.4 0.8 1.2 1.6
V − I
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
V
M13 (NGC 6205)
−0.4 0.0 0.4 0.8 1.2 1.6
V − I
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
V
−0.4 0.0 0.4 0.8 1.2 1.6
V − I
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
V
V1
V2
V6
V46
V47 V50
V55
V56
V8
V54
V31
V36
V32
V41
V58
V59
V57
V60
Figure 8.3: Color-Magnitude Diagrams of M13. The left panel shows the CMD with all the stars
in the FoV of the IAC images. Black and blue dots represent the member and non-member stars
respectively. The central panel shows only those stars with a high probability of being members of
the cluster. The right panel shows the member stars as measured from Hanle data exclusively. The
color markers are the same in all three panels. The blue circle corresponds to the only known RRab
star in the cluster. Green and magenta circles correspond to RRc and SX Phe stars respectively.
Red circles represent semi-regular variables. Brown circles correspond to RRd stars and the teal
circles to Type II Cepheid stars. The triangle markers correspond to newly discovered RRc (V54),
SX Phe (V55 and V56), W Uma (V57) and SR (V58, V59 and V60) stars. The blue squares
near the top of the RGB are 70 stars whose membership has been determined by means of radial
velocity from Gaia (Deras et al., 2019).
72 Chapter 8. A new study of the variable star population in the Hercules globular cluster (M13; NGC 6205)
8.6 Contributions made by the student to this article
• I cleaned, reduced and analyzed over 1800 science images of the GC.
• I obtained the instrumental magnitudes and transformed them to the standard system to obtain the light
curves of the star present in the GC.
• I conducted a search for variable stars.
• I estimated the Fourier coefficients and the physical parameters related to them.
• I estimated the distance to the cluster using different types of variable stars.
• I made Figures. 1, 3, 4, 6, 7, 8, part of 9, 12, 13, 14, 15, 18 and 19.
• I made Tables 1, 2, 3, 5 and 7.
• I wrote Sections 1, 2, 2.1, 2.2, 3, 3.1, 3.2, 3.2.1, 3.2.2, 3.2.3, 3.3, 3.4, 3.6, 4, 5, 6, 6.1, 6.2, 6.3, 6.4, 6.5 and
7.
• I co-wrote Sections 4.1, 5, 8, 8.1 and 9.
MNRAS 486, 2791–2808 (2019) doi:10.1093/mnras/stz642
Advance Access publication 2019 March 6
A new study of the variable star population in the Hercules globular
cluster (M13; NGC 6205)⋆
D. Deras,1† A. Arellano Ferro,1† C. Lázaro,2,3 I. H. Bustos Fierro,4 J. H. Calderón,4,5
S. Muneer6 and Sunetra Giridhar6
1Instituto de Astronomı́a, Universidad Nacional Autónoma de México, Ciudad Universitaria, C.P. 04510, México
2Departamento de Astrofı́sica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
3Instituto de Astrofı́sica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain
4Observatorio Astronómico, Universidad Nacional de Córdoba, Córdoba, Argentina
5Consejo Nacional de Investigaciones Cientı́ficas y Técnicas (CONICET), Córdoba, Argentina
6Indian Institute of Astrophysics, Koramangala 560034, Bangalore, India
Accepted 2019 March 1. Received 2019 February 28; in original form 2019 February 12
ABSTRACT
We present the results from VI CCD time-series photometry of the globular cluster M13 (NGC
6205). From the Fourier decomposition of the light curves of RRab and RRc stars we found an
average metallicity of [Fe/H]zw = −1.58 ± 0.09. The distance to the cluster was estimated
as 7.1 ± 0.1 kpc from independent methods related to the variable star families RR Lyrae, SX
Phe and W Virginis, from the luminosity of the theoretical ZAHB and from the orbit solution
of a newly discovered contact binary star. The RR Lyrae pulsation modes are segregated by the
red edge of the first overtone instability strip in this OoII-type cluster. A membership analysis
of 52 800 stars in the field of the cluster is presented based on Gaia-DR2 proper motions which
enabled the recognition of 23 070 likely cluster members, for 7630 of which we possess VI
photometry. The identification of member stars allowed the construction of a clean CMD and
a proper ZAHB and isochrone fitting, consistent with a reddening, age and distance of 0.02
mag, 12.6 Gyr and 7.1 kpc, respectively. We report seven new variables: one RRc, two SX
Phe stars, three SR and one contact binary. V31 presents double-mode nature and we confirm
V36 as RRd. 15 variable star candidates are also reported. The analysis of 18 stars in the field
of the cluster, reported as RR Lyrae from the Gaia-DR2 data base, reveals that at least seven
are not variable. We noted the presence of a high-velocity star in the field of the cluster.
Key words: stars: fundamental parameters – stars: variables: RR Lyrae – globular clusters:
individual: M13.
1 IN T RO D U C T I O N
M13 (NGC 6205) is a very bright globular cluster (V ≈ 5.8 mag) in
the constellation of Hercules (α = 16h41′41.′′24, δ = +36◦27′35.′′5,
J2000), and located in the halo (l = 59.◦01, b = 40.◦91) of the Galaxy.
The variable star population of M13 is not particularly rich. It
contains RR Lyrae stars (nine), SX Phe (four), CW stars (three),
variable red giants (16) and pulsars (five) that have been reported in
the literature. The Catalogue of Variable Stars in Globular Clusters
(CVSGC; Clement et al. 2001; 2015 edition) lists 53 variable stars
⋆ Based on observations made with the telescope IAC80, operated by the
Instituto de Astrofı́sica de Canarias in the Spanish Observatorio del Teide
on the island of Tenerife, and with the 2.0 m telescope at the Indian
Astrophysical Observatory, Hanle, India.
†E-mail: dderas@astro.unam.mx (DD); armando@astro.unam.mx (AAF)
although only 45 have been confirmed as variables. In the Catalogue
of Parameters for Milky Way Globular Clusters compiled by Harris
(1996) (2010 edition), the distance from the Sun and the reddening
for the cluster are given as 7.1 kpc and E(B − V) = 0.02, respectively.
One of the main characteristics of the colour–magnitude diagram
(CMD) of M13 is its well-known extreme blue horizontal branch
(HB).
Previous recent CCD photometric studies of M13 have been suc-
cessful in identifying new variables, studying individual cases and
filtering out some non-variables from the variable star designations
(Kopacki, Kołaczkowski & Pigulski 2003; Kopacki 2005) . With the
aim of determining the mean metallicity and distance to the cluster
from specific calibrations valid for different families of variable
stars, such as Fourier light curve decomposition, luminosity of
the HB and period–luminosity (PL) relations, the analysis of an
extensive new time-series of CCD images is undertaken in the
C© 2019 The Author(s)
Published by Oxford University Press on behalf of the Royal Astronomical Society
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2792 D. Deras et al.
present work. Given these new data, a systematic search for new
variables by several approaches may help updating the variability
census of M13. We aim to discuss the membership of the stars
in the field of M13, particularly of the variable star population,
using the recent proper motions and radial velocities available in
the Gaia-DR2.
The paper is structured in the following way. In Section 2,
we describe our observations, the data reduction process and the
transformation to the standard photometric system. In Section 3, we
discuss the population of variable stars present in M13, as well as the
strategies used to find new variables and their utility in estimating
stellar and mean cluster physical parameters. In Section 4, we
explain our methodology to estimate the values of the physical
parameters of the RR Lyrae stars. In Section 5, we mention different
values for the metallicity of the cluster found in the literature and
compare them with our own results. In Section 6, we determine the
distance to M13 using seven independent methods. In Section 7, we
discuss the strategy used to determine the membership of the stars
in the cluster. In Section 8 we discuss the overall properties of the
CMD of M13 as well as the structure of its HB, and in Section 9
we summarize our conclusions. Finally, in Appendix A we discuss
the presence of a high-velocity star in the field of the cluster.
2 DATA , O B S E RVAT I O N S A N D R E D U C T I O N S
The observations used for the present work were performed at two
different locations. The first set of data was obtained using the 2 m
telescope at the Indian Astronomical Observatory (IAO) in Hanle,
India, on six nights separated into two seasons. The first season
spans the nights of 2014 June 7–9, and the second season spans the
nights of 2014 August 3–5. The detector used was a SITe ST-002
thinned backside illuminated CCD of 2048 × 2048 pixels with a
scale of 0.296 arcsec pixel−1, translating to a field of view (FoV)
of approximately 10.1 × 10.1 arcmin2. The second set of data was
obtained with the 0.80 m telescope at the Instituto de Astrofı́sica
de Canarias (IAC) during 12 nights divided in four seasons. The
first season spans the nights of 2016 June 13–15, the second season
the nights of 2016 June 26–29, the third season the nights of 2016
July 2–4 and the fourth season the nights of 2016 July 16 and 18.
We used the CAMELOT camera with 2048 × 2048 pixels and
0.304 arcsec pixel−1, with a 10.4 × 10.4 arcmin2 FoV, and a back
illuminated detector CCD42-40 from E2V Technologies.
Table 1 summarizes the observation dates, exposure times and
average seeing conditions.
2.1 Difference image analysis
For the reduction of our data, we employed the software Difference
Imaging Analysis (DIA) with its pipeline implementation DanDIA1
(Bramich 2008; Bramich et al. 2013). With this, we were able to
obtain high-precision photometry for all the point sources in the
FoV of the two CCDs at the two locations. First, a reference image
is created by DanDIA by stacking the best images in each filter and
then it subtracts it from the rest of the images. The differential flux
for each star is then determined by means of the PSF calculated by
DanDIA from about 300–400 isolated stars in the FoV. Differential
fluxes are converted into total fluxes. The total flux ftot(t) in ADU/s
1DanDIA is built from the DanIDL library of IDL routines available at
http://www.danidl.co.uk.
Table 1. The distribution of observations of M13 for each filter. Columns NV
and NI give the number of images taken with the V and I filters, respectively.
Columns tV and tI provide the exposure time, or range of exposure times,
employed during each night for each filter. The average seeing is listed in
the last column.
Date NV tV (s) NI tI (s)
Avg. seeing
(arcmin)
20140607 16 40 17 7 1.6
20140608 24 40 24 7 1.7
20140609 2 40 4 7 1.7
20140803 16 30 18 7 1.6
20140804 24 30 26 7 1.7
20140805 20 30 22 7 1.6
20160613 6 600 6 400 1.5
20160614 96 100–300 94 30–200 1.7
20160615 86 80–100 67 20–30 1.7
20160626 25 100 26 50 2.6
20160627 94 60–80 93 15–30 1.2
20160628 94 70–80 94 20–30 1.7
20160629 56 150–200 33 60–80 2.4
20160702 79 80–100 80 15–20 1.3
20160703 113 40–80 116 10–20 1.2
20160704 61 80–100 58 20–40 1.8
20160716 41 120 45 80–100 1.8
20160718 100 100 99 20–30 1.7
Total: 953 – 922 – –
at each epoch t can be estimated as:
ftot(t) = fref +
fdiff(t)
p(t)
, (1)
where fref is the reference flux (ADU/s), fdiff(t) is the differential
flux (ADU/s) and p(t) is the photometric scale factor (the integral
of the kernel solution). Conversion to instrumental magnitudes was
achieved using:
mins(t) = 25.0 − 2.5 log
[
ftot(t)
]
, (2)
where mins(t) is the instrumental magnitude of the star at time t.
The above procedure has been described in detail in Bramich et al.
(2011).
2.2 Photometric calibrations
2.2.1 Relative calibration
To correct for possible systematic errors, we applied the method-
ology developed in Bramich & Freudling (2012) to solve for the
magnitude offsets Zk that should be applied to each photometric
measurement from the image k. In terms of DIA, this translates into
a correction for the systematic error introduced into the photometry
due to a possible error in the flux–magnitude conversion factor
(Bramich et al. 2015). In the present case the corrections were very
small, ∼ 0.001 mag for stars brighter than ∼V = 18.0.
2.2.2 Absolute calibration
Standard stars in the field of M13 are included in the online
collection of Stetson (2000)2 and we used them to transform in-
strumental vi magnitudes into the Johnson–Kron–Cousins standard
2http://www3.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/STETSON/st
andards
MNRAS 486, 2791–2808 (2019)
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M13 2793
Figure 1. Transformation relations obtained for the V and I filters between
the instrumental and the standard photometric systems. To carry out the
transformation, we made use of a set of standard stars (N = 368 for IAC
and N = 367 for Hanle) in the field of M13. The top panel corresponds to
the data from IAC and the bottom panel to the data from Hanle.
VI system. The mild colour dependence of the standard minus
instrumental magnitudes is shown in Fig. 1 for both the IAC and
Hanle observations. The transformation equations are explicitly
given in the figure itself.
3 VARIABLE STARS IN M13
All the previously known and newly discovered variable stars in
M13 are listed in Table 2 and have been identified in Fig. 2.
3.1 The search for new variable stars
Due to the high quality of our data, we were able to recover
16 280 light curves in V and 16 278 in I of the stars present in
our FoV. We carried out a search for new variable stars to enlarge
the variable star counts using our data and thereby refine the cluster
parameter estimates based on different approaches. We used three
search methods which will be briefly described below:
(i) We split the CMD of M13 into regions where it is common
to find variable stars, e.g. at the Instability Strip (IS) in the HB, the
Blue Straggler region (BS) and at the tip of the Red Giant Branch
(TRGB). We analysed the light curves of the stars in those regions
and looked for variability by determining their period (if any) and
plotting their apparent magnitudes with respect to their phase. We
discovered three new variables with this method: one RRc (V54)
and two SX Phe stars (V55 and V56).
(ii) Another approach used was via the string-length method
(Burke, Rolland & Boy 1970; Dworetsky 1983). Each light curve
was phased with periods between 0.02 and 1.7 d, in steps of 10−6
d, and the string-length parameter SQ was calculated in each case.
The best phasing is obtained with the true period and it produces
a minimum SQ. A plot of the minimum SQ for each star in our
collection of light curves (identified by the X-coordinate in the
reference image) is shown in Fig. 3, where all variables in Table 2
are identified. We note that most of the variables are located below an
arbitrary threshold at 0.4, hence we individually explored each light
curve below this value. Using this method, we identified a contact
binary (V57) and two long period giant stars (V58 and V59).
(iii) The third method consists in the detection of variations of
PSF-like peaks in stacked residual images from which we can see
the variable stars blink. All previous known variables were detected
and we noted 15 candidates that need confirmation.
3.2 The RR Lyrae stars
3.2.1 RRab and RRc stars
With the newly found RRc star (V54), the RR Lyrae population of
M13 consists of one RRab (V8), seven RRc, and two RRd (V31
and V36). Their VI light curves are shown in Fig. 4.
3.2.2 Multi-frequency RR Lyrae stars
The star V36 is a well-known multi-frequency variable. Kopacki
et al. (2003) identified three frequencies, very close to each other
and concluded that the star belongs to a group of RR Lyrae with non-
radial modes and period ratios larger than 0.95 (Olech et al. 1999).
We have attempted to reproduce the observed V and I light curves
of V36 as double-mode oscillations of two close frequencies, with
the adopted frequencies determined by minimization of the squared
residuals between the model and the observations. We found the
two frequencies and amplitudes reported in Table 2.
These periods correspond very closely to the first two frequencies
found by Kopacki et al. (2003). Fig. 5 shows the V-band model with
P1 = 0.31596 d and P2 = 0.30424 d. With the precision of our
photometry we were unable to identify a third period P3. Similar
periods and amplitudes were found from the analysis of the I-band
data.
The star V31 is also a double-mode star that apparently went
unnoticed by Kopacki et al. (2003). The VI light curves of V31
in Fig. 4 suggest the presence of more than one period. Similar to
V36, we identified in V31 two periods: P1 = 0.32904 d and P2 =
0.31936 d. Both V31 and V36 have a large period ratio P2/P1 ∼
0.97, implying that at least one of the modes is non-radial. The
two-frequency model fitting for V31 is shown in Fig. 5.
3.2.3 Bailey diagram and Oosterhoff type
The period–amplitude plane for RR Lyrae stars, also known as the
Bailey diagram, is shown in Fig. 6 for the VI band passes. The
periods and amplitudes are listed in Table 2. In most cases, we took
the amplitudes corresponding to the best fit provided by the Fourier
decomposition of the light curves. In cases where the light curve
showed Blazhko effect (V5, V9 and V34), the maximum amplitude
was measured and the star was plotted with a triangular marker.
The continuous and dashed black lines in the top panel of Fig. 6
are the loci for unevolved and evolved stars according to Cacciari,
Corwin & Carney (2005). The black parabola was obtained by
Kunder et al. (2013a) for RRc stars in 14 OoII clusters. In the
MNRAS 486, 2791–2808 (2019)
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Table 2. Data of variable stars in M13 in the FoV of our images.
Star ID Type 〈V〉 〈I〉 AV AI P HJDmax α (J2000.0) δ (J2000.0) Gaia-DR2 Source
(mag) (mag) (mag) (mag) (d) + 2450000
V1 CW 14.04 13.49 1.04 0.71 1.4590 7573.7624 16:41:46.47 36:27:27.59 1328057184175359488
V2 CW 12.99 12.32 0.92 0.66 5.1108 7555.5896 16:41:35.89 36:27:48.27 1328057867076964864
V5 RRc Bl 14.79 14.37 0.53 0.34 0.381784 7555.5102 16:41:46.36 36:27:39.76 1328057179882276480
V6 CW 14.081 13.41 – – 2.1129 7554.6234 16:41:47.96 36:29:09.50 1328059413265281920
V7 RRc 14.90 14.53 0.34 0.26 0.312668 7569.5321 16:41:37.11 36:26:28.71 1328057768297931904
V8 RRab 14.84 14.25 0.85 0.55 0.750303 7568.5076 16:41:32.64 36:28:01.92 1328058077530125568
V9 RRc Bl 14.82 14.37 0.53 0.36 0.392724 7566.5022 16:41:46.25 36:27:37.75 1328057184175404544
V111 SR 11.86 10.34 >0.06 >0.05 92.04 – 16:41:36.63 36:26:35.51 1328057763997754112
V151 SR 12.09 10.67 >0.01 >0.01 30.04 – 16:41:47.02 36:25:57.22 1328057081103004032
V171 SR 11.93 10.46 >0.05 >0.02 43.04 – 16:41:50.91 36:28:54.18 1328058696010941568
V18 L 12.29 10.95 >0.12 >0.05 41.254 – 16:41:24.07 36:25:30.52 1328057523479525760
V191 SR 12.00 10.53 >0.02 >0.002 30.0 – 16:41:31.99 36:28:29.82 1328058077536075264
V20 SR 12.06 10.54 >0.11 >0.06 40.04 – 16:41:23.52 36:30:17.04 1328105150377778048
V241 SR 11.94 10.36 >0.05 >0.03 45.04 – 16:41:41.94 36:26:51.76 1328057149822883328
V25 RRc 14.62 14.23 0.50 0.30 0.429529 7566.5022 16:41:42.70 36:27:30.86 1328057905736994560
V31 RRd 14.39 13.81 0.034 0.018 0.32904 7567.4298 16:41:46.26 36:28:55.11 –
0.048 0.020 0.31936
V32 SR 14.14 13.26 >0.03 >0.02 33.0 – 16:41:23.03 36:28:05.10 1328104669341378432
V331 SR 11.98 10.49 >0.11 >0.07 33.0 – 16:41:50.26 36:24:15.37 1328056703145773696
V34 RRc Bl 14.81 14.34 0.40 0.32 0.389497 7568.4786 16:41:49.08 36:27:07.58 1328057111162818560
V35 RRc 14.81 14.54 0.22 0.18 0.319991 7566.4249 16:41:41.45 36:27:47.15 1328057905737064576
V36 RRd 14.82 14.51 0.047 0.034 0.31596 7566.4299 16:41:43.48 36:26:25.97 1328057149822551424
0.058 0.032 0.30424
V37 SX Phe 17.18 – 0.07 0.06 0.049411 – 16:41:42.45 36:28:21.50 –
V38 SR 12.13 10.66 >0.16 >0.12 32.04 – 16:41:38.67 36:25:37.66 1328057012383752704
V391 SR 11.92 10.40 – – 56.04 – 16:41:42.52 36:26:55.73 1328057145522507648
V401 SR 12.10 10.67 >0.01 >0.01 33.04 – 16:41:49.69 36:27:48.89 1328058657351049344
V41 SR 17.18 11.98 >0.13 >0.09 42.54 – 16:41:45.69 36:27:57.36 1328057935796514560
V421 SR 11.88 10.39 >0.02 >0.03 40.04 – 16:41:35.49 36:27:27.35 1328057871377540224
V43 L 12.46 11.14 >0.08 >0.04 – – 16:41:27.07 36:28:00.11 1328058107595085440
V44 L 12.15 10.72 >0.04 >0.03 – – 16:41:39.16 36:27:21.86 1328057802657985792
V45 L 12.64 11.41 >0.02 >0.03 – – 16:41:41.11 36:27:22.65 1328057905737161344
V46 SX Phe 16.04 15.41 0.15 0.14 0.052186 7574.5437 16:41:41.34 36:27:05.10 1328057905737081856
V47 SX Phe 16.88 16.55 0.29 0.31 0.065256 7554.5293 16:41:38.76 36:26:20.90 1328057802657690880
V50 SX Phe 16.94 16.55 0.33 0.31 0.061754 7568.5076 16:41:32.45 36:28:51.70 1328058146249921024
V542 RRc 14.90 14.59 0.16 0.13 0.295374 7568.4743 16:42:07.65 36:29:36.53 1328058970889601408
V552 SX Phe 17.60 17.40 0.23 0.37 0.040505 7554.5817 16:41:48.47 36:26:13.58 1328057081095055488
V562 SX Phe 17.21 16.88 0.18 0.28 0.024140 7572.5045 16:41:45.63 36:26:54.87 1328057149815076736
V572 W Uma 18.60 17.98 0.52 0.46 0.285416 7553.38383 16:41:21.37 36:28:17.92 1328104669336332544
V582 L 13.77 12.71 >0.07 >0.03 – – 16:41:40.70 36:27:16.16 1328057905737008768
V592 L 12.27 10.89 >0.07 >0.05 – – 16:41:33.92 36:30:05.51 1328058249334814208
V602 ? 14.47 13.61 >0.10 >0.10 0.494497 – 16:41:44.53 36:27:52.58 1328057940089940736
Note. Bl: RR Lyrae with Blazhko effect. 1Saturated in our IAC images. 2New variable found in the present work. 3Time of minimum. 4Periods determined by
Osborn et al. (2017).
bottom panel, the black dashed locus was found by Arellano Ferro
et al. (2011, 2013) for the OoII clusters NGC 5024 and NGC 6333,
respectively. The black parabola was obtained in the present work
using a least-squares fit with the RRc stars. The blue solid and
segmented loci for unevolved and evolved stars, respectively, are
from Kunder et al. (2013b). The positions of the RRab and RRc stars
in this diagram are consistent with the definition of an OoII-type
globular cluster.
3.3 The W Virginis or CW stars
Three W Virginis stars are known in M13: V1, V2 and V6. Their
light curves are shown in Fig. 7 phased with the periods reported in
Table 2.
3.4 The SX Phe stars
Four SX Phe stars are known in M13; V37, V46, V47 and V50.
Their frequency spectra have been studied in detail by Kopacki
(2005). Further comments are needed for these stars. V37 is located
very near to two brighter stars and they are not properly separated
in our images (see the identification chart in Fig. 2(b) near the
top). As a consequence we could not retrieve the light curve of
the star. We have adopted the V and I values of Kopacki (2005)
to include the star into the discussions below. V46 was reported
by Kopacki (2005) as a V = 17.2 magnitude star with at least two
excited frequencies, and probably three. The coordinates given by
Kopacki (2005) point to the star marked in Fig. 2 and we found the
same period calculated by these authors. The light curve is shown in
MNRAS 486, 2791–2808 (2019)
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M13 2795
Figure 2. Identification chart of all known variables in M13 listed in Table 2 and the candidate stars listed in Table 3. The left panel is a field of 10.4 × 10.4
arcmin2. The right panel is 2.0 × 2.0 arcmin2. North is up and East is to the left.
Figure 3. Minimum value for the string-length parameter SQ calculated for
the 16 280 stars with a light curve in our V reference image, versus CCD
X-pixel coordinate. The colour code is as follows: blue circle corresponds
to a RRab star, green circles to RRc stars, brown circles to RRd stars, teal
circles to CW stars, red circles to semiregular variables and magenta circles
to SX Phe stars. The triangles correspond to newly discovered variables
in this work (one RRc, two SX Phe, three semiregular and one contact
binary). Red squares correspond to semiregular stars that were measured at
Hanle but are saturated in the IAC data and have been assigned an arbitrary
pixel coordinate. The dashed blue line is an arbitrary threshold set at 0.4,
below which most of the known variables are located. See Section 3.1 for a
discussion.
Fig. 8. We note that the mean V magnitude is 16.04, i.e. more than
one magnitude brighter than the one reported by Kopacki (2005).
Also, we were unable to find any secondary periodicity in V46,
hence we do not confirm its multi-frequency nature. Likewise, for
V47 we find a mean V = 16.88, versus 17.12 of Kopacki (2005). For
V50, the mean V magnitudes in both studies match within 0.01 mag.
We stress, however, that the identifications and periods found by us
do coincide with those of Kopacki (2005) within a few millionths
of a day, which rules out possible misidentifications.
In the present paper we have identified two more variables that
according to their period, light curve shape and position on the
CMD are classified as SX Phe stars: V55 and V56. The light curves
of the SX Phe stars are shown in Fig. 8. Their position in the CMD
and their use as distance indicators will be discussed in Section 6.3.
3.5 The new contact binary V57
We show in Fig. 9 the VI light curves of the new binary discovered in
our data. From the time of the minima in our light curves we derived
the ephemeris for the primary minimum: HJD = 2457553.38380
+ 0.285416 d E.
The morphology of the light curves of this short-period binary
is similar to the light curves of contact systems. In order to derive
some information on the physical parameters of the system, we have
modelled the VI light curves with the code BINAROCHE, some details
of which are described in Lázaro, Arévalo & Almenara (2009), with
the two light curves used simultaneously in the fit. For this work,
the code uses surface fluxes from the library of theoretical stellar
spectra of Lejeune, Cuisinier & Buser (1998) for [Fe/H] = −1.50.
As we do not have radial velocity curves, the masses of the stellar
components, which are essential parameters in the model, must be
estimated by some indirect reasoning. All that we have is the colour
(V − I) of the binary, which is a rather scarce information, but even
then we can reproduce the observed light curves with a reasonable
model of the system.
The observed V magnitude and colour in the maxima of the light
curves are:
Vbin,max = 18.40(V − I )bin,max = 0.60.
From the observed (V − I) colour out of eclipse, and assumed
E(B − V) = 0.02, we derive the intrinsic colour of the system out
of eclipse (V − I)bin, o = 0.57.
MNRAS 486, 2791–2808 (2019)
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Figure 4. Light curves in V and I filters of RR Lyrae stars in M13. The light curves are a combination of the data obtained at IAC (blue symbols) and at Hanle
(black symbols). The continuous red line represents the Fourier fit. Note that the vertical scale is not the same for all plots. Note the double-mode nature of
V31 and V36.
Figure 5. V31 and V36 data fitted with a two-frequency model.
MNRAS 486, 2791–2808 (2019)
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Figure 6. Bailey diagram for M13. Filled and open circles represent RRab
and RRc stars, respectively. Triangles correspond to stars with Blazhko
modulations. For a more detailed discussion, see Section 3.2.3.
From this combined colour of the binary, we attempted to derive
the intrinsic colours (V − I)1, 2 of the binary components. This can
be done if the relative contribution of the stars to the total light in V
and I are known, which can be derived from the model fit, and a (V −
I) − Teff calibration. For that purpose we have adopted the relation
of Casagrande et al. (2010), valid for the ranges in metallicity and
colour suitable to our data.
In order to calculate a model of the system, it is necessary to give
values to the masses of the components. This has been performed
including the relation between M/M⊙ and log Teff from the Padova’s
isochrone adopted for the cluster, and the relation (V − I) − Teff in
the simultaneous fit of the V, I light curves.
After some trials, the filling factors of the stars have been fixed to a
contact solution, while the parameters M1, q = M2/M1, Teff, 1, Teff, 2,
i (inclination angle), Ab, 1, Ab, 2 (bolometric albedo coefficients), β1,
β2 (gravity-darkening coefficients) are declared free and optimized.
From the best fit, the main derived parameters are:
M1 = 0.87 ± 0.02M⊙, q = M2/M1 = 0.72 ± 0.02
Teff,1 = 6400 ± 100 K, Teff,2 = 6285 ± 100 K
Requiv.,1(R⊙) ≃ 0.85, Requiv,2(R⊙) ≃ 0.73
i = 79◦ ± 2◦
Ab,1 = 0.5, Ab,2 = 0.5, β1 = 0.08, β2 = 0.08.
This solution suggests that V57 can be a new W UMa-type binary.
The adopted model gives a distance to the system d = 6.95 kpc
(for E(B − V) = 0.02 mag). This value is in good agreement with
those derived by the other methods, and gives some confidence on
the adopted model for the system.
From the observed V magnitude and the distance of the model,
the absolute V magnitude of the system is MV = +4.13, in good
agreement with that expected from the PL–colour relation of
Rucinski (1995) for W UMa systems in globular clusters:
MV = −4.43 log P + 3.63(V − Ic) − 0.31(σ = 0.29) = +4.17.
Figure 7. Light curves in VI filters of the three CW stars known in M13. The light curves are a combination of the data obtained at IAC (blue dots) and at
Hanle (black dots). Note that the scale is not the same for all plots.
MNRAS 486, 2791–2808 (2019)
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Figure 8. Light curves in V filter of the SX Phe in M13. The light curves are a combination of the data obtained at IAC (blue dots) and at Hanle (black dots).
Note that the vertical scale is not the same for all plots.
In Fig. 9 the observed and model light curves are shown. In
Fig. 10 we show the location of both stellar components in the
log Teff − log L diagram, together with the theoretical isochrone
values. In this diagram, the green hexagon and red square corre-
spond to the primary and secondary components, respectively. The
derived values of Teff (mean surface temperature) and Requiv (equiv-
alent volume radius) have been used to calculate the luminosity of
the stellar components shown in the diagram.
As the model relies on very limited data, we present this solution
as a preliminary model of the new binary. More multi-colour
photometric light curves, and radial velocity curves, would be
necessary to collect and analyse before we can be confident on
the nature of the system.
3.6 The giant variable SR or L stars
Numerous variable bright giants are known in this cluster. Unfor-
tunately, a few of them are saturated in our IAC images. A plot of
the V magnitude versus HJD reveals the long-term variations of two
previously unreported variables which we have called V58 and V59
and classified as L type. Also, a new variable V60 is found between
the HB and the RGB, not far from V32, which was considered
by Osborn et al. (2017) to be a non-typical SR (see Fig. 11). V60
displays a long-term variation (see Fig. 12) but a short-term variation
with a period of 0.494497 d is also evident in the data of 2014 and
2016, after the 2014 data are adequately shifted to the 2016 data. The
short-term variations may be due to the binary nature of the stars.
A blinking inspection of a series of images suggested light
variations of a group of fifteen stars not previously detected as
variables, which are listed in Table 3. Fig. 13 displays the CMD
of the cluster above the HB to illustrate the position of these
variable candidates. We confirm that their mid-term variations are
comparable to those observed in well-established SR variables (Fig.
12). The plots of the V magnitude as a function of HJD show a
conspicuous variation in most of them (Fig. 14). We have refrained
from assigning them a variable star number, which shall be assigned
in the CVSGC once the variation is confirmed. We will denote these
stars with a C designation (for candidates). Some of them are clearly
in the RGB and are cluster members, judging from their proper
motions and/or radial velocities according to Gaia-DR2, which shall
be further discussed in Section 7. Section 7. Five of these candidates
are found to be distributed above and to the blue side of the HB,
C7, C8, C9, C10 and C11 (empty triangles in Fig. 12), and their
proper motions and quality parameters reported in Gaia make their
membership dubious. The four candidates, C12, C13, C14 and C15,
are definitely not cluster members.
In their paper, Osborn et al. (2017) performed a thorough analysis
of the most luminous variables in the RGB of M13. They estimated
their cycle times between 30 and 90 d and point out that some
of them may have multiple periods. These authors also find that
the magnitude range of the variation and period increases with
luminosity. This result forecasts the difficulty of finding variable
stars in the lower parts of the RGB.
We call attention to these variables on the RGB, which are
confirmed members from their Gaia proper motions and/or radial
velocities. This also confirms that virtually all stars in M13 above V
< 12.5 mag and with (V − I) greater than about 1.2 mag do present
mid- to long-term variations.
Given the limited time distribution of our data, we have not
attempted to measure periods or characteristic cycle times for the
MNRAS 486, 2791–2808 (2019)
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M13 2799
Figure 9. Light curves of the contact binary V57. The bottom panels show
the adopted model fitted to the observations. See Section 3.5 for a detailed
discussion.
Figure 10. Position of the two components of V57 in the log Teff − log L
plane along with theoretical isochrone values. The green hexagon and red
square correspond to the primary and secondary components, respectively.
See Section 3.5 for a discussion.
Figure 11. Short-term (0.494497 d) variations of the peculiar semiregular
variable V60. Blue symbols are data from 2014 and have been shifted
in phase and magnitude to match black symbols from 2016, in order
to compensate for the obvious long-term variations seen in Fig. 13. See
Section 3.6 for a detailed discussion.
stars in the RGB. For the previously known SR stars, the given
periods in Table 2 are those of Osborn et al. (2017).
3.7 Comments on the RR Lyrae stars identified in Gaia-DR2
in the field of M13
Clementini et al. (2018) report the detection and characterization
of 140 784 RR Lyrae stars by a Specific Object Study pipeline in
Gaia-DR2 all over the sky. The sample of confirmed RR Lyrae
stars includes those found in 87 globular clusters including the
cluster being studied in the present work. They provide the multi-
band time series data (in the Gaia photometric system) and derive
from them the characteristic physical parameters of the listed RR
Lyrae stars. From the stars listed by Clementini et al. (2018), we
selected those inside the tidal radius of M13, 23 arcmin according
to the Catalogue of Milky Way Stellar Clusters by Kharchenko et al.
(2013). We found the 18 stars listed in Table 4. This table includes
the results of our analysis of their variability from our data. It can
be seen that only the four previously known RR Lyrae stars were
confirmed with our data, no variability was found in seven stars, six
stars were too faint for our observations or we were unable to build
a reliable light curve, and one is out of the FoV of our images.
4 R R LYRAE STARS: [FE/H] AND MV F ROM
L I G H T C U RV E F O U R I E R D E C O M P O S I T I O N
Since the light curves of RR Lyrae stars are periodic, it is natural to
use Fourier series to describe them with the following equation:
m(t) = A0 +
N
∑
k=1
Ak cos
(
2π
P
k(t − E0) + φk
)
(3)
where m(t) is the magnitude at time t, P is the period of pulsation,
and E0 is the epoch. When calculating the Fourier parameters, we
made use of a least-squares approach to estimate the best fit for
the amplitudes Ak and phases φk of the light-curve components.
MNRAS 486, 2791–2808 (2019)
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2800 D. Deras et al.
Figure 12. Light curves in V filter of the SR and L-type variables in M13. The light curves are a combination of the data obtained at IAC (blue dots) and at
Hanle (black dots). Note that the scale of the V-axis is not the same for all plots.
Figure 13. Upper section of CMD of M13 displaying, along with some
well-established variables, the 15 variable star candidates identified in the
FoV of the cluster by blinking techniques. These candidates are represented
by the empty, orange and cyan triangles and labelled as C stars, and their
light curves are shown in Fig. 14. The empty triangles correspond to five
variable star candidates that might not be truly cluster members and the four
orange triangles correspond to non-members. The full CMD of M13 is in
Fig. 18. See Section 3.6 for a discussion.
The phases and amplitudes of the harmonics in equation (3), i.e.
the Fourier parameters, are defined as φij = jφi − iφj, and Rij =
Ai/Aj. For the RR Lyrae stars, their Fourier coefficients are listed in
Table 5.
Over the last few years, our group has consistently used well-
tested calibrations and zero points to study over 30 globular clusters
in the Galaxy (Arellano Ferro, Bramich & Giridhar 2017). Below,
we list the specific set of equations used in this work. To calculate the
metallicity and absolute magnitude of the RRab stars, we employed
the calibrations of Jurcsik & Kovács (1996) and Kovács & Walker
(2001), respectively:
[Fe/H]J = −5.038 − 5.394P + 1.345φ
(s)
31 , (4)
MV = −1.876 log P − 1.158A1 + 0.821A3 + K. (5)
Note that the equation for the metallicity is given in the Jurcsik–
Kovács scale, but using [Fe/H]J = 1.431[Fe/H]ZW + 0.88 (Jurcsik
1995), it can be transformed to the standard Zinn–West scale
(Zinn & West 1984). We have adopted the value for K = 0.41
from Arellano Ferro, Giridhar & Bramich (2010).
In the case of the RRc stars, we employed the calibrations given
by Morgan, Wahl & Wieckhorst (2007) and Kovács & Kanbur
(1998), respectively:
[Fe/H]ZW = 52.466P 2 − 30.075P + 0.131φ
2(c)
31
− 0.982φ
(c)
31 − 4.198φ
(c)
31 P + 2.424, (6)
MV = 1.061 − 0.961P − 0.044φ
(s)
21 − 4.447A4. (7)
MNRAS 486, 2791–2808 (2019)
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Table 3. Data of variable star candidates in M13 in the FoV of our images. See discusion in Section 3.6.
Star ID Type V (mag) I (mag) AV (mag) AI (mag) α (J2000.0) δ (J2000.0) Membership
C1 SR 12.97 11.78 >0.04 >0.04 16:41:32.33 + 36:27:34.52 Yes
C2 SR 12.63 11.36 >0.07 >0.07 16:41:34.33 + 36:30:13.03 Yes
C3 SR 12.20 10.79 >0.10 >0.09 16:41:34.75 + 36:27:59.27 Yes
C4 SR 12.25 10.87 >0.07 >0.06 16:41:34.80 + 36:27:19.38 Yes
C5 SR 12.55 11.31 >0.07 >0.05 16:41:35.68 + 36:26:48.61 Yes
C6 SR 12.87 11.69 >0.44 >0.50 16:41:39.73 + 36:26:37.79 Yes
C7 Irregular 14.20 14.09 >0.06 >0.08 16:41:39.77 + 36:28:06.43 Uncertain
C8 Irregular 14.07 13.90 >0.09 >0.13 16:41:40.41 + 36:28:09.81 Uncertain
C9 Irregular 15.40 15.36 >0.09 >0.13 16:41:42.84 + 36:31:00.93 Uncertain
C10 Irregular 14.41 14.36 >0.11 >0.07 16:41:43.56 + 36:27:11.02 Uncertain
C11 Irregular 15.18 15.19 >0.20 >0.15 16:41:44.38 + 36:26:21.50 Uncertain
C12 Irregular 13.08 13.23 >0.09 >0.14 16:41:33.65 + 36:26:07.54 No
C13 Irregular 14.01 13.81 >0.04 >0.07 16:41:34.75 + 36:29:13.74 No
C14 Irregular 14.39 14.29 >0.14 >0.08 16:41:39.03 + 36:27:09.33 No
C15 Irregular 15.10 15.01 >0.08 >0.10 16:41:45.60 + 36:26:37.42 No
Figure 14. Light curves of 15 stars in the field of M13 not previously known as variables. Colours are as in Fig. 13. We have retained them as candidates
although their variations are rather conspicuous. A blow up of the blue symbols and the I light curves confirms the shorter term variations (not illustrated).
For convenience, one can transform the coefficients from cosine
series phases into sine series using the following relation:
φ
(s)
jk = φ
(c)
jk − (j − k)
π
2
. (8)
For comparison, we have transformed [Fe/H]ZW on the Zinn &
West (1984) metallicity scale into the UVES scale using the equa-
tion [Fe/H]UVES = −0.413 + 0.130 [Fe/H]ZW − 0.356 [Fe/H]2
ZW
(Carretta et al. 2009).
The values of MV reported in Table 6 have been transformed to
luminosities using the following equation:
log(L/L⊙) = −0.4(MV − M⊙
bol + BC). (9)
To calculate the bolometric correction, we made use of the for-
mula BC = 0.06[Fe/H]ZW + 0.06 reported in Sandage & Cacciari
(1990). We have adopted the value of M⊙
bol = 4.75 mag.
To estimate the effective temperature of the RRab stars, we
employed the calibration given by Jurcsik (1998):
log(Teff) = 3.9291 − 0.1112(V − K)0 − 0.0032[Fe/H], (10)
where
(V − K)0 = 1.585 + 1.257P − 0.273A1 − 0.234φ
(s)
31
+ 0.062φ
(s)
41 . (11)
For the RRc stars, the calibration of Simon & Clement (1993) was
used:
log(Teff) = 3.7746 − 0.1452 log(P ) + 0.0056φ
(c)
31 . (12)
Once we have estimated the effective temperature, the period
and the luminosity of the RR Lyrae stars, we can also esti-
mate their masses using: log(M/M⊙) = 16.907 − 1.47 log PF +
1.24 log(L/L⊙) − 5.12 log(Teff) as given by Van Albada & Baker
MNRAS 486, 2791–2808 (2019)
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Table 4. Results of our analysis of the RR Lyrae stars reported by Clementini et al. (2018) in M13. The types and periods were derived by Clementini et al.
(2018), and the remaining columns were extracted from Gaia-DR2.
Gaia-DR2 source Type Gmag P (d) RA DEC Result of our analysis
1328057179882276480 RRc 14.7406 0.381796 16:41:46.37 + 36:27:40.0 Previously known as V5, P = 0.381784d.
1328057940097496064 RRab 16.7973 0.414622 16:41:42.49 + 36:28:26.9 No variability detected in our observations.
1328057184175404544 RRc 14.7416 0.392737 16:41:46.27 + 36:27:37.9 Previously known as V9, P = 0.392724d.
1328057905737005696 RRab 16.4233 0.691576 16:41:39.91 + 36:27:48.3 No variability detected in our observations.
1328057321618952064 RRab 19.9812 0.669728 16:41:25.43 + 36:24:47.1 Fainter than the limit of our observations.
1328057940089939840 RRc 16.7692 0.342303 16:41:44.43 + 36:27:54.4 Unable to obtain a reliable light curve.
1328057768297931904 RRc 14.8746 0.312663 16:41:37.12 + 36:26:28.9 Previously known as V7, P = 0.312668d.
1328054297963963392 RRab 18.5601 0.644475 16:42:13.42 + 36:23:21.5 Out of the field of our observations.
1328056634424210560 RRab 19.8033 0.615953 16:41:33.90 + 36:24:47.1 Fainter than the limit of our observations.
1328055397466801664 RRab 19.9956 0.614019 16:42:03.18 + 36:26:22.1 Fainter than the limit of our observations.
1328057424694539776 RRab 18.9557 0.471707 16:41:28.79 + 36:25:42.4 No variability detected in our observations.
1328056703145793024 RRab 19.1703 0.570260 16:41:47.80 + 36:24:15.0 No variability detected in our observations.
1328055397466713728 RRab 20.3268 0.583670 16:42:04.43 + 36:26:10.3 Fainter than the limit of our observations.
1328056462625419776 RRab 20.3024 0.516029 16:41:29.53 + 36:22:49.5 Fainter than the limit of our observations.
1328057733938742528 RRab 18.7482 0.591195 16:41:25.72 + 36:28:08.6 No variability detected in our observations.
1328058077530125568 RRab 14.7583 0.750291 16:41:32.65 + 36:28:02.1 Previously known as V8, P = 0.750303.
1328056939364066176 RRab 18.8948 0.481375 16:41:41.01 + 36:24:52.6 No variability detected in our observations.
1328057012383745152 RRab 17.8061 0.703458 16:41:39.55 + 36:25:45.1 No variability detected in our observations.
Table 5. Fourier coefficients Ak for k = 0, 1, 2, 3, 4, and phases φ21, φ31 and φ41, for RRab and RRc stars. The numbers
in parentheses indicate the uncertainty on the last decimal place. Also listed is the deviation parameter Dm for V8 (see
Jurcsik & Kovács 1996).
Variable ID A0 A1 A2 A3 A4 φ21 φ31 φ41 Dm
(V mag) (V mag) (V mag) (V mag) (V mag)
RRab star
V8 14.843(1) 0.295(1) 0.145(2) 0.093(1) 0.039(1) 4.348(1) 8.846(2) 7.0565(4) 2.1
RRc stars
V5 14.790(1) 0.240(1) 0.010(1) 0.024(1) 0.014(1) 5.073(82) 4.638(35) 2.693(59)
V7 14.897(1) 0.156(1) 0.018(1) 0.004(1) 0.002(1) 4.761(34) 3.278(146) 1.806(318)
V9 14.819(1) 0.244(1) 0.017(1) 0.017(1) 0.013(1) 4.921(64) 4.532(65) 3.072(83)
V25 14.622(1) 0.186(1) 0.006(1) 0.009(1) 0.006(1) 3.755(177) 4.456(110) 2.463(176)
V34 14.806(1) 0.173(1) 0.009(1) 0.016(1) 0.010(1) 6.382(108) 4.781(63) 2.748(93)
V35 14.807(1) 0.088(1) 0.006(1) 0.002(1) 0.002(1) 5.153(121) 3.344(380) 2.547(389)
V54 14.897(1) 0.067(1) 0.010(1) 0.001(1) 0.001(1) 4.323(78) 5.574(5805) 6.405(5926)
Table 6. Physical parameters obtained from the Fourier fit for the RRab and RRc stars. The numbers in parentheses indicate the uncertainty on the last decimal
place. See Section 4 for a detailed discussion.
RRab star
Star [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V8a −1.603 ± 0.002 −1.536 ± 0.002 0.378 ± 0.016 3.794 ± 0.008 1.749 ± 0.006 0.68 ± 0.07 6.50 ± 0.05
RRc stars
Star [Fe/H]ZW [Fe/H]UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V7 −1.53(27) −1.44(30) 0.611(5) 3.866(1) 1.655(2) 0.524(6) 4.180(9)
V25 −1.81(15) −1.81(19) 0.542(6) 3.854(1) 1.683(3) 0.412(3) 4.571(13)
V35 −1.39(48) −1.28(48) 0.586(6) 3.867(1) 1.666(3) 0.517(9) 4.215(12)
V54b − − 0.652(6) 3.883(33) 1.639(2) 0.449(172) 3.804(10)
Weighted mean −1.72 −1.66 0.586 3.856 1.665 0.45 4.28
σ ± 0.12 ± 0.15 ± 0.003 ± 0.001 ± 0.001 ± 0.01 ± 0.01
Notes. aThe uncertainties in the parameters come from the uncertainties of the calibrations themselves.
bNot included in the averages of the physical parameters.
(1971) where logPF is the fundamental period, and their radii with
L = 4πR2σT4. These values are also reported in Table 6.
Stars with Bl designation in Table 2 are Blazhko variables. We
report their Fourier coefficients and physical parameters but they
were not taken into account in the physical parameter calculations.
MNRAS 486, 2791–2808 (2019)
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Since the Fourier decomposition for V54 yielded an anomalous
value for φ31, on which the metallicity, temperature and hence the
mass and radius depend, we decided not to include this star in
the average of these parameters. However, we included it in the
calculation of the average distance, since it does not depend on φ31.
The resulting physical parameters are summarized in Table 6.
4.1 An alternate formulation for [Fe/H]
We performed our calculations of [Fe/H] via the Fourier parameters
using equations (4) and (6) for the RRab and RRc stars, respectively,
mainly for the sake of homogeneity, since these formulations have
been systematically used by our group for a large number of clusters
(e.g. Arellano Ferro et al. 2017). However, new formulations have
been proposed by Nemec et al. (2013), in their equations (2) and
(4) for the RRab and RRc, respectively. We have used these new
formulations to calculate [Fe/H]ZW as −1.573 for the only RRab
star and −1.74 for the three RRc stars. These values are, within
the corresponding uncertainties of both formulations, in very good
agreement with the averages reported in Table 6. Nemec et al. (2013)
have noted the good agreement between the two formulations for
clusters with [Fe/H] < −1.0, as it seems to be the case for M13, and
that larger discrepancies are to be found for decreasing metallicities.
For [Fe/H] ∼ −2.0 the differences can be as large as ∼0.3 dex.
5 O N TH E M E TA L L I C I T Y O F M 1 3
M13 is a metal-poor cluster of the OoII type. From the Fourier
decomposition of the RR Lyrae light curves, we found an overall
average of 〈[Fe/H]ZW〉 = −1.58 ± 0.09. Other values for the
metallicity found in the literature include: −1.65 (Zinn 1985),
−1.65 (Djorgovski 1993), −1.63 (Marı́n-Franch et al. 2009), −1.58
(Carretta et al. 2009) and −1.53 (Harris 1996), which are in good
agreement with our own results.
6 O N TH E D I S TA N C E TO M 1 3
The distance to M13 has been a matter of controversy. In the
literature, we have found an assortment of values that range from
7.1 kpc (Harris 1996), 7.4 kpc (Paust et al. 2010; VandenBerg et al.
2013), 7.7 kpc (Hessels et al. 2007) to 8.0 kpc (Violat Bordonau,
Sanchez Bajo & Bennasar Andreu 2005). In the following, we
explain our own results for the determination of the distance to
M13 by using several independent distance indicators.
6.1 From the RR Lyrae stars
We started by adopting the reddening value reported by Harris
(1996) of E(B − V) = 0.02. With this value, and using the
independent calibrations for RRab and RRc stars to calculate Mv
mentioned in Section 4, we obtained the distances d = 7.6 kpc for
the only known RRab in the cluster and 〈d〉 = 6.8 ± 0.3 kpc
for the RRc stars. We have also made use of the PL relation
for RR Lyrae stars in the I filter (Catelan, Pritzl & Smith 2004)
MI = 0.471 − 1.132 log P + 0.205 log Z, with log Z = [M/H] −
1.765; [M/H ] = [Fe/H] − log(0.638f + 0.362) and log f = [α/Fe]
(Salaris, Chieffi & Straniero 1993), from which we derived a
distance of 7.0 ± 0.2 kpc. When taking these approaches, we made
use of all the RR Lyrae stars with the exception of V31 and V36 since
a double-mode effect appears to be present in their light curves.
Figure 15. PL relation in V for SX Phe stars in M13. Coloured solid lines
correspond to the PL relation derived by Cohen & Sarajedini (2012), the
long-dashed lines to the relation derived by Poretti et al. (2008) and the
short-dashed lines to the PL relation derived by Arellano Ferro et al. (2011).
The colours red, blue and green correspond to the fundamental, first overtone
and second overtone, respectively.
6.2 From the Type II Cepheids
There are three known Type II Cepheids in M13, namely V1, V2
and V6. We made use of Fourier decomposition of V1 and V2
in order to estimate an intensity weighted mean for the apparent
magnitude. Since the light curve of V6 is incomplete, we adopted
the value provided by Kopacki et al. (2003) of V = 14.078. We used
the PL relation for these type of stars MV = −1.64(±0.05) log P
+ 0.05(±0.05) as derived by Pritzl et al. (2003), from where we
found a distance of 7.1 ± 0.6 kpc.
6.3 From the SX Phe stars
In Fig. 15 we show the PL relation for the SX Phe stars in the log
P-V plane. There are four known SX Phe stars in M13 listed in the
CVSGCs (Clement et al. 2001), plus two that are newly reported
in this work (V55 and V56). Three versions of the PL relation
are shown in the figure. This relation is helpful for confirming
star membership in the cluster as well as for identifying pulsation
modes, and hence to estimate the mean distances to qualified cluster
members. Making use of the PL relation derived by Poretti et al.
(2008), MV = −3.65(±0.07) log P-1.83(±0.08), we found 〈d〉
= 7.2 ± 0.7 kpc. From this diagram we conjecture that V47, V50
and V55 are pulsating in the fundamental mode while V46 and V56
are likely second overtone pulsators.
6.4 From the binary V57
Via the model fitting for the contact binary V57, a distance of
6.9 ± 0.2 kpc was estimated (see Section 3.5).
6.5 Distance from the ZAHB fitting
According to VandenBerg et al. (2013), a proper fitting of the
theoretical Zero Age Horizontal Branch (ZAHB) and the Turn
Off (TO) to the observed stellar distributions gives a reliable
apparent magnitude modulus. The apparent modulus that we found
in this way for M13 was μ = 14.34 corresponding to a distance
d = 7.1 kpc. The average of the distance values obtained by the
aforementioned methods, listed in Table 7, is 〈d〉 = 7.1 ± 0.1 kpc.
MNRAS 486, 2791–2808 (2019)
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Table 7. Distance comparison to M13 from the different methods used in
this work.
Method Distance [kpc]
RRab Fourier decomposition 7.6a
RRc Fourier decomposition 6.8 ± 0.3
RRab/RRc I-band PL 7.0 ± 0.2
SX Phe PL 7.2 ± 0.7
Type II Cepheids PL 7.1 ± 0.6
Level of the ZAHB 7.1 ± 0.1
Eclipsing binary V57 6.9 ± 0.2
Weighted mean 7.1 ± 0.1
Note. aBased only on one RRab. Not included in the average.
6.6 Luminous red giants as distance indicators
The luminosity of the brightest stars at the TRGB can in principle
be used to estimate the distance to a stellar system. This method
was originally envisaged to determine distances to nearby galaxies
(Lee, Freedman & Madore 1993). As calibrated by Salaris & Cassisi
(1997), the bolometric magnitude of the tip of the RGB is:
M
tip
bol = −3.949 − 0.178 [M/H ] + 0.008 [M/H ]2, (13)
where [M/H ] = [Fe/H] − log(0.638f + 0.362) and log f = [α/Fe]
(Salaris et al. 1993).
The method is very sensitive to the selection of the stars to be
used and, on the other hand, it has to be considered that the brightest
stars in a given cluster may not be at the very tip of the RGB in the
CMD, but rather lower by a certain magnitude. Viaux et al. (2013)
argued, on the grounds of helium ignition delay in low-mass stars,
and the resulting extension of the red giant branch, that for the case
of M5 the brightest stars are between 0.04 and 0.16 mag below the
TRGB. The suggested offset was confirmed by the non-canonical
models of Arceo-Dı́az et al. (2015), who from the analysis of 25
globular clusters concluded that the theoretical TRGB is in average
about 0.26 ± 0.24 bolometric magnitudes brighter than the one
observed.
Using only the two brightest stars near the tip of the RGB in the
CMD (V11 and V42), and making use of equation (13) without
further assumptions, one finds a distance of 8.1 kpc, which is
too high relative to the distance found by all methods previously
discussed. We figured that, to reproduce the mean distance in
Table 7, 7.1 kpc, we need to assume that the true TRGB is some 0.25
mag brighter than V11 and V42, in agreement with the predictions
of Arceo-Dı́az et al. (2015). However, using the new calibration of
Mould, Clementini & Da Costa (2019), who from Gaia-DR2 data
calculated that the TRGB is close to MI ∼ −0.4 mag, we found
that in order to get a distance of 7.1 kpc for V11 and V42, we
need a correction of only 0.08 mag, in good agreement with the
calculations of Viaux et al. (2013).
7 STA R MEMBERSHIP USING Gaia
In this section to ascertain cluster membership we used an approach
based upon the high-quality astrometric data available in Gaia-
DR2 (Gaia Collaboration et al. 2018). The method is based on
the Balanced Iterative Reducing and Clustering using Hierarchies
(BIRCH) algorithm (Zhang, Ramakrishnan & Livny 1996) in a four-
dimensional space of physical parameters – positions and proper
motions – that detect groups of stars in that 4D space. A 4D Gaussian
ellipsoid is fitted for each group, and stars outside 3σ are rejected.
In order to decide if the remaining stars are members of the cluster,
their positions are plotted in celestial coordinates, in the vector
point diagram (VPD) of the proper motions and in the CMD (of
Gaia photometric system). If all three plots are consistent with
those of a globular cluster, the stars are considered members of the
cluster (Figs 16 and 17). More details about the method and some
applications on other clusters will be published elsewhere by Bustos
Fierro & Calderón. We applied this method on a field of 40 arcmin
of radius around the center of M13 containing 52 802 stars. We
found 23 070 stars most likely members of the cluster. In Fig. 16
we display the result of this membership determination.
We then cross-matched the member stars with our collection of
stars with light curves in the FoV of our M13 images. We found
7630 stars in common that were used to build the clean CMDs of
Fig. 18 (central panel).
We also cross-matched the member stars with the variables
reported by Clement et al. (2001) and the new variables discovered
in this work in order to confirm the membership status of all known
variables in M13. Fig. 17 plots the positions and proper motions of
all members and the variables.
8 TH E C MD O F M1 3
The CMDs in the mosaic of Fig. 18 illustrate the cleaning process
of field stars. The left panel shows all stars in the field of the cluster
with the photometric measurements in this work. Black symbols
are used for member stars identified as described in Section 7. The
central and right panels illustrate the distribution of member stars
for the IAC and Hanle data, respectively. The distribution of variable
stars and the isochrone and ZAHB fitting to the stellar distributions,
assuming the parameters derived in this paper, are also shown. Note
Figure 16. The red dots correspond to member stars and the grey dots to field stars according to the astrometric membership determination. Left panel: cluster
field; central panel: VPD of proper motions; right panel: CMD (Gaia photometric system).
MNRAS 486, 2791–2808 (2019)
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M13 2805
Figure 17. Projection on the sky of the proper motion vectors. Red arrows:
cluster members; green arrows: known variable stars; pink arrows are
newly discovered variables. Vectors have been enlarged 100 000 times for
visualization purposes.
that the positions of the variable stars are the same in all the CMDs
since their mean intensity weights were calculated using both sets
of data (IAC and Hanle).
8.1 The structure of the horizontal branch of M13
The HB of M13 displays mostly a prominent blue tail and the
redder components seem evolved and lie well above a ZAHB. The
RR Lyrae population is dominated by first overtone RRc pulsators.
Only one RRab is known and two double-mode or RRd stars are
also present. The distribution of RRab-RRc stars shows a clear
segregation around the first overtone red edge, represented as a
vertical dashed line identified in several other clusters, which seems
to be the rule among OoII clusters (Arellano Ferro et al. 2018). A
parameter that helps to describe the morphology of the HB is the
Lee index (Lee 1990) defined as L = (B-R)/(B+V+R), where B
and R are the number of stars to the blue or red sides of the IS,
respectively, and V is the number of variable stars within the IS.
In the case of M13 we found this value to be L = 0.95, which is
consistent with an OoII-type cluster with the two pulsation modes
well split.
9 SU M M A RY A N D C O N C L U S I O N S
In this work we have obtained high-precision photometry of over
16 000 point sources in the field of the globular cluster M13 in our
Figure 18. CMDs of M13. The left panel shows the CMD with all the stars in the FoV of the IAC images. Black and blue dots represent the member and
non-member stars (see Section 7). The central panel shows only those stars with a high probability of being members of the cluster. The right panel shows the
member stars as measured from Hanle data exclusively. The colour markers are the same in all three panels. The blue circle corresponds to the only known
RRab star in the cluster. Green and magenta circles correspond to RRc and SX Phe stars, respectively. Red circles represent semiregular variables. Brown
circles correspond to RRd stars and the teal circles to Type II Cepheid stars. The triangle markers correspond to newly discovered RRc (V54), SX Phe (V55
and V56), W Uma (V57) and SR (V58, V59 and V60) stars. The blue squares near the top of the RGB are 70 stars whose membership has been determined by
means of radial velocity from Gaia. The light blue isochrone is from VandenBerg et al. (2014), with Y = 0.25 and [α/H] = 0.4, corresponding to an age of
12.6 Gyr. The green isochrone is based on the models of Marigo et al. (2017) and corresponds also to an age of 12.6 Gyr and to a 〈[Fe/H]〉RR = −1.58. The
isochrones and ZAHB have been shifted to a distance of 7.1 kpc (see Table 7) and reddened by E(B − V) = 0.02, although the ZAHB (red line) needed an
extra 0.02 for a better fit. The dashed vertical line represents the red edge of the first overtone IS. See Section 8 for a discussion.
MNRAS 486, 2791–2808 (2019)
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V- and I-band reference images. We have been able to retrieve most
of the known variables cited in the literature. Of particular interest
is the RR Lyrae star population since the Fourier decomposition of
their light curves allowed us to determine the metallicity of M13
by two independent methods, from which we derived an average of
〈[Fe/H]ZW〉 = −1.58 ± 0.09.
From our two-colour photometry we constructed a clean CMD
based on the recent Gaia-DR2 (Gaia Collaboration et al. 2018)
proper motions and radial velocities, on which we overlaid two
isochrones that correspond to an age of 12.6 Gyr and a metallicity of
〈[Fe/H]〉RR =−1.65 and a theoretical ZAHB with the corresponding
metallicity. The isochrones come from different sets of model
atmospheres and yet they are practically indistinguishable from
one another.
We discovered seven new variables: one RRc (V54), two SX Phe
(V55 and V56), one contact binary (V57), and three SR (V58,
V59 and V60) stars. From the orbital solution of the contact
binary, we were able to derive the masses, radii and effective
temperatures of the primary and secondary components. We also
found 15 stars which seem to display mid- to long-term variations
but that have been retained as candidates waiting for confirmation
by more appropriate data. Based on its high radial velocity, we
identified a runaway star in the field of M13 which is discussed in
Appendix A. This star appears to be passing by and overtaking the
cluster.
We report the double-mode nature of the star V31 and confirm
the double-mode nature of V36. In our analysis of V36, we found
two frequencies that are consistent with the ones found by Kopacki
et al. (2003). In the case of V31, we also found two distinct periods
and at least one seems to be non-radial.
From the PL relation for the SX Phe stars, we found that it is
likely that V47, V50 and V55 are pulsating in the fundamental
mode and V46 and V56 in the second overtone.
The period of V8, the large positive value of L (long blue tail of
the HB), and the clear segregation between RRab and RRc stars on
the HB confirm the nature of M13 as an OoII-type cluster.
By using seven independent methods such as the MV values
obtained from the Fourier decomposition of RRab and RRc stars,
the PL relation for RR Lyrae stars in the I filter, the PL relation
for the SX Phe, the PL relation for the CW stars, the position
of the theoretical ZAHB and the orbital solution of the contact
binary V57, we were able to estimate the mean distance to M13, d
= 7.1 ± 0.1 kpc.
From Gaia-DR2 proper motion data we found 23 070 stars that
are likely cluster members. We also re-confirm the membership
status of all variables referred in this work and we give the Gaia-
DR2 identifier for all of them. From the analysis of the 18 stars
of the catalogue of Clementini et al. (2018) that are in the field
(see Table 4) we conclude that four of them were previously known
(namely V5, V7, V8 y V9), seven of them do not show variability,
six of them are fainter than the limit of our observations and one is
out of the FoV of our frames.
AC K N OW L E D G E M E N T S
We are grateful to Prof. Don VandenBerg for his valuable insights on
the ZAHB and isochrone models. We thank the staff of IAO, Hanle
and CREST, Hosakote, that made these observations possible. The
facilities at IAO and CREST are operated by the Indian Institute
of Astrophysics, Bangalore. This project was partially supported
by DGAPA-UNAM (Mexico) via grant IN106615-17. DD thanks
Figure 19. Light curves of two stars discussed by McDonald et al. (2018)
named L592 and L687. Using our own data we were able to detect periodic
brightness variations (see Note added in proof). The light curves are a
combination of the data obtained at IAC (blue dots) and at Hanle (black
dots).
CONACyT for the PhD scholarship. We have made extensive use
of the SIMBAD, ADS services, and of ’Aladin sky atlas’ developed
at CDS, Strasbourg Observatory, France. (Bonnarel et al. 2000) and
TOPCAT (Taylor 2005).
N OT E A D D E D IN PRO O F
While this paper was in print, Prof. Christine Clement called our
attention to two stars discussed by McDonald et al. (2018) named
L592 and L687 after Ludendorff (1905) with coordinates RA =
16:41:41.89, DEC = +36:28:18.2 and RA = 16:41:44.00, DEC =
+36:22:33.9 respectively. The light curves in Fig. 19 are based on
our data in the present paper and are phased with the periods 11.4
d and 11.5 d respectively, which match well the period estimation
of McDonald et al. of about 11 days for both stars. We clarify that
McDonald et al. listed L687 as IV-15 and Cudworth & Monet (1979)
noted that L687 and IV-15 are the same star. We suggest these stars
should be called V61 and V62 in the Catalogue of Variable Stars in
Globular Clusters.
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APPENDIX: A RUNAWAY STAR IN THE FIE L D
O F M 1 3
After the membership determination, we analysed the radial veloc-
ities distribution in the field of M13. There are 166 stars with radial
velocity measured in Gaia-DR2, 70 of which were identified as
members of the cluster. The radial velocity histogram in Fig. A1
shows a coherent concentration of the cluster members at Vr ∼
−250 km s−1 whereas the non-members are neatly separated at
lower velocities. In this figure, there is one noticeable High Radial
Velocity Star (HRVS) at Vr ∼ −310 km s−1. This HRVS is the
Gaia-DR2 source 1328140781417544832, RA (ICRS) 250.97068o,
DE (ICRS) 36.48989o, μα = (−6.468 ± 0.029) mas yr−1, μδ
= (−10.296 ± 0.035) mas yr−1 Vr = (−304.74 ± 1.67) km s−1,
̟ = (0.1708 ± 0.0198)mas (Fig. A2).
In Fig. A3 we display the positions and proper motion vectors
for the cluster members and the HRVS. If it was a member star
escaping the cluster, it is expected to have a relative proper motion
approximately radial from the centre of the cluster, but this is
not the case. Instead, all this evidence suggests that the HRVS
is passing by very close to the cluster and overtaking it in a rather
tangential way. It is also noticeably a star among the members
with discrepant radial velocity (MDRV); it corresponds to the
Gaia-DR2 source 1328033613402105728, RA (ICRS) 250.33156o,
DEC (ICRS) 36.35434o, μα∗ = (−4.732 ± 0.039) mas yr−1, μδ
= (−1.713 ± 0.054) mas yr−1, RV = (−73.02 ± 1.67)km s−1, ̟
= (0.2854 ± 0.0234)mas. If this star is indeed a cluster member,
we are unable to figure out the reason for the discrepancy with the
available information, if not it could be a field star that contaminates
the sample of member stars.
MNRAS 486, 2791–2808 (2019)
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2808 D. Deras et al.
Figure A1. Histogram of radial velocities for 166 stars measured with Gaia
in the FoV of M13. Purple bars at Vr ∼ −250 km s−1 are cluster members.
Light blue bars are non-members.
Figure A2. VPD of the members of M13. The blue dot shows the proper
motion of the HRVS.
Figure A3. Projection on the sky of the proper motion vectors. Red arrows:
cluster members; blue arrow: HRVS; green arrow: HRVS relative to the
mean proper motion of the cluster. Vectors have been enlarged 100 000
times for visualization purposes.
This paper has been typeset from a TEX/LATEX file prepared by the author.
MNRAS 486, 2791–2808 (2019)
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91
Chapter 9
NGC 6712: The variable star population of a
tidally disrupted globular cluster
9.1 Overview
NGC 6712 is a small metal-rich globular cluster located in the direction of the constellation of Scutum (α =
18h53′04.32′′, δ = −08◦42′21.5′′, J2000), and is currently located deep in the the Galactic bulge (l = 25.35◦,
b = −4.32◦) at a distance of 6.9 kpc (Harris, 1996). Its orbit is asymmetrical and penetrates deep into the
Galactic central regions (0.2-0.3 kpc away from the Galactic center (Dauphole et al., 1996)).
The multiple crossings of this cluster through the Galactic disc in its approximately 12 Gyrs of existence, makes
it prone to strong tidal perturbations (Andreuzzi et al. 2001; Paltrinieri et al. 2001). It is due to this interaction
with the Galactic disc that this cluster has probably lost 9.9 × 106 M⊙. Pryor et al. (1993) estimated a mass for
NGC 6712 as 105 M⊙, but an inverted mass function of this cluster found by de Marchi et al. (1999) suggests
that its original mass could have been up to 107 M⊙, since it only appears when clusters have lost most (up to ∼
99%) of their initial mass. This could potentially make NGC 6712 one of the most massive clusters in the early
stages of the Galaxy. This mass loss due to gravitational tides may have some influence in the proper motions
distribution of the stars in the cluster, which makes it particularly interesting to study in the light of the Gaia-DR2
data.
The variable star population of the cluster includes RRL stars, long-period semi-regular variables, short-period
eclipsing binaries and one X-ray source whose counterpart presents optical variability. The usage of variable
stars to estimate mean metallicity, reddening and distance for the globular cluster is well documented, so in the
present article we propose to undertake a VI CCD time-series analysis of NGC 6712 aimed to estimate the above
quantities.
9.2 Observations and transformation to the standard system
The data used in the present work were obtained from two sites. The first set of data was obtained with the 0.80
m IAC80 telescope at the Observatorio del Teide (Tenerife, Spain), during 10 non-consecutive nights, between
June 13th and July 16th of 2016 and the second set of data was obtained using the 2 m telescope at the Indian
Astronomical Observatory (IAO) in Hanle, India on 12 nights separated into four intermittent seasons between
June 2011 and April 2018. We obtained a total of 584 and 591 images in the V and I filter, respectively. For the
reduction of our data we followed the method described in Appendix B, with which we were able to recover the
light curves of about 11 294 stars in both V and I filters in our FoV.
92 Chapter 9. NGC 6712: The variable star population of a tidally disrupted globular cluster
Table 9.1: Data of member variable stars in NGC 6712 in the FoV of our images (Deras et al.,
2020).
Star ID Type
⟨
V
⟩ ⟨
I
⟩
AV AI P HJDmax α (J2000.0) δ (J2000.0) Gaia-DR2 Source
(mag) (mag) (mag) (mag) (days) + 2450000
V1 RRab 16.31 15.40 1.11 0.75 0.512039 7574.6054 18:52:59.85 -08:42:31.3 4203848946351463808
V2 SR 12.55 10.50 >0.15 >0.07 109.01 – 18:53:08.79 -08:41:56.8 4203849393028208384
V3 RRab 16.24 15.26 0.57 0.38 0.655956 7574.6678 18:53:02.29 -08:43:47.4 4203848877631995136
V4 RRab 16.47 15.41 0.56 0.36 0.611745 7574.6268 18:53:16.29 -08:42:38.1 4203849122517862144
V5 RRab 16.41 15.44 1.14 0.74 0.545362 7572.6357 18:53:08.68 -08:43:24.1 4203848778920519168
V6 RRab Bl 16.45 15.56 0.86 0.59 0.510871 7553.6906 18:53:05.35 -08:42:53.6 4203848985005261312
V7 M 16.986 11.976 >0.62 >0.23 193.02 – 18:52:55.39 -08:42:32.6 4203849672273972864
V8 SR 13.176 10.906 >0.48 >0.28 116.292 – 18:53:05.66 -08:41:12.4 4203849461747677056
V10 L 13.946 11.276 >0.21 – – – 18:52:57.35 -08:41:43.9 4203849874064515456
V12 RRab 16.30 – 1.21 – 0.502790 5284.7907 18:53:06.02 -08:41:33.4 4203849088158338944
V13 RRab 16.13 15.26 1.02 0.66 0.562655 7574.6637 18:52:57.77 -08:41:46.5 4203849809712937088
V15 L 13.836 11.086 >1.5 – – - 18:53:20.99 -08:42:48.2 4203837401469330944
V18 RRc 16.15 15.32 0.47 0.28 0.353543 7569.5876 18:53:02.44 -08:42:14.0 4203849053798612096
V19 RRc 16.00 15.19 0.40 0.24 0.412161 7586.6633 18:53:03.22 -08:41:39.2 4203849844072610816
V20 RRc 16.20 15.48 0.34 0.25 0.250521 5779.3246 18:53:04.16 -08:42:03.7 4203849083790434304
V21 L 13.586 11.266 >0.33 – – – 18:52:58.80 -08:42:06.2 4203849805344920576
V22 RRab3 16.17 15.24 0.58 0.41 0.654789 7586.6187 18:52:59.08 -08:41:20.5 4203849874064518016
V23 RRab4 16.23 15.26 0.31 0.19 0.642451 7572.6241 18:53:03.83 -08:42:39.4 4203848985005549056
V24 RRab 16.23 15.25 0.99 0.61 0.576454 7586.6334 18:53:04.37 -08:42:26.9 4203849053725057280
V26 RRc 16.16 15.64 0.12 0.09 0.334317 7569.6551 18:53:06.80 -08:41:31.2 4203849466042002944
V28 EW 18.23 17.42 0.26 0.08 0.434497 7573.67347 18:53:08.09 -08:40:52.2 4203849466042155264
V29 EW 18.45 16.95 - - 0.453571 7569.62267 18:53:04.17 -08:42:25.0 4203849053798521472
V305 EW 18.68 17.91 0.61 0.49 0.504748 7567.64827 18:53:02.51 -08:43:11.9 4203848881926249600
V315 EW 18.53 17.74 0.40 0.38 0.418762 7573.67347 18:53:03.49 -08:40:34.4 4203850221956827648
V325 SRs 13.726 11.566 >0.14 >0.06 31.1 7586.6252 18:52:59.32 -08:41:34.7 4203849805345042176
V335 SR 13.686 11.576 >0.24 – – – 18:53:06.20 -08:42:43.2 4203848985079013760
V345 SR 13.546 11.596 >0.25 – – – 18:53:01.99 -08:42:18.8 4203849049430686336
V355 SR 13.936 12.006 >0.17 – – – 18:53:09.02 -08:41:12.4 4203849461747685760
V365 SRs 13.666 11.796 >0.22 >0.11 17.8 5724.4243 18:53:09.39 -08:43:12.9 4203848808912547840
C8 SRs 13.846 11.646 >0.21 – 18.9 5780.3438 18:53:02.30 -08:42:38.4 4203849049430685312
1 The period is from Oosterhoff (1943). 2 The period is from Sloan et al. (2010). 3 Classified as RRab in this
work. 4 Reclassified as RRab in this work. 5 New variable discovered in this work. 6 Magnitude-weighted
mean. 7 Time of primary minimum light. 8 Member variable candidate. Bl denotes Blazhko effect.
In order to transform our light curves from the v instrumental magnitudes to the Johnson-Kron-Cousins standard
V system, we made use of 94 of these stars established as secondary standards by Stetson (2000) to transform our
IAC CCD photometry and 91 stars for the Hanle data set. Fundamental photometric and positional information
of all variable stars in the cluster is reported in Table 9.1 for member stars and in Table 9.2 for non-members.
9.3 Physical parameters of RRL stars in NGC 6712
We have performed a Fourier decomposition of the light curves of the RRL stars to obtain the amplitudes and
phase displacements of each harmonic, i.e the Fourier parameters. These are related to some of the physi-
cal parameters of the corresponding pulsating star by the semi-empirical calibrations mentioned in Chapter 5
(Eq. 5.4 and 5.8). From these calibrations we estimated the weighted mean of the metallicity of the cluster as
⟨
[Fe/H]ZW
⟩
=–1.25 ± 0.02 from the RRab stars and
⟨
[Fe/H]ZW
⟩
=–1.10 ± 0.16 from the RRc stars. The results
of this analysis are reported in Table 9.3.
9.4. On the reddening of NGC 6712 93
Table 9.2: Data of known, unclassified and newly discovered field variable stars (FV) in NGC
6712 in the FoV of our images (Deras et al., 2020).
Star ID Type
⟨
V
⟩ ⟨
I
⟩
AV AI P HJDmin α (J2000.0) δ (J2000.0) Gaia-DR2 Source
(mag) (mag) (mag) (mag) (days) + 2450000
V16 ? 14.983 14.163 >0.06 >0.06 – – 18:52:54.55 -08:39:17.0 4203850187670201088
V17 ? 14.893 14.233 >0.24 – – – 18:53:05.86 -08:41:23.4 4203849083790555520
V27 EW1 16.51 15.34 0.12 0.08 0.425714 7586.63252 18:53:08.88 -08:42:30.7 4203849019364947456
FV14 EW 18.25 17.14 0.54 0.48 0.354499 7574.63632 18:52:55.78 -08:43:49.1 4203847438817518976
FV24 EW 17.84 16.65 0.27 0.25 0.355390 7567.58732 18:52:57.45 -08:40:50.4 4203849874064515840
FV34 EW 14.44 13.66 0.38 0.35 0.445495 7574.60472 18:53:19.90 -08:44:10.1 4203837298410747776
FV44 EW 17.04 16.07 0.23 0.19 0.368598 7572.64282 18:53:20.86 -08:41:58.9 4203849259956801792
FV54 EW 18.93 17.96 0.54 0.54 0.347863 7572.63802 18:53:21.25 -08:47:46.5 4203835756498276608
FV64 EW 18.39 17.41 0.40 0.39 0.462812 7573.85352 18:53:21.90 -08:40:52.2 4203849362961930624
FV74 EW 17.88 16.74 0.72 0.51 0.344416 7567.65142 18:53:22.04 -08:45:06.3 4203836718651092352
FV84 SRs 14.173 10.743 >0.58 – 12.3 7555.6001 18:53:00.93 -08:44:18.1 4203848843272252160
FV94 SRs 13.453 10.173 >0.15 – 24.6 8217.4473 18:53:05.58 -08:39:27.6 4203850325109041792
1 Reclassified as EW in this work. 2 Time of primary minimum light. 3 Magnitude-weighted mean. 4 New
variable discovered in this work.
Another set of Fourier-based calibrations to estimate the metallicity of GCs from the light curves of RRL stars
has been proposed by Nemec et al. (2013) (Eq. 5.19 and Eq. 5.20 for RRab and RRc, respectively). We used
these calibrations to calculate the individual values [Fe/H]N13
UVES listed in column 4 of Table 9.3 which yielded
the weighted averages −0.82 ± 0.06 for the RRab stars and −0.96 ± 0.19 for the RRc stars. These values are in
good agreement with the ones derived using the formulations of Jurcsik et al. (1996) for RRab stars and Morgan
et al. (2007) for RRc stars within their respective uncertainties for clusters with [Fe/H]<–1.0. The light curves
of the RRL stars with their corresponding Fourier fit are shown in Fig. 9.1
9.4 On the reddening of NGC 6712
The reddening produced by the interstellar medium plays an important role in the determination of the distance to
a cluster. Although in his catalogue Harris (1996) reports a reddening of E(B−V) = 0.45, there is no consensus
in the literature as to which value is the more accurate. The scatter found in the several determinations available
in the literature may be due to the presence of differential reddening, as suggested by Janulis et al. (1992). The
inconsistencies in the estimation of the reddening impact on the determination of the distance to the cluster.
For this reason we decided to estimate the reddening using an independent method, based on the reddening at
minimum light of RRab stars, described below.
The method we used was originally proposed by Sturch (1966) and is based on the fact that the intrinsic color
(B−V)0 of RRab stars is constant near minimum light. To convert to E(V − I), we used the calibration derived
by Guldenschuh et al. (2005) where (V − I)0,min = 0.58 ± 0.02. Finally, we transformed to E(B − V) using
the relation E(V − I)/E(B − V) = 1.259 from Schlegel et al. (1998). This allowed us to estimate the individual
values of E(V − I) for six RRab stars, namely, V4, V5, V13, V22, V23 and V24. This method yielded a mean
value of E(B−V) = 0.35± 0.04 which we have adopted for our calculations throughout this work. The distance
estimated adopting this reddening is reported in Table 9.4.
9.5 Membership determination with Gaia
We carried out a membership analysis of stars in the FoV of our images of NGC 6712, using the high-quality
astrometric data by Gaia-DR2 (Gaia Collaboration et al., 2018). The method used for this analysis is that of
94 Chapter 9. NGC 6712: The variable star population of a tidally disrupted globular cluster
0.0 0.5 1.0
15.5
16.0
16.5
V
V1
0.0 0.5 1.0
φ
15.0
15.5
16.0
I 0.0 0.5 1.0
16.0
16.5
17.0
V3
0.0 0.5 1.0
φ
15.0
15.5
16.0
0.0 0.5 1.0
16.0
16.5
17.0
V4
0.0 0.5 1.0
φ
15.0
15.5
16.0
0.0 0.5 1.0
15.5
16.0
16.5
V
V5
0.0 0.5 1.0
φ
15.0
15.5
16.0
I 0.0 0.5 1.0
16.0
16.5
17.0
V6
0.0 0.5 1.0
φ
15.0
15.5
16.0
0.0 0.5 1.0
15.5
16.0
16.5
V12
0.0 0.5 1.0
φ
15.0
15.5
16.0
0.0 0.5 1.0
15.5
16.0
16.5
V
V13
0.0 0.5 1.0
φ
15.0
15.5
16.0
I 0.0 0.5 1.0
15.75
16.00
16.25
V18
0.0 0.5 1.0
φ
15.00
15.25
15.50
0.0 0.5 1.0
15.75
16.00
16.25
V19
0.0 0.5 1.0
φ
15.00
15.25
15.50
0.0 0.5 1.0
16.00
16.25
16.50
V
V20
0.0 0.5 1.0
φ
15.25
15.50
15.75
I 0.0 0.5 1.0
15.5
16.0
16.5
V22
0.0 0.5 1.0
φ
14.5
15.0
15.5
0.0 0.5 1.0
16.00
16.25
16.50
V23
0.0 0.5 1.0
φ
15.00
15.25
15.50
0.0 0.5 1.0
15.5
16.0
16.5
V
V24
0.0 0.5 1.0
φ
14.5
15.0
15.5
I 0.0 0.5 1.0
16.0
16.1
16.2
V26
0.0 0.5 1.0
φ
15.4
15.5
15.6
Figure 9.1: RRL stars in NGC 6712. Black and cyan symbols correspond to observations from
IAC and Hanle respectively. The red line represents the Fourier fit. Note that the scale in the
Y-axis is different for RRab and RRc stars. The grey dots in V12 come from Sandage et al. (1966)
and were used to complete our light curve in the V-band (Deras et al., 2020).
9.6. The color-magnitude diagram of NGC 6712 and its HB 95
Table 9.3: Physical parameters obtained from the Fourier fit for the RRab and RRc stars. The
numbers in parentheses indicate the uncertainty on the last decimal place (Deras et al., 2020).
RRab stars
Star ID [Fe/H]JK96
ZW [Fe/H]JK96
UVES [Fe/H]N13
UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V1 −1.23(4) −1.11(3) −0.82(6) 0.605(4) 3.819(9) 1.658(2) 0.68(7) 5.20(1)
V3 −1.30(5) −1.18(5) −0.77(10) 0.539(3) 3.800(20) 1.685(1) 0.65(11) 5.87(1)
V41 −1.26(5) −1.14(4) −0.80(8) 0.614(3) 3.803(20) 1.654(1) 0.63(12) 5.59(1)
V5 −1.30(3) −1.18(3) −0.94(5) 0.570(4) 3.815(9) 1.672(2) 0.69(8) 5.41(1)
V61 −0.80(5) −0.74(3) −0.26(5) 0.662(3) 3.823(20) 1.635(1) 0.62(12) 4.98(1)
V121 −1.12(42) −1.00(34) −0.64(6) 0.598(6) 3.822(10) 1.661(2) 0.68(8) 5.15(1)
V13 −1.28(2) −1.16(2) −0.90(3) 0.574(2) 3.812(8) 1.670(1) 0.67(6) 5.45(1)
V22 −1.16(4) −1.05(4) −0.53(5) 0.535(1) 3.803(10) 1.686(1) 0.63(7) 5.79(1)
V231 −1.04(11) −0.93(8) −0.39(16) 0.619(1) 3.801(27) 1.652(1) 0.60(19) 5.63(1)
V24 −1.21(3) −1.09(2) −0.78(4) 0.569(3) 3.812(8) 1.672(1) 0.66(6) 5.48(1)
Weighted Mean −1.25 −1.13 −0.82 0.551 3.812 1.679 0.66 5.64
σ ± 0.02 ± 0.02 ± 0.06 ± 0.039 ± 0.003 ± 0.001 ± 0.01 ± 0.08
RRc stars
V18 −1.17(23) −1.05(19) −0.97(10) 0.476(11) 3.866(1) 1.710(4) 0.51(1) 4.45(2)
V19 −1.64(45) −1.58(52) −1.60(16) 0.355(10) 3.857(1) 1.758(4) 0.52(1) 4.90(2)
V20 −0.96(19) −0.87(13) −0.87(7) 0.650(5) 3.878(1) 1.640(2) 0.61(1) 3.90(1)
Weighted Mean −1.10 −0.95 −0.96 0.572 3.869 1.671 0.56 4.09
σ ± 0.16 ±0.17 ± 0.19 ±0.070 ± 0.005 ± 0.028 ± 0.03 ± 0.24
1 Not included in the [Fe/H] averages.
Bustos Fierro et al. (2019) which uses proper motions of the stars and is outlined in Appendix C. For this GC,
Gaia detected 60 447 stars in a region of 8.5’ radius around the center of the cluster, out of which only 1529
seem to be members (Fig. 9.2). Of these 1529 likely members, we possess light curves for 1100 stars, that were
used to plot the CMD in Fig. 9.3.
9.6 The color-magnitude diagram of NGC 6712 and its HB
The location of NGC 6712 near the Galactic bulge, causes that its CMD is highly contaminated with field stars
(Fig. 9.3, left panel). The method described in Appendix C, allowed us to remove the field stars to produce a
much cleaner CMD (Fig. 9.3, right panel). Once the CMD was cleaned, we overlaid three isochrones; one with
a metallicity of [Fe/H]ZW = −1.25 (blue), which comes from the value obtained from the Fourier decomposition
of the light curves of the RRL stars, another with the metallicity of [Fe/H]UVES = −1.0 (red) reported by Harris
(1996) and one with [Fe/H]UVES = –0.86 (green) derived from the equations of Nemec et al. (2013), given in
Chapter 5. We shifted the isochrones to a distance of 8.1 kpc and reddened them by E(B − V) = 0.35. They
correspond to an age of 12 Gyrs. As with the isochrones, we also overlaid a ZAHB of metallicity [Fe/H]=-1.31
(VandenBerg et al., 2014) and shifted them to a distance of 8.1 kpc and reddened them by E(B − V) = 0.35.
The resulting positions of the isochrones and the ZAHB are consistent with our CMD.
Once the CMD has been cleaned from field stars, it displays a prominent red HB and a sparsely populated blue
tail which is typical of OoI-type clusters. The RRL population on the HB is dominated by fundamental mode
RRab pulsators (10) in comparison with first overtone RRc pulsators (4). The distribution of RRab and RRc
shows a clear mode segregation around the FORE.
96 Chapter 9. NGC 6712: The variable star population of a tidally disrupted globular cluster
Figure 9.2: Left panel: Proper motions map of the stars within a 8.5’ radius. Grey dots are field
stars and red dots are cluster members. Right panel: Projection on the sky of the proper motion
vectors. Grey arrows are used for member stars, red arrows for known cluster variables, blue
arrows for new variable stars reported in this work, red open squares for known field variables
and blue open squares for new field variables. The yellow grid corresponds to the Galactic l and
b coordinates. The large black arrow indicates the direction of the Galactic center. Vectors have
been enlarged 20000x for visualization purposes (Deras et al., 2020).
Table 9.4: Distance comparison to NGC 6712 from the different methods used in this work (Deras
et al., 2020).
Method Distance [kpc]
RRab Fourier decomposition 8.1 ± 0.3
RRc Fourier decomposition 8.0 ± 0.3
RRab / RRc I-band P-L 8.3 ± 0.3
9.6. The color-magnitude diagram of NGC 6712 and its HB 97
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
V − I
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
V
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
V − I
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
V
V2 V8
V10 V15
V16
V17
V21
V27
V28
V29
V30
V31
V32
V33V34
V35
V36
FV1
FV2
FV3
FV4
FV5
FV6
FV7
FV8
FV9
C
NGC 6712
0.6 0.7 0.8 0.9 1.0 1.1
V − I
15.8
16.0
16.2
16.4
16.6
V
V1
V3
V4
V4
V5
V5V6
V6
V12
V13
V18
V19
V20
V20
V22
V22
V23
V24
V26
V26
Figure 9.3: Color-Magnitude Diagrams of NGC 6712 in VI filters. The left panel shows the CMD
with all the measured stars in our FoV (black dots) and the likely star members (red dots). The
right panel shows the likely star members once the CMD has been cleaned of field stars. The right
panel also shows the known and newly discovered variables. The blue circles correspond to RRab
stars, the green circles to RRc stars, the red circles to long-period stars and the orange circles to
EW stars. Triangular markers are newly discovered stars. Markers with a black rim represent
non-member variable stars. The cyan square represents a candidate variable star. The isochrones
in the right panel are 12 Gyrs interpolations for [Fe/H] values −1.25 (blue), −1.0 (red) and −0.86
(green). All three isochrones were created using Y=0.25 and [α/Fe]=+0.4, from the model grid of
VandenBerg et al. (2014). The red box delimits the region where blue straggler stars are found.
The lower panel shows an expansion of the HB for clarity purposes. The open blue and green
circles correspond to the original position of the RRL stars before their magnitude and color were
corrected due a contaminating neighbouring star. The red tags accompany the new position after
the correction due to the presence of unseen companions within our PSF but that were detected
by Gaia. The dashed vertical line represents the FORE of the instability strip as estimated by
Arellano Ferro et al. (2016) with its uncertainty in grey (Deras et al., 2020).
98 Chapter 9. NGC 6712: The variable star population of a tidally disrupted globular cluster
9.7 Contributions made by the student to this article
• I cleaned, reduced and analyzed over 1100 science images of the GC.
• I obtained the instrumental magnitudes and transformed them to the standard system to obtain the light
curves of the star present in the GC.
• I conducted a search for variable stars.
• I estimated the reddening of the GC.
• I estimated the Fourier coefficients and the physical parameters related to them.
• I estimated the distance to the cluster using different types of variable stars.
• I made Figures. 1, 4, 5, 6, 7, 9 and 10.
• I made Tables 1, 3, 4, 5, 6, 7, 8 and 9.
• I wrote Sections 2, 2.1, 2.2, 3, 4.1, 4.2, 4.2.1, 5, 7.1, 7.2, 7.4 and 8.
• I co-wrote Sections 6, 8.1, 9 and 10.
MNRAS 493, 1996–2014 (2020) doi:10.1093/mnras/staa196
Advance Access publication 2020 January 22
NGC 6712: the variable star population of a tidally disrupted
globular cluster
D. Deras ,1‹ A. Arellano Ferro,1‹ C. Lázaro,2,3 I. H. Bustos Fierro ,4
J. H. Calderón,4,5 S. Muneer6 and Sunetra Giridhar6
1Instituto de Astronomı́a, Universidad Nacional Autónoma de México, Ciudad Universitaria, C.P. 04510 Ciudad de México, México
2Departamento de Astrofı́sica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
3Instituto de Astrofı́sica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain
4Observatorio Astronómico, Universidad Nacional de Córdoba, X5000IND Córdoba, Argentina
5Consejo Nacional de Investigaciones Cientı́ficas y Técnicas (CONICET), X5000BGR Buenos Aires, Argentina
6Indian Institute of Astrophysics, Koramangala 560034, Bangalore, India
Accepted 2020 January 20. Received 2020 January 8; in original form 2019 November 7
ABSTRACT
We present an analysis of VI CCD time-series photometry of globular cluster NGC 6712. Our
main goal is to study the variable star population as indicators of the cluster mean physical
parameters. We employed the Fourier decomposition of RR Lyrae light curves to confirm
that [Fe/H]UVES = −1.0 ± 0.05 is a solid estimate. We estimated the reddening to the cluster
as E(B − V) = 0.35 ± 0.04 from the RRab stars colour curves. The distance to the cluster
was estimated using three independent methods which yielded a weighted mean distance
<d > = 8.1 ± 0.2 kpc. The distribution of RRab and RRc stars on the horizontal branch shows
a clear segregation around the first overtone red edge of the instability strip, which seems to
be a common feature in OoI-type cluster with a very red horizontal branch. We carried out a
membership analysis of 60 447 stars in our field of view (FoV) using the data from Gaia-DR2
and found 1529 likely members; we possess the light curves of 1100 among the member stars.
This allowed us to produce a clean colour–magnitude diagram, consistent with an age of 12
Gyr, and enabled us to discover close unresolved contaminants for several variable stars. From
the proper motion analysis, we found evidence of non-member stars in the FoV of the cluster
being tidally affected by the gravitational pull of the bulge of the Galaxy. We found that the
RRab variable V6, shows a previously undetected Blazhko effect. Finally, we report 16 new
variables of the EW-type (9) and SR-type (7).
Key words: stars: fundamental parameters – stars: variables: RR Lyrae – globular clusters:
individual: NGC 6712.
1 IN T RO D U C T I O N
NGC 6712 is a small, sparse and metal-rich globular cluster
([Fe/H] = −1.02; Harris 1996) that can be found behind the Scutum
stellar cloud (α = 18h53
′
04.′′32, δ = −08◦42
′
21.′′5, J2000), and is
located in the populated region of the Galactic bulge (l = 25.35◦,
b = −4.32◦). Two versions of the Galactic orbit of the cluster can
be seen in fig. 4 of Allen (1990) and in fig. 18 of de Souza (2017).
The orbit is asymmetrical, it is confined to within 2.0 kpc from the
Galactic disc and penetrates deep into the Galactic centre, reaching
a Galactocentric distance of 0.2–0.3 kpc (Dauphole et al. 1996; de
Marchi et al. 1999). A recent estimation of the perigalacticon from
Gaia-DR2 proper motions is 0.45 ± 0.10 kpc (Baumgardt et al.
⋆ E-mail: dderas@astro.unam.mx (DD); armando@astro.unam.mx (AAF)
2019). According to Cudworth (1988), the latest Galactic plane
crossing could have happened as recently as ∼4000 yr ago, which
is much smaller than its half-mass relaxation time of 1 Gyr (Harris
1996). Hence, repeated visits to the bulge and constant interaction
with the disc in its 12 Gyr, makes it very likely that this cluster has
experienced several and strong tidal disruptions (Andreuzzi et al.
2001; Paltrinieri et al. 2001).
Theoretical calculations of the mass function lead to the suspicion
that the ∼105 M⊙ (Pryor & Meylan 1993) cluster we see today is
a remnant of the otherwise much more massive (∼107 M⊙) cluster.
This enormous mass-loss may be reflected in the proper motions
distribution, hence it is an added interest to explore the membership
status of the stars in the field of the cluster and their motions in the
light of the Gaia-DR2 data.
The variable star population of the cluster includes RR Lyrae,
long-period semiregular variables, short-period eclipsing binaries,
C© 2020 The Author(s)
Published by Oxford University Press on behalf of the Royal Astronomical Society
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NGC 6712: the variable star population 1997
and an X-ray source whose counterpart presents optical variability
(Homer et al. 1996; Pietrukowicz & Kaluzny 2004). The Catalogue
of Variable Stars in Globular Clusters (CVSGC; Clement et al.
2001), in its 2015 edition, lists 28 confirmed variables and sum-
marises the history of their discovery between 1917 and 2004. The
potential of variable stars to estimate mean metallicity, reddening,
and distance for the globular cluster is well known. High-quality
CCD photometry of the stars in the field of view (FoV) of the cluster
provide insights, not only on the pulsating properties of individual
variables and potential discovery of new ones, but also enables the
discussion of the colour–magnitude diagram (CMD) structure and
its relation with the stellar evolution patterns and age of the system.
In this study, we propose to undertake a VI CCD time-series analysis
of NGC 6712.
The paper is structured as follows: in Section 2, we detail the
observations and reduction process of our images; in Section 3, we
perform a membership analysis of the stars in our FoV to the cluster
based on the Gaia-DR2 proper motions; in Section 4, we describe
our systematic search for variables and report new discoveries; in
Section 5, we describe the Fourier approach to estimate the physical
parameters of the RR Lyrae stars; in Section 6, we discuss the
metallicity of the cluster employing alternative Fourier calibrations;
in Section 7, we present the cluster distance estimations by several
independent methods; in Section 8, we describe the structure of
the CMD and the Horizontal Branch (HB), obtained from our
photometry; in Section 9, we briefly comment about the interaction
of the cluster with the Galactic bulge; and in Section 10, we
summarise our conclusions.
2 DATA , O B S E RVAT I O N S , A N D R E D U C T I O N S
The data used in this work were obtained from two sites. The
first set of data was obtained with the 0.80 m IAC80 telescope
at the Observatorio del Teide (Tenerife, Spain), during 10 non-
consecutive nights, between 2016 June 13th and July 16th. We
used the CAMELOT camera with 2048 × 2048 pixels and 0.304
arcsec pixel−1, with a 10.4 × 10.4 arcmin2 FoV, and a back
illuminated detector CCD42-40 from E2V Technologies. The
second set of data was obtained using the 2 m telescope at the
Indian Astronomical Observatory (IAO) in Hanle, India on 12
nights separated into four intermittent seasons between 2011 June
and 2018 April. The detector used was a SITe ST-002 thinned
backside illuminated CCD of 2048 × 2048 pixels with a scale of
0.296 arcsec pix−1, which translates into a FoV of approximately
10.1 × 10.1 arcmin2. The log of our observations is given in
Table 1.
2.1 Difference image analysis
The technique of difference imaging analysis (DIA) with its pipeline
implementation DanDIA1 (Bramich 2008; Bramich et al. 2013) was
used for the reduction of our data. This allowed us to obtain high-
precision photometry for all the point sources in the FoV of the
CCD. The software first creates a reference image by stacking the
best images in each filter and then it subtracts them from the rest
of the images in our collection. Using the point spread function
(PSF) calculated by DanDIA from a sample of 300–400 isolated
stars, we can determine the differential flux for each point source in
1DanDIA is built from the DanIDL library of IDL routines available at
http://www.danidl.co.uk.
Table 1. The distribution of observations of NGC 6712 for each filter.
Columns NV and NI give the number of images taken with the V and I
filters, respectively. Columns tV and tI provide the exposure time, or range
of exposure times employed during each night for each filter. The average
seeing is listed in the last column.
Date NV tV (s) NI tI (s) Avg. Sitea
seeing
(arcmin)
2011-06-10 8 70–100 8 10 1.5 H
2011-06-11 7 70–150 10 15–40 2.1 H
2011-08-05 15 120–150 16 10–25 1.6 H
2011-08-06 16 110–200 17 12–30 1.7 H
2013-07-30 20 15–20 19 3–5 1.7 H
2015-03-10 4 45 6 10 2.3 H
2015-03-11 6 45 4 10 2.6 H
2015-03-12 4 45 4 10 2.2 H
2015-03-26 6 45 6 10 2.5 H
2015-03-27 9 45–60 12 10–12 1.8 H
2016-06-13 9 600–800 9 400–500 1.6 I
2016-06-15 23 80–100 22 20–30 2.0 I
2016-06-26 21 80–100 23 20–50 2.0 I
2016-06-27 89 40–80 88 10–20 1.2 I
2016-06-28 2 40–50 3 15 2.0 I
2016-06-29 40 20 45 60 2.8 I
2016-07-02 100 40–60 99 10–20 1.5 I
2016-07-03 94 20–80 90 5–20 1.2 I
2016-07-04 78 13–60 74 10–20 1.7 I
2016-07-16 24 50–80 24 13–30 1.8 I
2018-04-08 4 45 6 15 2.2 H
2018-04-08 5 20 6 7 2.4 H
Total: 584 – 591 – – –
Note. aH: Hanle; I: IAC.
the FoV. Then, the differential fluxes are converted into total fluxes.
The total flux ftot(t) in ADU/s at each epoch t can be estimated
as
ftot(t) = fref +
fdiff(t)
p(t)
, (1)
where fref is the reference flux (ADU/s), fdiff(t) is the differential
flux (ADU/s), and p(t) is the photometric scale factor (the integral
of the kernel solution). Conversion to instrumental magnitudes was
achieved using
mins(t) = 25.0 − 2.5 log
[
ftot(t)
]
, (2)
where mins(t) is the instrumental magnitude of the star at time t. A
more detailed description of this method can be found in Bramich
et al. (2011).
Systematic errors in our DIA photometry may be introduced
due to a possible error in the flux-magnitude conversion factor
(Bramich et al. 2015). To investigate their significance, we applied
the methodology developed in Bramich & Freudling (2012) to
solve for the magnitude offsets Zk that should be applied to each
photometric measurement from the image k. We found, however,
that in the present case the necessary corrections to our photometry
were negligible (<0.001 mag).
2.2 Transformation to the standard system
In order to transform our light curves from the v instrumental
magnitudes to the Johnson–Kron–Cousins standard V system, we
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Figure 1. Transformation relations obtained for the V and I filters between
the instrumental and the standard photometric systems. To carry out the
transformation, we made use of a set of 94 and 91 standard stars for IAC
(top panel) and Hanle (bottom panel), respectively, in the field of NGC 6712.
The Hanle I-band colour term is not significant, then a transformation of the
form I = 1.024(± 0.006)i − 1.354(± 0.093) was employed.
made use of the standard stars in the field of NGC 6712 which are
included in the online collection of Stetson (2000).2 These stars
are generally distributed in the cluster periphery. Unfortunately,
this collection does not have the standard values for the I filter. Dr.
Raúl Michel from the Observatorio Astronómico Nacional, México,
made available to us his bulky observations of stars in the field of
NGC 6712, that have been transformed into the I-standard system
using the equatorial standards of Landolt (1983). We used 94 of
these stars as standards to transform our IAC CCD photometry.
For the Hanle set of data, we identified 91 standard stars. The mild
colour dependence of the standard minus instrumental magnitudes is
shown in Fig. 1 for our observations. The transformation equations
in both VI filters are explicitly given in the figure itself. We note
that for the Hanle I-band transformation, the colour term is not
significant and found that a linear transformation of the form I =
1.024(± 0.006)i − 1.354(± 0.093) matches better the IAC and the
Hanle I light curves. The time-series VI photometry obtained in
this work is reported in Table 2, of which only a small portion is
included in the printed version of the paper. The full table shall be
2http://www3.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/STETSON/st
andards
Table 2. Time-series VI photometry for the variables stars observed in this
work.a
Variable Filter HJD Mstd mins σm
star ID (d) (mag) (mag) (mag)
V1 V 2455723.42031 16.611 17.723 0.006
V1 V 2455723.42540 16.601 17.714 0.007
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
V1 I 2455723.41738 15.622 16.608 0.008
V1 I 2455723.42325 15.631 16.617 0.012
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
V2 V 2457555.59991 12.610 15.120 0.001
V2 V 2457555.60211 12.611 15.121 0.001
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
V2 I 2457555.60101 10.564 13.386 0.001
V2 I 2457555.60322 10.553 13.375 0.001
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Note. aThe standard and instrumental magnitudes are listed in columns 4
and 5, respectively, corresponding to the variable stars in column 1. Filter
and epoch of mid-exposure are listed in columns 2 and 3, respectively. The
uncertainty on mins is listed in column 6, which also corresponds to the
uncertainty on Mstd. A full version of this table is available at the CDS data
base.
available in electronic form in the Centre de Donnés astronomiques
de Strasbourg data base (CDS).
3 STA R MEMBERSHIP USING G AIA
To determine cluster membership of the stars in our FoV, we
used the high-quality astrometric data available in Gaia-DR2 (Gaia
Collaboration 2018). We employed the procedure developed by
Bustos Fierro & Calderón (2019), which consists of two stages: the
first stage is based on the Balanced Iterative Reducing and Cluster-
ing using Hierarchies (BIRCH) algorithm (Zhang, Ramakrishnan &
Livny 1996) in a four-dimensional space of physical parameters –
gnomonic projection of celestial coordinates and proper motions –
that detects groups of stars in this 4D space; in the second stage an
analysis of the projected spatial distribution of stars with different
proper motions allows the extraction of most of the members in
the outskirts of the cluster or with large proper motions dispersion.
Finally, it is checked that the extracted cluster members have their
positions in celestial coordinates, the vector point diagram (VPD)
of the proper motions, and the CMD (of Gaia photometric system)
consistent with the characteristics of a globular cluster. We used a
tidal radius of 8.5 arcmin from Harris (1996, 2010 edition) to set
a boundary for the cluster and found 60 447 measured stars which
we then cross-matched with our own data set. We found 1529 likely
cluster members, for 1100 of which we possess light curves. In
Fig. 2, we display the VPD resulting from this membership analysis.
The axes in the VPD correspond to the components of the proper
motion vectors, therefore every proper motion is represented by
a point. Since the cluster members share a common motion, they
appear as a small concentration different from the wider distribution
of field stars. Further comments on the cluster dynamics will be
given later in Section 9.
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NGC 6712: the variable star population 1999
Figure 2. Left-hand panel: Vector point diagram of the stars within a 8.5 arcmin radius. Grey dots are field stars and red dots are cluster members. Right-hand
panel: Projection on the sky of the proper motion vectors. Grey arrows are used for member stars, red arrows for known cluster variables, blue arrows for new
variable stars reported in this work, red open circles for known field variables, and blue open circles for new field variables. The yellow grid corresponds to
the Galactic l and b coordinates. The large black arrow indicates the direction of the Galactic centre. Vectors have been enlarged 20 000× for visualisation
purposes. See Section 9 for further discussion.
4 VA R I A B L E S TA R S I N N G C 6 7 1 2
4.1 Search for new variables
In Fig. 3 we present an identification chart with all known and newly
found variables at present, including those that may not be cluster
members.
In order to identify the variable star population in NGC 6712, we
used several methods that we describe below. Our first approach was
the use of the string-length method (Burke, Rolland & Boy 1970,
Dworetsky 1983). We phased each light curve in our data with
periods between 0.02 and 1.7 d in steps of 10−6 d. In each case,
the string-length parameter SQ was calculated. The best phasing
occurs when SQ is minimum, and corresponds to the best period
in our data. We then created a plot of the obtained minimum SQ
versus X-coordinate in our reference image for each light curve in
our collection (Fig. 4). All variables in Tables 3 and 4 are identified.
We note that most of the variables are located below an arbitrary
threshold at 0.4, hence we individually explored each light curve
below this value. Using this method, we identified two new cluster
variables (V30 and V31) and seven new field variables (FV), FV1–
FV7, all of which seem to be contact binaries or of the EW-type.
The second method was to explore the regions of the CMD where
variable stars are expected to be found. In particular, we explored
the stars near the tip of the red giant branch (TRGB) and we found
four new semiregular or SR variables (V32–V35), an SR-type field
variable (FV8), and an SR candidate (which we will denote as C)
for which we are unsure of its variation.
The last method used was to plot the rms versus mean magnitude
of all the stars in our FoV (Fig. 5). We performed the search in
regions where variable stars such as RR Lyrae- and SR-type stars
are usually located. With this approach, we identified two more
SR-type variable stars (V36 and FV9).
In Fig. 3, we present an identification chart with all known and
newly found variables at present, including those that may not be
cluster members. It is important to mention that in our data set
we were not able to recover the light curves of V9 and V14 since
they are out of our FoV, V11 since it lies near the lower border
of our reference image and was not measured, and V25 since it
is below the limit of our photometry. Notice that V25 is an X-ray
source (LXMB), with a faint optical counterpart. We also found
that several of the known variable stars are non-cluster members,
namely, V9, V11, V14, V16, V17, V27, V28, and V29, according
to the method used to determine stellar membership described in
Section 3.
4.2 RR Lyrae stars in NGC 6712
In the 2015 edition of the CVSGC, there are listed eight RRab, six
RRc, and one RR Lyrae without mode classification. The present
data and a careful inspection of the resulting light curves, made
necessary it to reclassify some stars, namely V22, V23, and V27,
as RRab, RRab, and EW, respectively. The details of these new
classifications are explained in Appendix A. As a result, the RR
Lyrae population of NGC 6712 is formed by 10 RRab and 4
RRc stars. Our observations generally cover the light curves phase
reasonably well, with the exception of V12, the reason being its
nearly half-day periodicity that operates against a better phase
coverage. In order to confirm its light-curve shape and pulsation
mode type, we complemented our V data with V data of Sandage,
Smith & Norton (1966). While the scatter of these observations
is large, they confirm the RRab-type nature of this star. In spite
of this, V12 shall not be considered for the physical parameters
calculations. With the inclusion of the data from Hanle, we were
also able to detect a mild modulation in the amplitude of the light
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Figure 3. Identification chart for all known and newly discovered variables in NGC 6712. The bottom two diagrams show the central regions of the cluster and
some stars are identified in more than one panel: The field sizes of the three panels are 10.4 × 10.4, 3.3 × 3.3, and 1.3 × 1.3 arcmin2, respectively. Variables
labelled ‘FV’ are field stars according to our proper motion analysis (see Section 3). The marker V25 shows no star but this is an X-ray source (LMXB) (Homer
et al. 1996) not visible in our images.
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NGC 6712: the variable star population 2001
curve of V6. This suggests the presence of a previously undetected
Blazhko effect. The light curves of all RR Lyrae are shown
in Fig. 6.
4.2.1 Bailey diagram and Oosterhoff type
The period–amplitude plane for RR Lyrae stars, also known as the
Bailey diagram, is a useful tool as it clearly segregates pulsation
modes, helps defining the Oosterhoff type of a given cluster, and
identifies RRab stars that may be advanced in their evolution
towards the asymptotic giant branch (AGB). The diagram for NGC
6712 is shown in Fig. 7 for the VI band passes. The periods and
amplitudes are listed in Table 3. The amplitudes were measured
from the corresponding fit provided by the Fourier decomposition
of their light curves (see Section 5). The average of the periods
of the RRab stars in NGC 6712 is <Pab > = 0.58 ± 0.02, which
classifies it as a OoI-type cluster. The RRab and RRc stars in this
diagram occupy the loci that has been observed to be consistent
with the definition of a OoI-type globular cluster (black solid lines
in V and I filters, respectively). V6 is classified as an RRab-type star,
nevertheless, it is located at an odd position on the Bailey diagram,
which can be explained due to its Blazhko nature. We note that V5 is
located along the evolved star sequence, however, its position of the
CMD lies very close to the Zero Age Horizontal Branch (ZAHB)
and without a formal analysis of a possible secular period change,
we cannot state its evolutionary status relative to the rest of the RR
Lyrae stars.
4.3 Semiregular variable stars in NGC 6712
There are seven known semiregular or long-period variables, clas-
sified as SR or L in the CVSGC, namely, V2, V7, V8, V10, V14,
V15, and V21. Unfortunately V14 lies outside our FoV and was
not measured. Without a sufficiently long time base and dense
data set, the classification of long-term semiregular variables is
always dubious. Our Hanle and IAC data sets span 8 yr, which
enables to peer the long-term light behaviour. On the other hand,
the more continuous pace of the observations at the IAC, enabled
us to analyse even the short-term behaviour. Additionally, we found
seven semiregular variables and have tagged them as SR-type stars.
Five of these stars are cluster members (V32–V36) and two are not
(FV8 and FV9). In Fig. 8, the light curves of the above stars are
displayed. The insets in the figure show the short-term light-curve
behaviour and sometimes suggest a periodicity. A proper period
search in all these light curves allowed us to find reliable periods
in five of these stars, listed in Tables 3 and 4. The corresponding
phased light curves are displayed in Fig. A1
4.4 Eclipsing binaries in the field of NGC 6712
The catalogue of Clement et al. (2001) lists two W Ursae
Majoris-type binaries or EW stars (V28 and V29), first reported
by Pietrukowicz & Kaluzny (2004). We were able to recover their
light curves from our data and to identify nine new eclipsing binaries
present in our FoV. The V and I light curves are shown in Fig. 9.
The light curves present a morphology typical of the W UMa
contact systems, except V30 which presents a steeper descent to
minimum and is probably not a contact binary. V27 was previously
misidentified as a RRc star (see Section A6).
From the study of cluster membership using Gaia-DR2, most of
the new binaries seem to be field stars, and so their reddening and
Figure 4. Minimum value for the string-length parameter SQ calculated for
the 11 294 stars with a light curve in our V reference image, versus CCD X-
pixel coordinate. The blue circles correspond to an RRab stars, green circles
to RRc stars, red circles and triangles correspond to semiregular variables,
and orange circles and triangles to EW stars. The cyan square denotes an SR
variable candidate. Triangles are used for the newly discovered variables.
The dashed blue line is an arbitrary threshold set at 0.4, below which most
of the known variables are located. See Section 4.1 for a discussion.
Figure 5. The rms magnitude deviations as a function of the mean
magnitudes V and I. The blue circles correspond to an RRab stars, green
circles to RRc stars, red circles correspond to semiregular variables, and
orange circles and triangles to EW stars. The cyan square denotes an SR
variable candidate. Triangles are used for the newly discovered variables.
MNRAS 493, 1996–2014 (2020)
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Figure 6. RR Lyrae stars in NGC 6712. Black and cyan symbols correspond to observations from IAC and Hanle, respectively. The red line represents the
Fourier fit. Note that the scale in the Y-axis is different for RRab and RRc stars. The grey dots in V12 come from Sandage et al. (1966) and were used to
complete our light curve in the V band (see Section 5).
distances are unknown. In order to get some information from their
photometric light curves, we have assumed they are contact systems.
Based on Gaia-DR1, Chen et al. (2018) derived period–luminosity
relations for W UMa-type systems, of the form Mλ(max)C =
aλ log P + bλ, claiming 8 per cent distance accuracy, where P is
the orbital period. We have used the 3D Dust Mapping (Green et al.
2014; Green 2019)3 facility to get the increase of E(B − V) with dis-
tance, looking for the best agreement between Mλ(max)C and Mλ =
mλ(max) − Aλ − 5 log d(pc) + 5, for each one of the systems in
the VI bands. The E(B − V) in the direction of the cluster changes
from ∼0.13 at 2 kpc, to ∼0.37–0.40 at 7.9 kpc. The results are
given in the columns 2 and 3 of Table 5, and are consistent with the
3http://argonaut.skymaps.info
Gaia-DR2 membership study, supporting the hypothesis that most
of them are closer than the cluster.
With a value for E(B − V), we can derive the intrinsic colours
(V − I)o of the binaries, used to estimate the effective temperatures
of the components from the calibration (V − I)o − Teff of Huang
et al. (2015), and the relative contribution of each component to
the total flux in V and I, calculated in the light-curves fits. The VI
light curves have been modelled with the code BINAROCHE (Lázaro,
Arévalo & Almenara 2009), weighting in the fits the deviations of
both components from the mass–radius relations of Awadalla &
Hanna (2005) for W UMa binaries. The derived parameters of the
fits are given in Table 5, while the observed and model light curves
are shown in Fig. 9. The collected data of V30 cover only one
of the eclipses, which has been arbitrarily considered to be the
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NGC 6712: the variable star population 2003
Figure 7. Bailey diagram for NGC 6712. Filled black and red symbols
represent RRab and RRc stars, respectively. The open symbols denote the
original position of the RR Lyrae before a correction to their amplitude
was performed due to contamination of a neighbour star. The triangular
marker (V6) is a Blazhko RRab star. The continuous and dashed black lines
in the top panel are the loci for unevolved and evolved stars according to
Cacciari, Corwin & Carney (2005). The red parabolas were calculated by
Arellano Ferro et al. (2015) from RRc stars in five OoI clusters. In the
bottom panel, the black solid and segmented loci for unevolved and evolved
stars, respectively, are from Kunder et al. (2013). See Section 4.2.1 for a
more detailed discussion.
primary, with a steeper descent than other light curves, and could
be a detached or semidetached binary.
Given the assumptions implicit in the analysis, and the quality of
the light curves, we prefer not to put an error bar in the tabulated
values. A realistic estimation of the uncertainties can be of about ±
300 K in the Teff,1 effective temperatures, affected by the E(B − V)
used to deredden the observed (V − I) colour. The ratios T2/1 =
Teff,2/Teff,1 and q = M2/M1 are determined by the photometric quality
of the light curves, and its uncertainty can be of about ±0.02. The
most uncertain parameter is the primary mass M1, but with these
values the models put the systems at distances similar to those given
in column 3.
5 R R LYRAE STARS: [FE/H] AND MV F RO M
L I G H T- C U RV E F O U R I E R D E C O M P O S I T I O N
The light curves of RR Lyrae stars exhibit periodic oscillations and
are hence well represented by a Fourier series of the following
form:
m(t) = A0 +
N
∑
k=1
Akcos
(
2
P
k(t − E0) + φk
)
, (3)
where m(t) is the magnitude at time t, P is the period of pulsation,
and E0 is the epoch. To estimate the Fourier parameters, we followed
a least-squares approach, where the quantities to be minimized are
the amplitudes Ak and phases φk of the light-curve components. The
Fourier parameters, i.e. the amplitudes and phases of the harmonics
in equation 3 are defined as Rij = Ai/Aj and φij = jφi − iφj,
respectively. The estimated Fourier coefficients for the RR Lyrae
stars in our data are listed in Table 6.
We have employed the calibrations of Jurcsik & Kovács (1996,
hereafter JK96) and Kovács & Walker (2001) to calculate the
metallicity and absolute magnitude of the RRab stars. The specific
set of equations used in this work is listed below.
[Fe/H]J = −5.038 − 5.394 P + 1.345 φ
(s)
31 , (4)
MV = −1.876 log P − 1.158 A1 + 0.821 A3 + K. (5)
These calibrations have an associated uncertainty of 0.14 dex
and 0.04 mag for the metallicity and the absolute magnitude,
respectively. Since the metallicity is given in the Jurcsik–Kovács
scale, we can transform it to the standard Zinn–West scale (Zinn &
West 1984) with the following equation: [Fe/H]J = 1.431[Fe/H]ZW
+0.88 (Jurcsik 1995). For equation (5), we adopted the zero-point
K = 0.41 from the analysis of Arellano Ferro, Giridhar & Bramich
(2010).
For the RRc stars, we adopted the calibrations given by Morgan,
Wahl & Wieckhorst (2007) and Kovács & Kanbur (1998), respec-
tively
[Fe/H]ZW = 52.466 P 2 − 30.075 P + 0.131 φ
2(c)
31
− 0.982 φ
(c)
31 − 4.198 φ
(c)
31 P + 2.424, (6)
MV = 1.061 − 0.961 P − 0.044 φ
(s)
21 − 4.447 A4. (7)
These calibrations have an associated uncertainty of 0.14 dex
and 0.042 mag for the metallicity and the absolute magnitude,
respectively. One can transform the coefficients from cosine series
phases into sine series using the following relation, if needed:
φ
(s)
jk = φ
(c)
jk − (j − k)

2
. (8)
The Fourier coefficients of the RR Lyrae stars identified in
NGC 6712 are listed in Table 6, and in Table 7 we report the
physical parameters derived from the Fourier decomposition. In
their paper, JK96 propose a test that validates the usability of the
Fourier coefficients obtained from the Fourier decomposition of
the light curves of RRab stars. They suggest that the values of the
iron abundance [Fe/H] obtained are only valid if their deviation
parameter or Dm does not exceeds the value of 3.0. The Dm value
for each RRab star is listed in Table 6 in column 10.
The iron abundance in the scale of Zinn & West (1984) can be
converted into the high-dispersion spectroscopy (HDS) or UVES
scale of Carretta et al. (2009) via the equation [Fe/H]UVES =
−0.413 + 0.130[Fe/H]ZW − 0.356[Fe/H]2
ZW, and are also listed
in Table 7.
The values of MV reported in Table 7 have been transformed to
luminosities using the following equation:
log(L/L⊙) = −0.4
(
MV − M⊙
bol + BC
)
. (9)
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Table 3. Data of member variable stars in NGC 6712 in the FoV of our images.
Star ID Type <V > <I > AV AI P HJDmax α (J2000.0) δ (J2000.0) Gaia-DR2 Source
(mag) (mag) (mag) (mag) (days) +2450000
V1 RRab 16.31 15.40 1.11 0.75 0.512039 7574.6054 18:52:59.85 −08:42:31.3 4203848946351463808
V2 SR 12.55 10.50 >0.15 >0.07 109.0a – 18:53:08.79 −08:41:56.8 4203849393028208384
V3 RRab 16.24 15.26 0.57 0.38 0.655956 7574.6678 18:53:02.29 −08:43:47.4 4203848877631995136
V4 RRab 16.47 15.41 0.56 0.36 0.611745 7574.6268 18:53:16.29 −08:42:38.1 4203849122517862144
V5 RRab 16.41 15.44 1.14 0.74 0.545362 7572.6357 18:53:08.68 −08:43:24.1 4203848778920519168
V6 RRab Bl 16.45 15.56 0.86 0.59 0.510871 7553.6906 18:53:05.35 −08:42:53.6 4203848985005261312
V7 M 16.98g 11.97g >0.62 >0.23 193.0c – 18:52:55.39 −08:42:32.6 4203849672273972864
V8 SR 13.17g 10.90g >0.48 >0.28 116.29c – 18:53:05.66 −08:41:12.4 4203849461747677056
V10 L 13.94g 11.27g >0.21 – – – 18:52:57.35 −08:41:43.9 4203849874064515456
V12 RRab 16.30 – 1.21 – 0.502790 5284.7907 18:53:06.02 −08:41:33.4 4203849088158338944
V13 RRab 16.13 15.26 1.02 0.66 0.562655 7574.6637 18:52:57.77 −08:41:46.5 4203849809712937088
V15 L 13.83g 11.08g >1.5 – – – 18:53:20.99 −08:42:48.2 4203837401469330944
V18 RRc 16.15 15.32 0.47 0.28 0.353543 7569.5876 18:53:02.44 −08:42:14.0 4203849053798612096
V19 RRc 16.00 15.19 0.40 0.24 0.412161 7586.6633 18:53:03.22 −08:41:39.2 4203849844072610816
V20 RRc 16.20 15.48 0.34 0.25 0.250521 5779.3246 18:53:04.16 −08:42:03.7 4203849083790434304
V21 L 13.58g 11.26g >0.33 – – – 18:52:58.80 −08:42:06.2 4203849805344920576
V22 RRabd 16.17 15.24 0.58 0.41 0.654789 7586.6187 18:52:59.08 −08:41:20.5 4203849874064518016
V23 RRabe 16.23 15.26 0.31 0.19 0.642451 7572.6241 18:53:03.83 −08:42:39.4 4203848985005549056
V24 RRab 16.23 15.25 0.99 0.61 0.576454 7586.6334 18:53:04.37 −08:42:26.9 4203849053725057280
V26 RRc 16.16 15.64 0.12 0.09 0.334317 7569.6551 18:53:06.80 −08:41:31.2 4203849466042002944
V28 EW 18.23 17.42 0.26 0.08 0.434497 7573.6734b 18:53:08.09 −08:40:52.2 4203849466042155264
V29 EW 18.45 16.95 – – 0.453571 7569.6226b 18:53:04.17 −08:42:25.0 4203849053798521472
V30f EW 18.68 17.91 0.61 0.49 0.504748 7567.6482b 18:53:02.51 −08:43:11.9 4203848881926249600
V31f EW 18.53 17.74 0.40 0.38 0.418762 7573.6734b 18:53:03.49 −08:40:34.4 4203850221956827648
V32f SRs 13.72g 11.56g >0.14 >0.06 31.1 7586.6252 18:52:59.32 −08:41:34.7 4203849805345042176
V33f SR 13.68g 11.57g >0.24 – – – 18:53:06.20 −08:42:43.2 4203848985079013760
V34f SR 13.54g 11.59g >0.25 – – – 18:53:01.99 −08:42:18.8 4203849049430686336
V35f SR 13.93g 12.00g >0.17 – – – 18:53:09.02 −08:41:12.4 4203849461747685760
V36f SRs 13.66g 11.79g >0.22 >0.11 17.8 5724.4243 18:53:09.39 −08:43:12.9 4203848808912547840
Ch SRs 13.84g 11.64g >0.21 – 18.9 5780.3438 18:53:02.30 −08:42:38.4 4203849049430685312
Notes. aThe period is from Oosterhoff (1943).
bTime of primary minimum light.
cThe period is from Sloan et al. (2010).
dClassified as RRab in this work. See Section A3.
eReclassified as RRab in this work. See Section A4.
fNew variable discovered in this work.
gMagnitude-weighted mean.
hMember variable candidate. Bl denotes Blazhko effect.
To calculate the bolometric correction, we used the equation
BC = 0.06[Fe/H]ZW + 0.06 as derived by Sandage & Cacciari
(1990). We have adopted the value of M⊙
bol = 4.75 mag.
The effective temperature of the RRab stars was estimated with
the calibration given by Jurcsik (1998)
log(Teff) = 3.9291 − 0.1112(V − K)0 − 0.0032[Fe/H], (10)
where
(V − K)0 = 1.585 + 1.257P − 0.273A1 − 0.234φ
(s)
31
+ 0.062φ
(s)
41 . (11)
For the effective temperature in RRc stars, we used
log(Teff) = 3.7746 − 0.1452log(P) + 0.0056φ
(c)
31 , (12)
as derived by Simon & Clement (1993). We can also estimate
the masses of the RR Lyrae stars using log(M/M⊙) = 16.907 −
1.47log(PF) + 1.24log(L/L⊙) − 5.12log(Teff) as given by van Al-
bada & Baker (1971) where PF is the fundamental period. The stellar
mean radii can be obtained from L = 4πR2σT4. These values are
also reported in Table 7.
6 FU RT H E R C O M M E N T S O N TH E
M E TA L L I C I T Y O F N G C 6 7 1 2
NGC 6712 is a moderately metal-poor OoI-type cluster from which
Harris (1996) reports a metallicity of [Fe/H] = –1.02. For our
estimation of the metallicity, we did not use the RRab stars V4,
V6, and V23 since the value of their Dm parameter is larger than
3.0. For V6, its light curve also shows a mild Blazhko-like amplitude
modulation which affects the value of its physical parameters. In
the case of V12, we recall that its light curve is incomplete due
to its period being nearly half-day and although its Dm value
is smaller than 3.0, the uncertainties in its physical parameters
are very large, and therefore was not taken into account for the
averages. Following the Fourier decomposition method described in
Section 5, we found an average [Fe/H]JK96
UVES = −1.13 ± 0.02 given
in HDS scale (Carretta et al. 2009).
It has been recognised that the calibration of JK96 produces
metallicities too rich by ∼0.3 dex for extremely metal-poor stars
(e.g. JK96; Smolec 2005). For more metal-rich stars, the differences
between spectroscopic values and those from the JK96 calibration
are small (Smolec 2005). The present results offer a good opportu-
nity to test the prediction of JK96 calibration for the high metallicity
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NGC 6712: the variable star population 2005
Figure 8. SR stars in NGC 6712. Colour symbols are as in Fig. 6. The full time span is of 9 yr and the long-term variation is evident. The insets display a
blow up of the IAC data (black symbols) for which a smoother and continuous pace is available, and the variation in a shorter time span is shown. Stars with
no inset lack Hanle data due to saturation. See Section 4.3.
extreme, being NGC 6712 among the most metal-rich globular
clusters. Nemec et al. (2013, N13) proposed new formulations to
derive photometric values on the HDS scale for the RRab and RRc
stars, using non-linear model between HDS values of [Fe/H] and the
Fourier parameter φ31, in a broader metallicity range, particularly
at the lower end of [Fe/H]. For completeness, we reproduce here
their formulations.
For the RRab stars
[Fe/H] = b0 + b1 P + b2φ
(s)
31 (Kp) + b3φ
(s)
31 (Kp) P
+ b4
(
φ
(s)
31 (Kp)
)2
, (13)
with the coefficient values b0 = −8.65 ± 4.64, b1 = −40.12 ± 5.18,
b2 = 5.96 ± 1.72, b3 = 6.27 ± 0.96, and b4 = −0.72 ± 0.17,
and an rms error of 0.084 dex. The Fourier parameter φ
(s)
31 (Kp), is
calculated from the light curve in the Kepler photometric system,
thus, our Fourier parameters from the V light curve listed in
Table 6 were transformed into the Kepler system via the relation
φ
(s)
31 (Kp) = φ
(s)
31 + 0.151 (Nemec et al. 2011) before applying the
calibration.
For the RRc stars
[Fe/H] = b0 + b1 P + b2φ
(c)
31 + b3φ
(c)
31 P + b4P + b5
(
φ
(c)
31
)2
,
(14)
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Table 4. Data of known, unclassified, and newly discovered field variable stars (FV) in NGC 6712 in the FoV of our images.
Star ID Type <V > <I > AV AI P HJDmin α (J2000.0) δ (J2000.0) Gaia-DR2 Source
(mag) (mag) (mag) (mag) (days) +2450000
V16 ? 14.98d 14.16d >0.06 >0.06 – – 18:52:54.55 −08:39:17.0 4203850187670201088
V17 ? 14.89d 14.23d >0.24 – – – 18:53:05.86 −08:41:23.4 4203849083790555520
V27 EWa 16.51 15.34 0.12 0.08 0.425714 7586.6325b 18:53:08.88 −08:42:30.7 4203849019364947456
FV1c EW 18.25 17.14 0.54 0.48 0.354499 7574.6363b 18:52:55.78 −08:43:49.1 4203847438817518976
FV2c EW 17.84 16.65 0.27 0.25 0.355390 7567.5873b 18:52:57.45 −08:40:50.4 4203849874064515840
FV3c EW 14.44 13.66 0.38 0.35 0.445495 7574.6047b 18:53:19.90 −08:44:10.1 4203837298410747776
FV4c EW 17.04 16.07 0.23 0.19 0.368598 7572.6428b 18:53:20.86 −08:41:58.9 4203849259956801792
FV5c EW 18.93 17.96 0.54 0.54 0.347863 7572.6380b 18:53:21.25 −08:47:46.5 4203835756498276608
FV6c EW 18.39 17.41 0.40 0.39 0.462812 7573.8535b 18:53:21.90 −08:40:52.2 4203849362961930624
FV7c EW 17.88 16.74 0.72 0.51 0.344416 7567.6514b 18:53:22.04 −08:45:06.3 4203836718651092352
FV8c SRs 14.17d 10.74d >0.58 – 12.3 7555.6001 18:53:00.93 −08:44:18.1 4203848843272252160
FV9c SRs 13.45d 10.17d >0.15 – 24.6 8217.4473 18:53:05.58 −08:39:27.6 4203850325109041792
Notes. aReclassified as EW in this work. See Section A6.
bTime of primary minimum light.
cNew variable discovered in this work.
dMagnitude-weighted mean.
with the coefficient values b0 = 1.70 ± 0.82, b1 = −15.67 ± 5.38,
b2 = 0.20 ± 0.21, b3 = −2.41 ± 0.62, b4 = 18 ± 8.70, and b5 =
0.17 ± 0.04, and an rms error of 0.13 dex. The Fourier parameter
φ
(c)
31 is straight that given in Table 6.
We used these formulations to calculate the individual values
[Fe/H]N13
UVES listed in column 4 of Table 7 and the weighted averages
−0.82 ± 0.06 for the RRab stars and −0.96 ± 0.19 for the RRc stars.
The weighted average of these two values is −0.85 ± 0.05, which is
in good agreement with the value reported by Ferraro et al. (1999a)
of [Fe/H]N13
UVES = −0.88. These values of the iron-to-hydrogen ratio,
in the HDS scale, should be compared to the values of [Fe/H]JK96
UVES
given in column 3 of Table 7, i.e. the transformation of the values in
the ZW scale, into the HDS or UVES scale established by Carretta
et al. (2009). In conclusion, the N13 calibrations give slightly
higher iron-to-hydrogen abundances for this metal-poor cluster than
the JK96 calibration, however, the difference is small and both
determinations agree reasonably well with the value of −1.0 dex
adopted in the compilation of Harris (1996). Later, we shall return to
the metallicity issue when we compare isochrones for a given metal-
licity with the observed star distribution in the CMD (see Section 8).
7 O N TH E D I S TA N C E TO N G C 6 7 1 2
7.1 The reddening of NGC 6712
The determination of an accurate value for the interstellar reddening
is fundamental in the estimation of a distance to NGC 6712. In his
catalogue, Harris (1996, 2010 edition) reports a reddening of E(B
− V) = 0.45 although there is no consensus in the literature with
respect its actual value. In order to obtain an independent estimate
for the reddening, we followed the method used by Sturch (1966),
according to which, RRab stars have a constant intrinsic colour
(B − V)0 near minimum light, between phases 0.5 and 0.8. We
also made use of the calibration of (V − I)0 in this range of phases
derived by Guldenschuh et al. (2005) as (V − I )o,min = 0.58 ± 0.02
mag. This allowed us to estimate the individual values of E(V − I)
for six RRab stars, namely, V4, V5, V13, V22, V23, and V24. We
then converted to E(B − V) using the ratio E(V − I)/E(B − V) =
1.259 derived by Schlegel, Finkbeiner & Davis (1998). This yielded
a mean reddening E(B − V) = 0.35 ± 0.04 which we have adopted
for our calculations. The light curves were phased with the periods
listed in Table 3 and the values for the reddening of each star used
are in Table 8. Pieces of evidence of differential reddening in the
region of NGC 6712 were offered by Janulis & Smriglio (1992),
which can explain the inconsistencies found in the literature for the
several reddening estimations, which will impact the determination
of the distance to the cluster. See for instance table 3 in Janulis &
Smriglio (1992) and a further discussion in Section 7.4.
7.2 From the RR Lyrae stars
From the Fourier decomposition of the light curves of the RRab and
RRc stars and the empirical calibrations for the absolute magnitude
MV given in Section 5, we found a distance of 8.1 ± 0.3 kpc
and a distance of 8.0 ± 0.3 kpc, respectively. As an independent
estimation for the distance, we also made use of the P–L relation for
RR Lyrae stars in the I band (Catelan, Pritzl & Smith 2004), MI =
0.471 − 1.132 log P + 0.205 log Z, with log Z = [M/H] − 1.765;
[M/H ] = [Fe/H] − log(0.638 f + 0.362) and log f = [α/Fe], from
where we adopted [α/Fe] = + 0.3 (Salaris, Chieffi & Straniero
1993). With the aforementioned relation, we derived a distance of
8.3 ± 0.3 kpc. The value of the extinction coefficient adopted was
Av = 3.1E(B − V). We used all the RR Lyrae stars listed in Table 6.
7.3 Luminous red giants as distance indicators
A method that was originally developed to determine distances to
nearby galaxies (Lee, Freedman & Madore 1993) can in principle
be used to determine the distance to a globular cluster, using the
luminosity of the brightest stars near the TRGB. The bolometric
magnitude of the TRGB can be estimated from the cluster metallic-
ity via the equation (Salaris & Cassisi 1997)
M
tip
bol = −3.949 − 0.178 [M/H ] + 0.008 [M/H ]2, (15)
where [M/H ] = [Fe/H] − log(0.638 f + 0.362) and log f = [α/Fe]
(Salaris et al. 1993).
When using this method, one should be aware of two things:
first, this method strongly depends on an adequate selection of the
stars used, and second, that the brightest stars in a given cluster
may not actually be at the tip of the RGB in the CMD, but rather
beneath it by a certain magnitude. Viaux et al. (2013) argued that,
in low-mass stars, the neutrino magnetic dipole moment, may delay
MNRAS 493, 1996–2014 (2020)
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NGC 6712: the variable star population 2007
Figure 9. EW stars in NGC 6712. Black and cyan symbols correspond to observations from IAC and Hanle, respectively. Red continuous curves are the model
solution. The stars V30, V31, and FV1–FV7 are newly discovered variables. See Section 4.4.
Table 5. Physical parameters of EW stars in NGC 6712 in the FoV of our images (Section 4.4). Stars labelled
FV are not cluster members.
Star ID E(B − V) d (kpc) M1 (M⊙) q Teff,1 (K) T2/1 R1 (R⊙) R2 (R⊙) i (◦)
V27 0.21 2.8 1.37 0.93 5530 0.94 1.26 1.12 39.6
V28 0.36 6.3 1.30 0.40 7480 0.98 1.33 0.83 45.0
V30 0.37 7.9 0.48 0.60 9750 0.60 0.88 0.74 77.5
V31 0.37 6.3 0.45 1.0 9700 0.72 0.86 0.86 68.3
FV1 0.35 4.0 0.92 0.99 6200 0.99 1.00 1.00 74.8
FV2 0.29 4.0 1.19 0.44 5840 0.92 1.16 0.80 63.6
FV3 0.15 1.2 1.24 0.98 6220 0.98 1.26 1.25 68.0
FV4 0.19 3.2 0.95 0.87 5850 0.98 1.01 0.96 56.1
FV5 0.41 7.0 0.77 0.99 7110 0.98 0.90 0.91 80.0
FV6 0.36 6.3 1.40 0.85 7300 0.75 1.29 1.13 83.4
FV7 0.21 3.2 0.80 1.0 5800 0.90 0.92 0.92 79.0
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Table 6. Fourier coefficients Ak for k = 0, 1, 2, 3, 4 and phases from the cosine series φ
(c)
21 , φ
(c)
31 , and φ
(c)
41 , for RRab and RRc stars.
The numbers in parentheses indicate the uncertainty on the last decimal place. Also listed is the deviation parameter Dm for the RRab
stars (see JK96).
Star ID A0 A1 A2 A3 A4 φ
(c)
21 φ
(c)
31 φ
(c)
41 Dm
(V mag) (V mag) (V mag) (V mag) (V mag)
RRab stars
V1 16.311(2) 0.381(3) 0.182(3) 0.111(3) 0.071(3) 4.120(25) 8.286(37) 6.185(55) 1.4
V3 16.238(12) 0.221(2) 0.092(2) 0.050(2) 0.022(3) 4.305(36) 8.790(57) 7.336(114) 2.1
V4 16.428(1) 0.209(2) 0.097(2) 0.056(2) 0.018(2) 4.211(27) 8.660(49) 7.197(126) 5.7
V5 16.253(2) 0.379(3) 0.202(3) 0.128(3) 0.093(2) 4.037(20) 8.349(30) 6.482(58) 2.5
V6 16.257(2) 0.292(2) 0.118(2) 0.052(3) 0.020(2) 4.251(27) 8.740(53) 6.711(134) 6.3
V12 16.343(3) 0.422(4) 0.219(4) 0.142(4) 0.104(5) 3.999(30) 8.369(45) 6.351(60) 1.7
V13 16.133(1) 0.345(2) 0.182(2) 0.114(1) 0.076(1) 4.086(12) 8.438(18) 6.608(28) 1.3
V22 16.166(1) 0.226(1) 0.102(1) 0.051(1) 0.022(1) 4.315(18) 8.931(31) 7.344(66) 2.3
V23 16.225(1) 0.140(1) 0.041(1) 0.013(1) 0.008(1) 4.442(38) 9.013(119) 7.755(233) 8.9
V24 16.228(2) 0.329(2) 0.180(2) 0.111(2) 0.071(1) 4.156(18) 8.564(29) 6.821(37) 1.2
RRc stars
V18 16.151(1) 0.226(2) 0.012(2) 0.019(2) 0.012(2) 5.775(132) 4.629(92) 2.901(140) –
V19 16.000(1) 0.192(2) 0.014(2) 0.012(2) 0.012(2) 7.1176(125) 4.836(145) 4.159(135) –
V20 16.202(3) 0.190(3) 0.011(3) 0.008(3) 0.006(3) 3.156(322) 2.119(475) 6.918(465) –
V26 16.162(1) 0.047(2) 0.007(2) 0.018(2) 0.015(1) 7.7262(260) 4.7131(1024) 4.989(1000) –
Table 7. Physical parameters obtained from the Fourier fit for the RRab and RRc stars. The numbers in parentheses indicate the
uncertainty on the last decimal place. See Section 5 for a detailed discussion.
RRab stars
Star ID [Fe/H]JK96
ZW [Fe/H]JK96
UVES [Fe/H]N13
UVES MV log Teff log(L/L⊙) M/M⊙ R/R⊙
V1 − 1.23(4) − 1.11(3) − 0.82(6) 0.605(4) 3.819(9) 1.658(2) 0.68(7) 5.20(1)
V3 − 1.30(5) − 1.18(5) − 0.77(10) 0.539(3) 3.800(20) 1.685(1) 0.65(11) 5.87(1)
V4a − 1.26(5) − 1.14(4) − 0.80(8) 0.614(3) 3.803(20) 1.654(1) 0.63(12) 5.59(1)
V5 − 1.30(3) − 1.18(3) − 0.94(5) 0.570(4) 3.815(9) 1.672(2) 0.69(8) 5.41(1)
V6a − 0.80(5) − 0.74(3) − 0.26(5) 0.662(3) 3.823(20) 1.635(1) 0.62(12) 4.98(1)
V12a − 1.12(42) − 1.00(34) − 0.64(6) 0.598(6) 3.822(10) 1.661(2) 0.68(8) 5.15(1)
V13 − 1.28(2) − 1.16(2) − 0.90(3) 0.574(2) 3.812(8) 1.670(1) 0.67(6) 5.45(1)
V22 − 1.16(4) − 1.05(4) − 0.53(5) 0.535(1) 3.803(10) 1.686(1) 0.63(7) 5.79(1)
V23a − 1.04(11) − 0.93(8) − 0.39(16) 0.619(1) 3.801(27) 1.652(1) 0.60(19) 5.63(1)
V24 − 1.21(3) − 1.09(2) − 0.78(4) 0.569(3) 3.812(8) 1.672(1) 0.66(6) 5.48(1)
Weighted mean − 1.25 − 1.13 − 0.82 0.551 3.812 1.679 0.66 5.64
σ ±0.02 ±0.02 ±0.06 ±0.039 ±0.003 ±0.001 ±0.01 ±0.08
RRc stars
V18 − 1.17(23) − 1.05(19) − 0.97(10) 0.476(11) 3.866(1) 1.710(4) 0.51(1) 4.45(2)
V19 − 1.64(45) − 1.58(52) − 1.60(16) 0.355(10) 3.857(1) 1.758(4) 0.52(1) 4.90(2)
V20 − 0.96(19) − 0.87(13) − 0.87(7) 0.650(5) 3.878(1) 1.640(2) 0.61(1) 3.90(1)
Weighted mean − 1.10 − 0.95 − 0.96 0.572 3.869 1.671 0.56 4.09
σ ±0.16 ±0.17 ±0.19 ±0.070 ±0.005 ±0.028 ±0.03 ±0.24
Note. aNot included in the [Fe/H] averages.
Table 8. Reddening estimations from RRab stars.
Star ID E(B − V)
V4 0.409 ± 0.005
V5 0.330 ± 0.015
V13 0.308 ± 0.007
V22 0.335 ± 0.003
V23 0.326 ± 0.023
V24 0.388 ± 0.008
Mean 0.349
σ ±0.036
the helium ignition and result in the extension of the RGB, and
that, in the case of M5 the brightest stars are between 0.04 and
0.16 mag below the TRGB. The suggested offset was confirmed
by the non-canonical models of Arceo-Dı́az et al. (2015); from
the analysis of 25 globular clusters these authors concluded that
the theoretical TRGB is in average about 0.26 ± 0.24 bolometric
magnitudes brighter than the one observed. Using equation (15) and
the SR-type star members in the cluster, we estimated its distance
and found that if we use a correction of 0.1 mag, V8 and V15,
which are near the TRGB following the course of the two extreme
isochrones in Fig. 10, yield values of 8.0 and 8.2 kpc, respectively,
which are in good agreement with the estimated distances obtained
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NGC 6712: the variable star population 2009
Figure 10. CMDs of NGC 6712 in VI filters. The left-hand panel shows the CMD with all the measured stars in our FoV (black dots) and the likely star
members (red dots). The right-hand panel shows the likely star members once the CMD has been cleaned of field stars following the method described in
Section 3. The right-hand panel also shows the known and newly discovered variables. We possess the light curves of 1100 stars (members and non-members).
The blue circles correspond to RRab stars, the green circles to RRc stars, the red circles to long-period stars, and the orange circles to EW stars. Triangular
markers are newly discovered stars. Markers with a black rim represent non-member variable stars. The cyan square represents a candidate variable star. The
isochrones in the right-hand panel are 12 Gyr interpolations for [Fe/H] values −1.25 (blue), −1.0 (red), and −0.86 (green). See Section 8 for a discussion on
the average metallicity. All three isochrones were created using Y = 0.25 and [α/Fe] = + 0.4, from the model grid of VandenBerg et al. (2014). The isochrones
and the ZAHB (black line) have been shifted to a distance of 8.1 kpc and reddened by E(B − V) = 0.35. The red box delimits the region where blue straggler
stars are found (see Section 8.2 for details). The lower panel shows an expansion of the HB for clarity purposes. The open blue and green circles correspond
to the original position of the RR Lyrae before their magnitude and colour were corrected due a contaminating neighbouring star. The red tags accompany the
new position after the correction. The dashed vertical line represents the FORE of the instability strip as estimated by Arellano Ferro et al. (2016) with its
uncertainty in grey.
by the other methods discussed previously. This probably suggests
that the true TRGB in NGC 6712 is located at about 0.1 mag above
these stars.
7.4 Cluster distance inconsistencies
The RR Lyrae star light-curve Fourier decomposition and the I-band
P–L relationship lead to a distance of about 8.1 ± 0.2 kpc (Table 9)
if a reddening E(B − V) = 0.35 is assumed (Section 7.1 and Table 8).
This value is over 1 kpc larger than 6.9 kpc for E(B − V) = 0.45
adopted by Harris (1996), and the kinematic value 6.95 ± 0.39 kpc
obtained by Baumgardt et al. (2019) from the analysis of
Gaia-DR2 proper motions and radial velocities. We must stress
Table 9. Distance comparison to NGC 6712 from the
different methods used in this work.
Method Distance (kpc)
RRab Fourier decomposition 8.1 ± 0.3
RRc Fourier decomposition 8.0 ± 0.3
RRab/RRc I-band P–L 8.3 ± 0.3
that if E(B − V) = 0.45 is adopted, our Fourier and P–L values
would result in a distance of 7.0 ± 0.2 kpc. The Galactic dust maps
and calibrations of Schlegel et al. (1998) and Schlafly & Finkbeiner
(2011) give E(B − V) values of 0.397 ± 0.007 and 0.342 ± 0.006,
respectively. The 3D Dust Mapping facility mentioned in Section
MNRAS 493, 1996–2014 (2020)
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Table 10. Data of RR Lyrae stars (in bold face) in the field of NGC 6712, which show flux contamination from an adjacent star (within the PSF) from
Gaia-DR2.
Variable Gaia-DR2 ID G GBP − GRP VGaia IGaia V − IGaia Vmix 
V Imix 
I AmpV AmpI
V4 4203849122517862144 16.154 1.239 16.446 15.362 1.084 16.408 0.038 15.347 0.015 0.58 0.35
V4 4203849122443611648 20.053 20.070 20.032 0.038
V5 4203848778920519168 15.911 1.005 16.110 15.242 0.869 15.954 0.157 15.166 0.076 1.34 0.81
V5 4203848778846478080 18.114 18.132 18.093 0.038
V6 4203848985005261312 16.147 16.164 16.126 0.038 15.976 0.188 15.938 0.188 1.06 0.66
V6 4203848985005260672 17.954 17.972 17.934 0.038
V20 4203849083790434304 16.042 0.999 16.240 15.377 0.863 16.069 0.171 15.293 0.083 0.48 0.26
V20 4203849088158325760 18.143 18.161 18.123 0.038
V22 4203849874064518016 15.945 1.115 16.186 15.216 0.969 16.138 0.048 15.196 0.021 0.61 0.40
V22 4203849878432420352 19.528 19.545 19.507 0.038
V26 4203849466042002944 15.980 0.861 16.132 15.392 0.740 15.874 0.257 15.249 0.142 0.15 0.10
V26 4203849466042002816 17.545 17.563 17.525 0.038
4.4 suggests the increase trend of E(B − V) with distance in a given
direction, and suggests for NGC 6712, E(B − V) ∼ 0.37–0.40 at
7.9 kpc. Thus, we consider very unlikely that the reddening is above
0.40, in which case the distance yielded by our methods is hardly
below 7.5 ± 0.2 kpc. However, we must add that, if a reddening of
0.40 is taken, the isochrones and ZAHB of the CMD will appear
unacceptably shifted to the red (see Section 8). Thus, the RR Lyrae
distance and our approaches lead to a distance, given the respective
uncertainties, marginally larger than the kinematic result. Finally,
we want to point out to two independent photometric reddening
and distance determinations for NGC 6712: E(B − V) = 0.33 and
7.9 ± 1.0 kpc (Ortolani et al. 2000) and E(B − V) = 0.33 ± 0.05
and ∼8.0 kpc (Paltrinieri et al. 2001). Both values were obtained by
considering that the colour of the RGB at the level of the HB of the
globular cluster NGC 6712 is a close analogue to NGC 6171. This
difference in colour yields 
(B − V)NGC6712–NGC6171 = 0.0, meaning
that these two have the same reddening.
8 TH E C O L O U R – M AG N I T U D E D I AG R A M O F
N G C 6 7 1 2
Given its position near the Galactic bulge, NGC 6712 presents a
highly contaminated CMD. This can be seen in Fig. 10, left-hand
panel. From their BV photometry and constraining to stars within
a radius r <47 arcsec, Ortolani et al. (2000) managed to clean the
CMD, but no membership analysis was actually carried out.
With the method described in Section 3, we were able to remove
the field stars and obtained a remarkably clean CMD, see Fig. 10,
right-hand panel. This allowed us to overlay three isochrones, one
with a metallicity of [Fe/H] = −1.25 (blue), which comes from the
value obtained from the Fourier decomposition of the light curves
of the RR Lyrae stars, another with the metallicity of [Fe/H] = −1.0
(red) reported by Harris (1996) and one with [Fe/H] = –0.86 (green)
derived from the equations of N13, given in Section 6. The three
isochrones were shifted to a distance of 8.1 kpc and have been
reddened by E(B − V) = 0.35. The isochrones were calculated
from the models of VandenBerg et al. (2014) with Y = 0.25 and
[α/Fe] = +0.4, and they correspond to an age of 12 Gyr. It is
evident that the two isochrones with extreme metallicities, bracket
the one for [Fe/H]= −1.0, which, despite the scatter at the RGB,
particularly near the tip, seems to produce the best fit. In other words,
the observed CMD supports the conclusion that [Fe/H] = −1.0
is indeed a solid estimation. The iron-to-hydrogen calibrations of
JK96 and of N13, discussed in Sections 5 and 6, respectively,
agree, within their own uncertainties, with this value of the
metallicity.
Like the isochrones, a ZAHB of metallicity [Fe/H] = −1.31
(VandenBerg et al. 2014) was placed in the CMD. No significant
differences were observed for other metallicities. The resulting
position of the ZAHB is consistent with the distance of 8.1 kpc.
8.1 The structure of the horizontal branch of NGC 6712
Once the CMD has been cleaned from field stars using the method
described in Section 3, it displays a prominent red HB and a sparsely
populated blue tail which is typical of OoI-type clusters. The RR
Lyrae population on the HB is dominated by fundamental mode
RRab pulsators (10) in comparison with first overtone pulsators (4).
The distribution of RRab and RRc shows a clear mode segregation
around the First Overtone Red Edge (FORE) if one accepts that
the FORE may be shifted to the blue by a few hundredths of a
magnitude, of the otherwise established position (see Fig. 10), which
given the uncertainties, it is quite likely. We can also see that V22
(RR Lyrae with no previous classification) and V23 (originally
classified as RRc) clearly fall on the right side of the FORE of the
instability strip which along with their period, light-curve shape,
and position in the period–amplitude diagram (Fig. 4.2.1) confirm
their nature as RRab stars.
While plotting all the positions of the Gaia sources in our FoV,
we noticed that two sources, not resolved in our photometry, fall
within the FWHM of the PSF of some of the RR Lyrae stars,
namely V4, V5, V6, V20, V22, and V26. The data of the two
Gaia sources for each variable are listed in Table 10. Hence, their
magnitude and colour are contaminated by the extra flux of the
neighbour. A correction to their positions on the HB was performed
by measuring the combined flux in the G, GBP, and GRP band,
of both sources, and converting them to the Johnson VGAIA and
IGAIA magnitudes. For this purpose, we used the relations provided
by J.M. Carrasco (2018: Gaia team), available in section 5.3.7 of
the Gaia-DR2 documentation.4 The magnitudes of the two sources
were combined to calculate Vmix and Imix, and the corrections 
V =
VGAIA − Vmix and 
I = IGAIA − Imix were then added to the intensity-
weighted means <V > and <I > (Table 3) calculated from our
light curves, i.e. to estimate their apparent magnitude without the
flux contamination of the neighbouring star. This also helped us
4http://gea.esac.esa.int/archive/documentation/GDR2/index.html
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NGC 6712: the variable star population 2011
to recalculate the amplitude for these stars and, in doing so, their
position in the Bailey diagram (Section 4.2.1). In the lower panel of
Fig. 10, we plot the corrected positions of the aforementioned RR
Lyrae stars.
At this stage, we can comment on the RR Lyrae pulsating-mode
distribution and their relation with the HB overall structure. A pa-
rameter that is useful in describing the morphology of the HB is the
Lee parameter (Lee 1990) defined as L = (B − R)/(B + V + R)
where B, V, and R are the number of stars to the blue, inside, and to
the red of the IS. For NGC 6712, we estimated from the clean CMD
of Fig. 10, L = −0.44. For the sake of comparison, Paltrinieri et al.
(2001) reported L = −0.66, although they estimated this value by
constraining the stars in their FoV to a radius <5 arcmin so it might
still include some foreground stars on the HB which would make
up for the difference with respect to our estimated value. It has
been highlighted in several recent studies, e.g. Arellano Ferro et al.
(2019, their fig. 8), that the RRab and RRc stars may be cleanly
separated by the FORE or that a mixture of modes can be found
in the bimodal region, also sometimes referred to as the either–or
region. And, that this condition seems to be correlated with the
overall structure parameter L and the Oosterhoff type. There are
six OoI-type clusters with very red HB’s (L < −0.5), Rup 106,
NGC 362, NGC 1261, NGC 6171, NGC 6362, and NGC 6712; for
NGC 362 there is no detailed study of the mode distribution and
Rup 106 has no known RRc stars. In the bottom panel of Fig. 10,
we note that V13 might be the only RRab star sitting in the bimodal
region. However, given the uncertainties in the stellar colours and
the positioning of the FORE, the two modes could be considered
as being segregated, in which case, in NGC 6712, like in the other
three, NGC 1261, NGC 6171, and NGC 6362, the modes are well
separated. This property is also observed in all OoII-type clusters
thus far studied but only in some OoI-type clusters of intermediate
values for L.
8.2 Comments on the blue-straggler population of NGC 6712
Paltrinieri et al. (2001), based on their V − (B − V) CMD,
identified 108 candidate blue-straggler stars (BSS) in the central
region of NGC 6712. The cluster being a very compact object,
finding a large number of BSS does not comes as a surprise, since
they may originate in stellar collisions (Ferraro et al. 1999b). The
BSS region in our CMD does not look so much populated. We
identified 61 cluster members in this region after cleaning for the
non-members through the proper motion analysis (Section 3). We
also found two contact binaries among these BSS, V30 and V31.
Thus, we are inclined to believe that NGC 6712 does not have a
large population of BSS as previously suggested by Paltrinieri et al.
(2001)
9 C O M M E N T S O N TH E N G C 6 7 1 2 C L U S T E R
TIDAL IDENTITY
As we have briefly summarised in the introduction, there are
numerous pieces of evidence that favour the idea that NGC 6712
was once was a very massive globular cluster in the Galaxy, and
that after numerous close encounters with the Galactic bulge, and
the constant interaction with the disc, it has been tidally stripped of
nearly 99 per cent of its mass (Andreuzzi et al. 2001; Paltrinieri et al.
2001). To what extent do these tidal disrupting forces imprint their
evidence in the remaining cluster core? The unprecedented accurate
proper motions in the Gaia-DR2 are the best observations available
at present to investigate this question. The proper motion vectors,
projected on the plane of the sky, of the 1529 cluster member stars
identified in Section 3 are shown in the bottom panel of Fig. 2,
where it is evident that the whole cluster moves as a compact entity,
and the stars in solidarity maintain their identity. This is contrary to
what is seen for instance in Pal 13 (Yepez et al. 2019, their fig. 12),
a rather loose cluster being tidally disrupted.
However, we note in the bottom panel of Fig. 2, that some
apparent field variables (labelled in the figure) have proper motions
that are consistently nearly perpendicular to the cluster motion,
parallel to the Galactic longitude and towards the Galactic centre.
This suggests that these stars participate in the local Galactic
rotation, as expected in field stars, and not in the present orbital
motion of the cluster.
It is also fair to say that based solely on the proper motions of
the stars in our FoV, we are unable to tell the difference between
the ones belonging to the field and the ones that are actually being
or starting to be stripped away from the cluster.
The cluster, in its current stage within its tidal radius, keeps itself
rather compact. This allows it to better withstand its gravitational
interaction with the Galactic bulge and the Galactic plane now than
in the past, when it was much more massive and extended. It is
possible that its compactness allows the cluster to preserve its tidal
identity for many more encounters to come.
1 0 S U M M A RY A N D C O N C L U S I O N S
In this paper, we have performed high-precision VI CCD photometry
of 11 294 stars in the FoV of globular cluster NGC 6712. Our main
goal was to obtain physical parameters of the RR Lyrae star popu-
lation via Fourier decomposition of their light curves. This method
yielded metallicities of [Fe/H]zw = –1.25 ± 0.02 and [Fe/H]zw = –
1.10 ± 0.16 for the RRab and RRc stars, respectively, with a
weighted mean of [Fe/H]zw = –1.23 ± 0.02 using the calibrations
by JK96 and Morgan et al. (2007). For comparison, we estimated
the metallicities using the calibration derived by Nemec et al. (2011,
N13), which are given in the HDS scale [Fe/H]N13
UVES = −0.82 ± 0.06
for the RRab stars and [Fe/H]N13
UVES = −0.96 ± 0.19 for RRc stars
yielding a weighted mean of [Fe/H]N13
UVES = –0.85 ± 0.05. The
above values are, within the uncertainties, in good agreement with
the value reported by Harris (1996) of [Fe/H] = –1.02.
NGC 6712 is currently located inside the bulge of the Galaxy
and therefore, it is highly contaminated by field stars. A proper
motion analysis of 60 447 stars in our FoV was carried out in order
to determine membership status and yielded 1529 likely members
or 2.5 per cent, for which we possess the light curves of 1100. This
allowed us to plot a remarkably clean CMD and to overlay three
theoretical isochrones using the models of VandenBerg et al. (2014)
and the metallicity found from the Fourier decomposition of the light
curves of the RR Lyrae stars. We found that these isochrones are
consistent with an age of 12 Gyr. We also used a theoretical ZAHB
with similar metallicity to the isochrones to fit the observed HB.
The cleaning of the CMD also allowed us to decrease the estimated
number of BSS members in NGC 6712 down to 61, since after the
membership analysis many of them turned out to be field stars. The
distance derived by the methods mentioned in Section 7 yielded a
value of <d > = 8.1 ± 0.2 kpc, if a colour excess E(B − V) = 0.35
is assumed. Pieces of evidence have been discussed that this value
could be as large as 0.40, in which case the same methods yield a
distance of 7.5 ± 0.2 kpc.
Based on their light-curve morphology, their position on the
Bailey diagram, and the period analyses of the stars V22, V23,
and V27, we were able to classify properly V22 as an RRab star,
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to reclassify V23 as an RRab and V27 as an EW star. Furthermore,
V22 and V23 are both on the right side of the FORE, where only
RRab stars are expected to be found.
From the average of the periods of the RRab stars in the cluster
(<P > = 0.58), and the distribution of RRab and RRc star in the
period–amplitude diagram, we confirm that NGC 6712 is a OoI-type
cluster. We detected a mild Blazhko effect in the amplitude of V6.
We additionally found nine new EW variables (V30 and V31 and
FV1–FV7) and seven SR-type variables (V32–V36 and FV8–FV9).
For five of these SR-type (V32, V36, FV8, FV9 and a candidate C),
we were able to derive periods ranging between 12 and 31 d.
There are numerous pieces of evidence in the literature that NGC
6712 is a remnant of a much more massive cluster that has been
tidally disrupted by its numerous encounters with the Galactic bulge.
We were able, from the proper motion analysis, to find evidence that
the present cluster is in fact a compact core that seems to retain its
identity in spite of the gravitational interaction with the Galactic
bulge and disc. The proper motion of some of the field variable
stars are parallel to the Galactic longitude and point towards the
Galactic centre, suggesting that they are field stars participating in
the Galactic rotation, not necessarily being gravitationally stripped
away. As of now, we do not have a method to properly differentiate
star members that are being currently stripped away from the
cluster from the ones that belong to the field. It is possible that
what we are seeing now is nothing more than the remaining
core.
AC K N OW L E D G E M E N T S
We are grateful to Dr. Raúl Michel Murillo for kindly providing
the standard stars in the field of NGC 6712. We thank the staff
of the 2.0 m telescope at the Indian Astrophysical Observatory
(IAO), Hanle and the Centre for Research and Education in Science
and Technology (CREST), Hosakote that made these observations
possible. We also thank the staff of the telescope of 0.80 m of
the Instituto de Astrofı́sica de Canarias (IAC80) at the Spanish
Observatorio del Teide. This project was partially supported by
Dirección General de Asuntos de Personal Académico, Universidad
Nacional Autónoma de México (DGAPA-UNAM) (Mexico) via
grant IN106615-17. Dan Deras thanks Consejo Nacional de Ciencia
Y Tecnologı́a (CONACYT) for the PhD scholarship. We have made
extensive use of the SIMBAD and ADS services.
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APPENDIX: FIELD VARIABLE AND PECULIAR
STARS
A1 The case of V7
This variable was discovered by Sawyer (1953) and classified as
a Mira-type star by Sloan et al. (2010), who were able to derive a
pulsation period of 193.0 d. Our data span only 30 d but it displays
a clear variation (see Fig. 8). These are evolved stars that during
their evolution on the AGB can produce and expel dust into the
interstellar medium, affecting their local reddening. In the case of
V7, its location on the CMD is extremely shifted to the red probably
due to its own dust shrouding. Hence, it is not labelled in Fig. 10 due
to scale reasons. Its CMD coordinates are (4.701,16.98). According
to the membership analysis carried out in Section 3, the star is a
cluster member.
A2 The case of V14, V16, and V17
None of these three stars is a cluster member according to the
analysis described in Section 3, in agreement with the proper motion
analysis performed by Cudworth (1988) for V17. The SR star V14
is not in the FoV of our images but in terms of variability, V14
is a clear variable (Rosino 1966). V16 and V17 were flagged as
variables by Harwood (1960), but Rosino (1966) and Sandage et al.
(1966) did not detect any variability. Unfortunately, the light curves
of V16 and V17 were not published by previous authors. The light
curves in our data are shown in Fig. 8, where subtle but clear signs
of long-term variations can be seen.
A3 The case of V22
Cudworth (1988) noted that this star showed low-amplitude varia-
tions and that given its position on the CMD, it could presumably
be catalogued as an RR Lyrae. In the CVSGC (Clement et al. 2001),
the star is designated as RR type and its period is not provided. V22
is also located near the locus where RRab stars in OoI-type clusters
are expected to be found in the period–amplitude diagram (Fig. 7).
The shape of the light curve (see Fig. 6), its amplitude, position on
the HB (Fig. 10) and period (P = 0.654789 d), clearly indicate its
RRab nature.
A4 The case of V23
This star, discovered by Tuairisg et al. (2003, P1 in their paper),
has been classified as RRc. Nevertheless, its reported period in the
CVSGC of P = 0.3896 d does not phase the curve well. Moreover,
using our 7 yr time-base data, we found a period of P = 0.642451 d
which is more consistent with an RRab star. Also, like V22, its
position on the HB (Fig. 10) and on the Bailey diagram (Fig. 7)
along with the shape of its light curve (Fig. 6) are all consistent with
the star being an RRab.
A5 The case of V25
This is a low-mass X-ray binary discovered by Homer et al. (1996).
According to these authors, the optical counterpart shows variability
and has a mean magnitude V = 20.34 ± 0.07, i.e. too faint for our
CCD photometry. The nearest optical source in our images, to the
coordinates listed by the CVSGC, is a V ∼ 16 mag star which shows
no variations.
A6 The case of V27
This star was discovered and classified as RRc by Pietrukowicz &
Kaluzny (2004, NGC 6712 07 in their paper). These authors noted
that the star showed an ‘unstable light curve with some cycle-to-
cycle changes’. We found that once it is phased with a period of
P = 0.425714 d (which is approximately twice the reported value
by Pietrukowicz & Kaluzny 2004), the light curve of V27 (Fig. 6)
shows alternating deep and shallow minima typical of EW stars.
Hence, the star should be considered a short-period contact eclipsing
binary.
A7 Semiregular red variables
For five stars among the SR variables, a period that seems to phase
all the available data was identified and are given in Tables 3 and
4. These stars and their phased light curves are shown in Fig. A1.
Their amplitudes are of a few tenths of a magnitude and their periods
range between 12 and 31 d. This suggests a classification as of the
semiregular or SRs type.
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Figure A1. SR stars that could be phased with a single period, given in Tables 3 and 4.
This paper has been typeset from a TEX/LATEX file prepared by the author.
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119
Chapter 10
Conclusions
As individual objects, GCs constitute isolated laboratories in which many aspects of stellar evolution and dy-
namics can be studied. When considered as a system, the GCs in the surrounding halo of a galaxy provide a
fossil record of the prevailing dynamical and chemical conditions at the time of its formation.
Our main goal in this work has been to identify systematic differences in three GCs based on their different
metallicities: one metal-poor (NGC 6205), one metal-intermediate (NGC 6712) and one metal-rich (NGC 6171),
as well as their relation to the structure of their HBs. The complete analysis of each cluster is contained and
discussed at length in their respective papers in Chapter 6 through Chapter 9, and they shall be repeated for the
sake of completeness but only discussed briefly from a broader perspective in the following sections.
10.1 Physical parameters of RR Lyrae stars in NGC 6171 (M107)
In this article, we presented the results of the first ever determination of physical parameters of RR Lyrae stars in
NGC 6171 by means of Fourier decomposition of their light curves. By performing high-precision photometry,
we were able to extract the light curves of 3827 stars in our reference image, 20 of which are RR Lyrae, 1 SX Phe,
and 1 Lb variables. Of the three Lb variables, the variability of two is newly announced in this work (V27 and
V28). An estimation of the reddening via the minimum color of RRab stars suggests a mean value of E(B − V)
= 0.45. However, historical determinations of E(B − V) display a large scatter between 0.25 and 0.46. This fact
may seem to suggest that some differential reddening may be present. Taking individual reddening values for
the RRab stars and an average of 0.45 for the RRc stars, and considering the results from the P-L relations for
the RR Lyrae and SX Phe stars, we found an overall average distance of 5.4 ± 0.1 kpc. For RRab and RRc, the
metallicities are [Fe/H]ZW = –1.33 ± 0.12 and –1.02 ± 0.19, respectively. It is worth noticing that the results
from the RRc stars are closer to the spectroscopic results, which are indisputably the most accurate and are not
affected by the reddening. The above distance and iron values tend to be at the lower end of the range of previous
estimates, as shown in Table 6 in the paper. Particularly, [Fe/H]ZW of –1.33 from the RRab seems a bit too
low. We should highlight, however, that the isochrone for this metallicity fits the observations very well. If the
isochrone for [Fe/H]ZW = –1.02 from the RRc stars is shifted upward to match the data, it would imply an even
shorter distance of 5.0 kpc. Let us refer to the Mv versus [Fe/H]ZW relation for RR Lyrae. From the homogeneous
Fourier decomposition of RR Lyrae stars in a sample of 24 clusters of both OoI and OoII types, Arellano Ferro
et al. 2017 found that the slope of the relationship is different when calculated from RRab and from RRc stars.
In Figure 6 of the paper, the two trends for the RRab and RRc stars are shown and the corresponding position
of NGC 6171 given the values in Table 4 in the paper. This shows that despite the difference in the metallicity
from the two stellar groups, the present results are consistent with the homogeneous Fourier analysis in a larger
sample of globular clusters.
120 Chapter 10. Conclusions
10.2 Period changes of RR Lyrae stars of NGC 6171 (M107)
Pulsation period changes have been analyzed via the times of maximum light for 22 RRLs in NGC 6171. Archival
data collected from the literature, previously unpublished data spanning 19 years, and recent CCD observations
enable a span of up to 82 years for most of the sample stars, which makes this work the first significant study
of period changes in NGC 6171. Secular period variations were found for four stars, three with significant
decreasing periods and one (V17) with increasing period. No signs of irregular period variations were found in
the RRLs of this cluster, but instead, they all have either a remarkably stable period or a secular period change
that can be represented by a parabolic O − C diagram. The overall average of the period change rates found in
NGC 6171 is not significantly different from zero, as expected from the canonical evolutionary models of the
HB for a cluster with a red HB. Notwithstanding this fact, individual stars with large positive and negative period
changes have been found, a trend also observed in M3 and M5. In NGC 6171, we have found a single case with
positive β (V17) that seems to be consistent with the period change rate expected in a truly advanced stage of
evolution toward the AGB. On the contrary, a few cases emerged with values of β significantly negative, which
cannot be reconciled with post-HB evolutionary predictions and may be examples of pre-core-helium-burning
stars on their contraction towards the ZAHB. The majority of the RRLs in this cluster, both RRab and RRc,
display a stable period for at least 82 years, well within the uncertainties of the O − C approach. Under the
paradigm that period changes are a consequence of stellar evolution, it must be concluded that these stable stars
are evolving very slowly and their putative period changes are, given the data presently available, undetectable
by the approach described in this work.
10.3 A new study of the variable star population in the Hercules gloubular clus-
ter (M13; NGC 6205)
In this work we have obtained high-precision photometry of over 16 000 point sources in the field of the globular
cluster M13 in our V- and I-band reference images. We have been able to retrieve most of the known variables
cited in the literature. Of particular interest is the RR Lyrae star population since the Fourier decomposition
of their light curves allowed us to determine the metallicity of M13 by two independent methods, from which
we derived an average of [Fe/H]ZW = –1.58 ± 0.09. From our two-color photometry we constructed a clean
CMD based on the recent Gaia-DR2 (Gaia Collaboration et al., 2018) proper motions and radial velocities,
on which we overlaid two isochrones that correspond to an age of 12.6 Gyr and a metallicity of [Fe/H]RR =
–1.65 and a theoretical ZAHB with the corresponding metallicity. The isochrones come from different sets of
model atmospheres and yet they are practically indistinguishable from one another. We discovered seven new
variables: one RRc (V54), two SX Phe (V55 and V56), one contact binary (V57), and three SR (V58, V59 and
V60) stars. From the orbital solution of the contact binary, we were able to derive the masses, radii and effective
temperatures of the primary and secondary components. We also found 15 stars which seem to display mid- to
long-term variations but that have been retained as candidates waiting for confirmation by more appropriate data.
Based on its high radial velocity, we identified a runaway star in the field of M13 which is discussed in Appendix
A of the paper. This star appears to be passing by and overtaking the cluster. We report the double-mode nature
of the star V31 and confirm the double-mode nature of V36. In our analysis of V36, we found two frequencies
that are consistent with the ones found by Kopacki et al. (2003). In the case of V31, we also found two distinct
periods and at least one seems to be non-radial. From the P-L relation for the SX Phe stars, we found that it is
likely that V47, V50 and V55 are pulsating in the fundamental mode and V46 and V56 in the second overtone.
The period of V8 (0.75 d), the large positive value of L (long blue tail of the HB), and the clear segregation
between RRab and RRc stars on the HB confirm the nature of M13 as an OoII-type cluster. By using seven
independent methods such as the MV values obtained from the Fourier decomposition of RRab and RRc stars,
10.4. NGC 6712: The variable star population of a tidally disrupted globular cluster 121
the P-L relation for RR Lyrae stars in the I filter, the P-L relation for the SX Phe, the P-L relation for the CW
stars, the position of the theoretical ZAHB and the orbital solution of the contact binary V57, we were able to
estimate the mean distance to M13, d = 7.1 ± 0.1 kpc. From Gaia-DR2 proper motion data we found 23 070
stars that are likely cluster members. We also re-confirm the membership status of all variables referred in this
work and we give the Gaia-DR2 identifier for all of them. From the analysis of the 18 stars of the catalogue of
Clementini et al. (2018) that are in the field (see Table 4 in paper) we conclude that four of them were previously
known (namely V5, V7, V8 and V9), seven of them do not show variability, six of them are fainter than the limit
of our observations and one is out of the FoV of our frames.
10.4 NGC 6712: The variable star population of a tidally disrupted globular
cluster
In this paper, we have performed high-precision VI CCD photometry of 11 294 stars in the FoV of globular
cluster NGC 6712. Our main goal was to obtain physical parameters of the RR Lyrae star population via Fourier
decomposition of their light curves. This method yielded metallicities of [Fe/H]ZW = –1.25± 0.02 and [Fe/H]ZW
= –1.10± 0.16 for the RRab and RRc stars, respectively, with a weighted mean of [Fe/H]ZW = –1.23± 0.02 using
the calibrations by Jurcsik et al. (1996) and Morgan et al. (2007). For comparison, we estimated the metallicities
using the calibration derived by Nemec et al. (2011) and Nemec et al. (2013), which are given in the HDS scale
[Fe/H]N13
UVES = –0.82 ± 0.06 for the RRab stars and [Fe/H]N13
UVES = –0.96 ± 0.19 for RRc stars yielding a weighted
mean of [Fe/H]N13
UVES = –0.85 ± 0.05. The above values are, within the uncertainties, in good agreement with
the value reported by Harris (1996) of [Fe/H] = –1.02. NGC 6712 is currently located inside the bulge of the
Galaxy and therefore, it is highly contaminated by field stars. A proper motion analysis of 60 447 stars in our FoV
was carried out in order to determine membership status and yielded 1529 likely members or 2.5 per cent, for
which we possess the light curves of 1100. This allowed us to plot a remarkably clean CMD and to overlay three
theoretical isochrones using the models of VandenBerg et al. (2014) and the metallicity found from the Fourier
decomposition of the light curves of the RR Lyrae stars. We found that these isochrones are consistent with an
age of 12 Gyr. We also used a theoretical ZAHB with similar metallicity to the isochrones to fit the observed
HB. The cleaning of the CMD also allowed us to decrease the estimated number of BSS members in NGC 6712
down to 61, since after the membership analysis many of them turned out to be field stars. The distance derived
by the methods mentioned in Section 7 yielded a value of
⟨
d
⟩
= 8.1 ± 0.2 kpc, if a color excess E(B − V) =
0.35 is assumed. Pieces of evidence have been discussed that this value could be as large as 0.40, in which case
the same methods yield a distance of 7.5 ± 0.2 kpc. Based on their light-curve morphology, their position on
the Bailey diagram, and the period analyses of the stars V22, V23, and V27, we were able to classify properly
V22 as an RRab star, to reclassify V23 as an RRab and V27 as an EW star. Furthermore, V22 and V23 are both
on the right side of the FORE, where only RRab stars are expected to be found. From the average of the periods
of the RRab stars in the cluster (
⟨
P
⟩
= 0.58), and the distribution of RRab and RRc star in the period-amplitude
diagram, we confirm that NGC 6712 is an OoI-type cluster. We detected a mild Blazhko effect in the amplitude
of V6. We additionally found nine new EW variables (V30 and V31 and FV1-FV7) and seven SR-type variables
(V32-V36 and FV8-FV9). For five of these SR-type (V32, V36, FV8, FV9 and a candidate C), we were able to
derive periods ranging between 12 and 31 d. There are numerous pieces of evidence in the literature that NGC
6712 is a remnant of a much more massive cluster that has been tidally disrupted by its numerous encounters
with the Galactic bulge. We were able, from the proper motion analysis, to find evidence that the present cluster
is in fact a compact core that seems to retain its identity in spite of the gravitational interaction with the Galactic
bulge and disc. The proper motion of some of the field variable stars are parallel to the Galactic longitude and
point towards the Galactic centre, suggesting that they are field stars participating in the Galactic rotation, not
necessarily being gravitationally stripped away. As of now, we do not have a method to properly differentiate
122 Chapter 10. Conclusions
star members that are being currently stripped away from the cluster from the ones that belong to the field. It is
possible that what we are seeing now is nothing more than the remaining core.
10.5 On the search for new variable stars
In order to have a larger sample of variable stars using our data and thereby refine the cluster parameter estimates
based on different approaches, we carried out a search using three methods which will be briefly described below:
• Firstly, we analyzed the light curves of the stars in regions of the CMD where it is common to find variable
stars i.e., on the IS at the level of the HB, the BS region and at the TRGB.
• A second approach used was via the string-length method described in Chapter 5. Each light curve was
phased with periods between 0.02 d and 1.7 d, in steps of 10−6 d, and the string-length parameter SQ
was calculated in each case. We defined an arbitrary threshold on a plot of SQ vs. X-coordinate in the
reference image and we individually explored each light curve below this value. The reasoning behind this
method is that small SQ values may correspond to periods of pulsation determined with a high degree of
accuracy. It is particularly useful for finding variable stars outside the common locations mentioned in the
previous method.
• The third method consists in the detection of variations of PSF-like peaks in stacked residual images from
which we can potentially see previously undetected variable stars blink.
These methods complement each other and were tailored for each GC studied in this work. A total of 25 new
variables were discovered in the three GCs studied: two SR in NGC 6171; one RRc, two SX Phe, one contact
binary and three SR in NGC 6205; and nine EW and seven SR in NGC 6712.
10.6 On the determination of membership of stars
Gaia-DR2 has proved to be a handy tool in determining the membership of stars to the parental cluster based
on their proper motions as shown by the results obtained in Chapter 8 and Chapter 9. A couple of caveats are in
order when using this method which will be further described in Appendix C:
• It is possible that field stars with similar proper motions to those in the cluster in question may slip through
(any given GC is prone to this effect). A solution to this could be to use the radial velocity of the stars as
a constraint, but at present not all of them have been measured by Gaia-DR2.
• Particularly in the case of clusters whose orbit crosses near the Galactic bulge (as it is in the case of NGC
6712), they will experience some degree of tidal disruptions due to the Galactic gravitational pull and
their orbital crossings. In consequence, the proper motions of the stars on the outskirts of the GC (which
are more weakly-gravitationally bound to it) will be altered giving the appearance of not belonging to the
cluster by exhibiting a difference with respect to the mean value of the more tightly-bound stars at the core.
These "stray" stars may show up on the CMD in the form of scatter around its well-known evolutionary sequences,
provided that the photometry is accurate and the reddening is well estimated. Although the astrometry provided
by Gaia is of exquisite accuracy up to 1 kpc, the data from Gaia-DR2 is still being currently refined since there
are a lot of stars with measured negative parallaxes or simply they just were not measured. This has an impact
on the determination of the star membership due to the fact that most GCs lie at greater distances than 1 kpc
from the Sun (5.4 kpc, 7.1 kpc and 8.1 kpc for NGC 6171, NGC 6205 and NGC 6712, respectively). According
10.7. On the classification of Galactic GCs and their Mv-[Fe/H] relation 123
to their website1, Gaia-DR3 is expected to be released on the second half of 2021 which will provide much
improved data on proper motions and will include radial velocities leading to a tighter constraint in membership
determination. As a byproduct, this will further help to determine with higher accuracy the physical parameters
of binary systems like the ones newly found in NGC 6205 and NGC 6712.
10.7 On the classification of Galactic GCs and their Mv-[Fe/H] relation
Fig. 10.1 is an updated version of Fig. 8 from Arellano Ferro et al. (2019) and it shows where the clusters studied
in this work lie according to their estimated values of L and [Fe/H]ZW. All the clusters plotted were studied with
the same approach of determining the metallicity in the Zinn-West scale through the Fourier decomposition of
the light curves of their RRL stars. The GCs are divided according to their Oosterhoff type: OoI (in blue) and
OoII (in green). The GCs inside the region between the two slopes are thought to be of extragalactic origin
(Bono et al., 1994). All the studied OoII-type clusters show a very clear segregation of modes of pulsation of its
RRL star population around the FORE between RRab and RRc stars on their HBs. Also, they possess a larger
fraction of blue stars on the HB than OoI clusters, as it can be interpreted from the values of L. NGC 6205 is a
clear example of this, as its CMD displays a prominent extended blue HB and where the separation between the
only known RRab and the seven RRc stars is apparent. On the other hand, the OoI-type clusters may or may not
show this segregation and they typically show a redder and more compact HB. In this work we have confirmed
that NGC 6171 and NGC 6712 are OoI clusters in which this segregation is present and both GCs show indeed
redder and sparse HBs. Particularly, NGC 6712 is a borderline case, since either a few RRab are sitting in the
either-or region, or the FORE is shifted to the red by only about 0.04 mag (see Fig. 9.3 in Chapter 9) which is
the uncertainty in the location of the FORE. From the GCs studied by our group, only a few OoI clusters with
L between 0.0 and 0.7 have been found to have the inter-mode region populated by RRab and RRc stars: NGC
3201, NGC 5904 (M5), NGC 6402 (M14) and NGC 6934.
Several authors agree that the origin of the Oosterhoff dichotomy is the result of the presence of a hysteresis
zone on the HB, that can be simultaneously shared by RRab and RRc stars. As clearly illustrated by the pre-HB
evolutionary models of Silva Aguirre et al. (2008), low-mass stars undergo mass loss during the RGB phase.
Depending on the remaining mass, the stars will start their evolution on the HB i.e., their phase of core helium
burning, in bluer or redder positions. Less massive stars start in the blue part of the ZAHB, while more massive
stars may even clump in the red extreme. The post-ZAHB evolutionary tracks are illustrated in Fig. 5.1. When
along their evolution the stars cross de IS, they pulsate as RRL, either as RRc (less massive) or RRab (more
massive). The hysteresis and the fundamental and first overtone instability strips are illustrated in Fig. 3.4.
This means that RRL stars will cross the IS twice in two directions before they start ascending on the AGB and
at some point during their evolution on the HB they will switch their modes of pulsation. RRL stars crossing
on opposite directions will coincide in this hysteresis zone. Therefore whether the hysteresis zone, or either-or
region, is occupied by both RRc and RRab stars or that the two modes are neatly separated by the FORE, is a
consequence of the initial mass distribution on the ZAHB, which is in turn set by the mass loss mechanisms at
the RGB. Hence, the Oosterhoff type of a given cluster seems to be defined by the mass-loss conditions which
are probably related to some other parameter, most likely the metallicity as it is illustrated in Fig. 10.1.
In Fig. 10.2 we show the distribution of clusters in the Mv-[Fe/H] plane obtained from the RRab stars (upper
panel) and the RRc stars (lower panel). This figure is an updated version of Fig. 2 of Arellano Ferro et al. (2017)
and includes the values of the GCs studied in this work. The solid black lines are the linear fits to the data and
1https://www.gaia.ac.uk/news/archive/gaia-data-release-3
124 Chapter 10. Conclusions
correspond to the equations:
Mv = 0.205(±0.025)[Fe/H]ZW + 0.892(±0.045) (10.1)
for the RRab solutions, and
Mv = 0.074(±0.015)[Fe/H]ZW + 0.680(±0.025) (10.2)
for the RRc solutions.
It is evident that the luminosity of the HB decreases with increasing metallicity. This is the result of an increase
in the opacity κ in the stellar interior due to element nucleosynthesis. Eq. 10.1 can be compared with the
other well known calibrations such as Mv = 0.22(±0.05)[Fe/H] + 0.89(±0.07) by Gratton et al. (2003) or Mv =
0.214(±0.047)[Fe/H] + 0.88(±0.07) of Clementini et al. (2003) (dotted line), to which our relation is consistent
within the respective uncertainties. Harris (1996) also provided a comparable relation Mv = 0.16[Fe/H] + 0.84
(long dashed line) or the one obtained by Kains et al. (2012) Mv=0.16(±0.01)[Fe/H]ZW + 0.85(±0.02) which
is considerably flatter. The most remarkable feature of Fig. 10.2 is that the cluster distribution for the RRc stars
(lower panel) is significantly flatter than the distribution for the RRab stars, and that it has a much smaller scatter.
Eq. 10.1 and Eq. 10.2 intersect at [Fe/H]ZW = –1.62 but the slopes diverge at higher and lower metallicities with
differences at [Fe/H]ZW = –2.2 and [Fe/H]ZW = –1.1 being as large as 0.076 dex and 0.068 dex respectively. We
note that a visual inspection of Figure 9 in Kains et al. (2012) may also suggest different slopes for the RRab and
RRc stars.
As mentioned in the previous paragraphs, all the metallicity determinations of the GCs studied by our group were
estimated using the Zinn-West scale. In Chapters 8 and 9 we additionally employed a calibration based on the
HDS to also determine the metallicity of the GCs NGC 6205 and NGC 6712. There are transformation equations
that help us to go from the ZW scale to the HDS scale, and it would be of interest to see if once all the metallicities
are transformed into the HDS scale, the discrepancy in the slopes still holds. If it does not hold, it would mean
that the slopes are the result of uncertainties in the calibrations used. If it does, it would suggest the presence of
unknown variables that are playing a role in this slope difference. It is possible that one of this variables is the
effective temperature as it seems that the luminosity of RRc stars is less sensitive to it (Giuseppe Bono, private
communication). If this is indeed the case, we would need to derive new calibrations that account for this new
term. As for the absolute calibration of the Mv-[Fe/H] relation itself, it is still an open problem and Bono (2003)
provides an overview on the possible causes that can influence it such as evolutionary effects, the lack of linearity
in the transition from metal-poor to metal-rich RRL stars, reddening, mean magnitudes or RRL stars, metallicity
and microturbulent velocity. To date, it is difficult to argue what the reason for this slope difference may be but
it seems clear that the luminosity in the RRc stars is less sensitive to variations of metallicity than in the RRab
stars, which would have a measurable impact in the distance scale when calculated from RR Lyrae stars.
10.7. On the classification of Galactic GCs and their Mv-[Fe/H] relation 125
Figure 10.1: The HB structure parameter L as a function of [Fe/H]ZW. Circles and squares are
used for inner and outer halo clusters, respectively. The black-rimmed symbols represent globular
clusters where the fundamental and first overtone modes are well segregated around the FORE,
as opposed to filled non-rimmed symbols. The upper and lower solid lines are the limits of the
Oosterhoff gap according to Bono et al. (1994). Empty symbols are clusters yet to be studied by
our group (Arellano Ferro et al., 2019).
126 Chapter 10. Conclusions
Figure 10.2: Empirical Mv-[Fe/H] relation from the homogeneous Fourier decomposition of RR
Lyrae light curves in families of OoI, OoII and OoIII globular clusters. Data points are plotted
with 1-σ error bars. The upper panel shows the cluster distribution where MV and [Fe/H]ZW were
calculated from the light curve decomposition of RRab stars. The solid black line is the least-
squares fit to the data (except the OoIII clusters, and NGC 288, NGC 6366 and NGC 7492 which
are based only on one star). Dotted, short, and long dashed lines are the relations of Clementini
et al. (2003), Chaboyer (1999) and that adopted by Harris (1996) respectively. The lower panel
refers to RRc stars. The red line is the reproduction of the calibration for the RRab stars from the
upper panel to stress the significantly different slope (Arellano Ferro et al., 2017).
127
Appendix A
The standard photometric system
It is very often that in the study of variable stars, in order to achieve adequate coverage of the light curves, par-
ticularly of long period stars, the observer finds convenient to use several telescopes at different sites, with the
consequent employment of the wide variety of astronomical instrumentation. Even when using the same equip-
ment, the atmospheric conditions may change from one night to the next and will certainly alter the quality of
the images. A proper combination of multi-site data requires an accurate transformation of each one of the in-
strumental system into the international Standard Photometric System in order to obtain the same measurements
irrespective of the instrumentation used or the weather conditions at the observing site.
Defining a standard system also means defining standard magnitudes in each color for a number of stars, which
are called standard stars. Photometric standard stars can be defined as a set of stars whose light output has
been measured very carefully in specific photometric system passbands, and have shown no signs of variability.
After measuring those standard stars, one can derive what are called transformation equations, which are used to
transform raw instrumental magnitudes to standard magnitudes of a given system. It is important to mention that
the more standard stars we select, the color and magnitude distributions will be richer and our transformation
equations will be more accurate. The standard minus the instrumental magnitude differences are known to show
a mild dependence on the color. The general form of the transformation equations are:
V = v + A + B(v − i) (A.1)
I = i + C + D(v − i) (A.2)
where V and I are the magnitudes in the standard system, and v and i are in the instrumental system. A, B, C
and D are the corresponding transformation coefficients.
The first photometric system was defined by Johnson et al. (1953) for the UBV filters. Vega (α Lyrae) was
chosen to be the zero point for their system, i.e., VVega = U − BVega = B − VVega = 0.0. In the same year,
Kron et al. (1953) defined a photometric system in the RI filters, but it was Cousins (1976) who modified it and
extended it. This gave birth to the Johnson-Kron-Cousins photometric system. Fig. A.1 shows the passbands in
the Johnson-Kron-Cousins filters UBVRI. For this work only V and I filters were used.
In 1973 Arlo Landolt published a list of stars carefully transformed to the Johnson UBV system (Landolt, 1973).
He extended that list to include stars in the Cousins RC and IC bands in 1983 (Landolt, 1983). In 1992 he pub-
lished one of the most important works in astronomical photometry (Landolt, 1992). This included an extensive
list of stars in the Johnson UBV and Cousins RC and IC systems. Most observers now use those lists to transform
their system to that of the Johnson-Cousins systems.
128 Appendix A. The standard photometric system
Typically between 30 and 200 standard stars per GC are used to transform our instrumental system into the
Johnson-Kron-Cousins photometric system (Landolt, 1992). Standard stars in the field of the clusters are mainly
taken from the work of Stetson (2000)1. The specific mathematical form of the transformation equations in V
and I filters for each of the GCs studied in this work, are given in their respective published papers in Chapters
6, 8 and 9.
Figure A.1: Transmission of Johnson-Kron-Cousins filters.2
1http://www3.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/STETSON/standards
2https://www.aip.de/en/research/facilities/stella/instruments/data/johnson-ubvri-filter-curves
129
Appendix B
DanDIA
For the reduction of our data, we employed the software Difference Imaging Analysis (DIA) with its pipeline
implementation DanDIA1 (Bramich 2008; Bramich et al. 2013). With this, we were able to obtain high-precision
photometry for all the point sources in the FoV of the CCDs from where our data were obtained. The processing
of the images acquired at the different telescopes using the DIA technique goes through eight stages, which will
be briefly described below as they appear in the DIA documentation:
S1 - Preprocess
The program makes sure that all the raw sky images as well as the calibration frames such as bias and flats,
and their headers, are compatible with the pipe line implementation. When all the images are compatible, the
program calculates the read-out noise and the average gain of the chip for each filter, which will be used in the
latter stages of the reduction process. This step also organizes the images by types and generates a master bias
and a master flat that will be applied for the corresponding corrections.
S2 - Cosmic
The program removes any cosmic rays from the reduced images produced in the previous stage that might have
been detected by the CCD. The program is based on an algorithm employing Laplacian edge detection2, which
is fully described in van Dokkum (2001). It was not used in the processing of our images.
S3 - Reference
The program creates a reference image by stacking the images with the best seeing from each season (although
the images to be combined can also be selected manually). It starts with registering the coordinate system of the
first image that will be combined and detects all the bright stars on the rest of the images. Then, it matches the
stars between image pairs, and derives a linear transformation between images. Finally, interpolates each image
to the coordinate system of the first image. The program also estimates the sky background, PSF FWHM, and
ellipticity, for the newly created reference image. The algorithm used to find stars on each image to be combined
is based on the algorithm described in the paper by Stetson (1987) and the algorithm used to match stars between
images is based on the similar triangles technique and is described in the paper by Pál et al. (2006).
1DanDIA is built from the DanIDL library of IDL routines available at http://www.danidl.co.uk
2https://www.owlnet.rice.edu/~elec539/Projects97/morphjrks/laplacian.html
130 Appendix B. DanDIA
S4 - PSF
The program measures the position and fluxes of all the stars in the FoV of the reference image for each filter.
Then, it selects ∼300-400 isolated stars in order to estimate the shape of the Point Spread Function (PSF). This
program was originally developed by Diolaiti et al. (2000).
S5 - Register
This program aligns all images for each filter with the corresponding reference image created in stage S3. The
bright stars for each image are detected and then are matched to bright stars detected on the reference image.
Once they are matched, a linear transformation is used from the coordinate system of the current image to that
of the reference image. This transformation is then applied to the current image using image interpolation.
S6 - Subtract
The program determines the kernel solutions that best match the reference image to the set of science images from
one filter, and subsequently produces the corresponding difference images. The theory behind the algorithm used
to determine the kernel solution for a pair of images is described in the paper by Bramich (2008), and the method
is based on modelling the kernel as a discrete pixel array, rather than using a set of modified Gaussian functions
as kernel basis functions. The program allows for both a spatially invariant kernel solution and a spatially variant
kernel solution. The spatially variant kernel solution is implemented by solving for individual pixel kernels in a
grid of image subregions and then interpolating the solutions to obtain the kernel at any one pixel.
S7 - Diphot
This program measures the difference in flux and its uncertainty for each star on the set of difference images.
The set of stars that are measured is taken from the star list created in Stage 4. For a given star at a specific
epoch, the difference flux is measured by optimally scaling the PSF model for the star to the difference image
at the coordinates of the star as derived from the reference image, and including an additive constant in the
fit to compensate for imperfect background subtractions in the difference image. The PSF model employed is
constructed by extracting the reference image PSF model at the coordinates of the star and then convolving this
PSF model with the relevant kernel solution, also extracted at the coordinates of the star. The fitting of the PSF
model to the residual flux on the difference image is iterated. In the first iteration, the program uses the image
pixel values to calculate the difference image pixel variances. In the second iteration and beyond, the program
will use the combined image and difference image models to calculate the difference image pixel variances. For
the third iteration and beyond, a 4σ clip algorithm will be applied to the residuals to identify and reject outlier
difference image pixel values. At each iteration, the single worst outlier pixel value is rejected, and the iteration
of the fitting process is terminated only when no more difference image pixels are rejected.
S8 - Post calibration
This step corrects for possible systematic errors, using the methodology developed in Bramich et al. (2012) to
solve for the magnitude offsets Zk that should be applied to each photometric measurement from the image k.
This translates into a correction for the systematic error introduced into the photometry due to a possible error
in the flux-magnitude conversion factor (Bramich et al., 2015).
131
Appendix C
Star membership determination using
Gaia-DR2
Stellar clusters are groups of stars that are linked together due to gravitational force and can occupy a region
in space that can vary from a few tens of light years (for open clusters) to a few hundreds of light years (for
globular clusters). Since the stars within them are gravitationally bound and move through space as a whole,
they are expected to share similar physical and kinematic properties in contrast to those considered to be field
stars. Nevertheless, when we observe these stellar structures, they will always be contaminated by foreground
and background field stars, since we are constrained to observe only their projected positions on the sky. This
means that without an appropriate method, it’s nearly impossible to differentiate field stars from member stars.
As described on the Gaia website1 (Gaia Collaboration et al., 2018): "Gaia is an ambitious mission to chart
a three-dimensional map of our galaxy, the Milky Way, in the process revealing its composition, formation
and evolution. Gaia will provide unprecedented positional and radial velocity measurements with the accuracy
needed to produce a stereoscopic and kinematic census of about one billion stars in our galaxy and throughout
the Local Group. This amounts to about 1 per cent of the Galactic stellar population."
One of the key features of Gaia is the measurement of the proper motions of the stars within its FoV. This
parameter is what we will be using to determine the star membership. To achieve our goal we have employed the
method developed by Bustos Fierro et al. (2019), which consists of two stages that will be outlined as described
by them:
C.1 First Stage
• Download from the Gaia-DR2 a cone around the cluster of about two cluster tidal radii, for example from
Kharchenko et al. (2013), and compute the gnomonic projection of the celestial coordinates: (Xt, Yt)
(Astropy Collaboration et al. 2013; Astropy Collaboration et al. 2018).
• Pre-process the data (Xt, Yt, µα∗, µδ) using RobustScaler in ScikitLearn, in order to remove outliers and
normalize the different dimensions. This normalization is necessary because of the very different range
of the values of coordinates and proper motions.
• Run the BIRCH clustering algorithm (Zhang et al., 1996) in the 4D space: [Xt, Yt, µα∗, µδ].
• Fit a 4D Gaussian distribution with the candidate members of every cluster identified by the algorithm.
• Discard candidate members that lie outside the 3σ ellipsoid and fit a new 4D Gaussian distribution.
1https://sci.esa.int/web/gaia
132 Appendix C. Star membership determination using Gaia-DR2
• Retain as candidate members of the cluster only the stars within the 3σ ellipsoid of the last fitted Gaussian
distribution.
• Plot the space distribution, the vector point diagram - hereafter VPD - and the color-magnitude diagram
in Gaia-DR2 photometric system (Gmag, BPmag - RPmag) - hereafter CMD - with the possible members
of the clusters found.
• Verify that the space distribution, the VPD, and the CMD are consistent with those of a stellar cluster.
This is the only step in this stage that is still performed by a human being.
Notice that after this first stage, there will be sharply defined borders although globular clusters do not physically
possess them, these artificial borders are the result of the 3σ cut. This means that it is possible that there are
members outside this cut that will need to be identified in the second stage.
C.2 Second Stage
• Select rings of proper motions centred in the mean proper motion of the detected members, with increasing
radii.
• Compute the radial density projected on the sky of all the stars within each ring of proper motions.
• Determine for every radial density profile the mean density of the background and the radius of the central
overdensity.
• From the stars that were not classified as members in the first stage, extract as new members those that
lie within each ring of proper motions in the VPD and inside the corresponding central overdensity in the
field.
The authors performed a membership analysis of 8 globular clusters (NGC 1261, NGC 3201, NGC 6139, NGC
6205, NGC 6362, NGC 6397, and NGC 6712) where they report to have extracted between 37% and 90% of the
cluster members. Although the percentage of extracted star members may seem small, it is important to note that
not all the stars in the FoV of the targets have measured proper motions by Gaia, so not all the cluster members
will be extracted.
133
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