UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
PROGRAMA DE POSGRADO EN ASTRONOMÍA
INSTITUTO DE ASTRONOMÍA
LA CONEXIÓN GALAXIA-HALO:
RELACIONES DE ESCALA
TESIS
QUE PARA OPTAR POR EL GRADO DE:
MAESTRA EN CIENCIAS (ASTRONOMÍA)
PRESENTA:
BRISA LLANETH MANCILLAS VAQUERA
TUTOR:
DR. VLADIMIR ANTÓN ÁVILA REESE,
INSTITUTO DE ASTRONOMÍA-UNAM
MÉXICO, D.F., OCTUBRE 2014
 
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Resumen
A través de un modelo donde se cargan discos en equilibrio centŕıfugo en halos de
materia oscura fŕıa y considerando su contracción adiabática por la galaxia formada,
exploramos de manera (semi)emṕırica las relaciones de escala y otras correlaciones de
la población de galaxias de disco en el Universo local. El modelo considera también
la formación de bulbos y la transformación de gas en estrellas por inestabilidades
del disco. Las simulaciones cosmológicas numéricas proveen las condiciones iniciales:
la distribución de masa y los parámetros de giro λ y concentración C de los halos.
La condición inicial astrof́ısica la da la relación de la fracción bariónica con la masa
del halo, fbar–Mh. Esta relación contiene información de los principales procesos de
incorporación y pérdida de bariones por parte de las galaxias. Nuestro objetivo es
explorar los efectos de esta relación sobre las correlaciones globales y distribuciones
internas de masa de las galaxias de disco en función de su escala.
Se generó un catálogo sintético de decenas de miles de galaxias formadas en
halos de masas entre 1010 y 1014 M⊙. Las relaciones estelares y bariónicas de radio-
masa y Tully-Fisher (TF) predichas, aśı como correlaciones de los cocientes de masa
bulbo/total (B/T) y gas/estrellas (Rgas) con M∗, reproducen bien las observaciones
correspondientes que hemos compilado, pero sólo hasta M∗ ≈ 3 − 5 × 1010 M⊙;
para masas mayores en promedio los radios y Vmax son muy grandes, B/T muy
bajo y Rgas muy alto. Esto es debido a que fbar disminuye considerablemente con
Mh . No obstante, hay que tomar en cuenta que halos muy masivos ensamblan
fracciones importantes de sus masas en fusiones lo cual implica que sus galaxias
sufrirán también fusiones e interacciones.
Introduciendo artificialmente el efecto de inestabilidad por las fusiones, los bulbos
son mayores y el contenido de gas y el parámetro de giro disminuyen. Nuestro en-
foque es heuŕıstico ya que el efecto artificial es afinado para reproducir las relaciones
observadas de radio-M∗, B/T–M∗ y Rgas–M∗ a las altas masas. El método ofrece
ii
iii
entonces una sofisticada herramienta para estimar el parámetro de giro inicial de los
bariones de los que se forman las galaxias de disco, λbar. Hasta log(Mh/M⊙)∼ 11.5,
λbar promedio y su dispersión son las mismas que las de los halos, pero a masas
mayores λbar promedio disminuye con Mh (y M∗ por ende). Las relaciones predichas
de TF estelar y bariónica concuerdan con las observaciones. No hay un problema
con el punto cero, siendo que por construcción nuestros modelos están de acuerdo
con la función de masa bariónica (y estelar) observada. A bajas masas, la TF estelar
sufre un doblez que aparentemente también lo muestran las galaxias enanas; a nivel
de TF bariónica este doblez es menor. A altas masas, los modelos muestran también
un cierto doblez. La dispersión intŕınseca de los modelos es mayor a la que presen-
tan las observaciones. Este podŕıa ser un potencial problema. La dispersión de la
relación radio–M∗ es alta y producida por la dispersión ”cosmológica” del parámetro
λbar aśı como por las dispersiones de fbar y C. En el caso de la relación TF, la dis-
persión es menor y producida principalmente por la dispersión ”cosmológica” de C.
Se encuentran también algunas tendencias débiles de los residuales de las relaciones
de escala con propiedades de las galaxias como Rgas, B/T y la densidad superficial
Σe. Los residuales de la relación Mbar–Mh muestran una mayor correlación con estas
propiedades.
El modelo nos permite explorar el efecto de la relación global fbar–Mh y su dis-
persión sobre la distribución interna de masas de halo, disco y bulbo. Encontramos
que el cociente de velocidades de disco a total a 2.2 radios de escala es siempre
del tipo submáximo. La fracción de masa ”luminosa” a un radio efectivo (1Re)
de las galaxias tiene su máximo (alrededor de 0.5 en promedio) para galaxias con
Vmax ≈ 200 km/s, Σe ≈ 500 M⊙/pc
2, y M∗ ≈ 5 × 1010 M⊙. Para valores menores
que estos, las galaxias de disco (B/T≤ 0.5) son más y más dominadas por materia
oscura a 1Re, pero para valores mayores, también; es una clara huella de la forma
de la relación fbar–Mh. Esta huella se ve también en las correlaciones con M∗ o Σe
de los cocientes de velocidad del disco (o disco + bulbo) a velocidad total o de la
pendiente del perfil externo de las curvas de rotación. Estas correlaciones se podrán
obtener con resultados de futuros catastros espectroscopicos que barran un amplio
intervalo de masas y densidades superficiales de galaxias de disco.
Contents
1 Introduction 1
1.1 Antecedentes y planteamiento . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objetivos y Metodoloǵıa de la Tesis . . . . . . . . . . . . . . . . . . . 7
1.3 Contenido . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Observational correlations of disk galaxies 11
2.1 The (semi-emprical) baryonic-to-halo mass relation . . . . . . . . . . 12
2.2 The size-stellar mass relation . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 The Tully-Fisher relation . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 The bulge-to-total mass ratio vs M∗ . . . . . . . . . . . . . . . . . . . 26
2.5 The gas-to-stellar mass ratio vs. stellar mass . . . . . . . . . . . . . . 28
3 The Semi-empirical Method 30
3.1 The static model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 A disk-halo model . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.2 A disk-bulge-halo model . . . . . . . . . . . . . . . . . . . . . 34
3.1.3 Stellar and gas fractions . . . . . . . . . . . . . . . . . . . . . 39
3.2 The procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Disk galaxy correlations: the case of isolated galaxies 47
4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 A semi-empirical picture of local disk galaxy correlations 59
5.1 Disk galaxy correlations . . . . . . . . . . . . . . . . . . . . . . . . . 61
iv
CONTENTS v
5.1.1 The size–mass relation and inference of the spin parameter of
galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.2 The B/T–M∗, Rgas–M∗ and Σe–M∗ correlations . . . . . . . . 66
5.1.3 The Tully-Fisher relations . . . . . . . . . . . . . . . . . . . . 71
5.1.3.1 The intrinsic scatter around the TF relations . . . . 75
5.2 The source of scatter in the disk scaling relations . . . . . . . . . . . 76
5.3 On the scatter of the Mbar–Mh and M∗–Mh relations . . . . . . . . . 82
5.4 Correlations among the residuals of the scaling relations . . . . . . . 83
6 The inner mass distributions of the semi-empirical disk galaxies 86
6.1 Galaxy-to-total maximum velocity ratios . . . . . . . . . . . . . . . . 86
6.2 The inner baryon and dark matter mass fractions . . . . . . . . . . . 91
6.3 The outer slope of the circular velocity curves . . . . . . . . . . . . . 93
7 Summary and conclusions 97
A The empirical HI-to-M∗ and H2-to-M∗ correlations for local late-
and early-type galaxies 104
Chapter 1
Introduction
1.1 Antecedentes y planteamiento
Las galaxias de disco son el tipo más frecuente de estos objetos en el Universo local y
todo apunta a que fueron aún más abundantes en el pasado (v.gr. Bruce et al., 2012;
Buitrago et al., 2013; Mortlock et al., 2013, ver predicciones teóricas en Avila-Reese
et al. 2014 y más referencias ah́ı ). Una serie de evidencias observacionales sugieren
que la formación de discos es el proceso genérico de ensamblaje de las galaxias,
mientras que las componentes esferoidales se desarrollan a partir de los discos en
procesos violentos de fusiones e interacciones, aśı como producto de su evolución
secular. Como resultado de las fusiones se espera la formación de esferoides (bulbos)
prominentes del tipo ”clásicos” (por ejemplo las galaxias eĺıpticas), mientras que el
segundo tipo de mecanismos lleva a los aśı llamados pseudobulbos, mismos que lucen
como prolongaciones del disco interno, compartiendo propiedades con el mismo (ver
una reseña en Kormendy and Kennicutt, 2004).
Desde la perspectiva teórica, la formación y evolución de galaxias requiere del
marco cosmológico. El vertiginoso desarrollo de la cosmoloǵıa en los útimos lustros
ofrece ahora un contexto maduro y cimentado en un gran cuerpo de observaciones
para calcular la evolución de las galaxias (ver reseñas al respecto en Avila-Reese,
2006; Frenk and White, 2012). El paradigma cosmológico actual se resume en el
aśı llamado modelo de materia oscura fŕıa con constante cosmológica Λ (ΛCDM
por sus siglas en inglés). De acuerdo al mismo, las estructuras cósmicas emergen
de tenues perturbaciones en densidad que a su vez provienen de las fluctuaciones
1
CHAPTER 1. INTRODUCTION 2
cuánticas del universo inflacionario. Estas perturbaciones evolucionan gravitacional-
mente hasta separarse de la expansión y colapsan en estructuras autogravitantes que
llamamos halos. En el contexto del modelo ΛCDM se propone que la materia está
dominada por una componente invisible (no interactúa con la radiación) pero que
produce gravedad, llamada ”materia oscura”. La materia oscura, según sondeos
cosmológicos y astronómicos es ≈ 5 veces más abundante que la materia ordinaria,
llamada bariónica. Y ambas son sólo el 30% de lo que está actualmente compuesto
el Universo; el restante 70% es la aśı llamada enerǵıa oscura, un medio repulsivo que
produce la expansión acelerada del Universo; la determinación más reciente de los
parámetros cosmológicos fueron dados por la misión Planck y una combinación de
varios sondeos (Planck Collaboration et al., 2013). La dinámica de la enerǵıa oscura
se describe bien por la constante cosmológica aunque su explicación f́ısica es aún un
gran reto.
En el contexto del modelo ΛCDM, el gas bariónico es atrapado gravitacional-
mente en los pozos de potencial de la materia oscura; el gas se enfŕıa radiativamente
precipitándose hacia el centro de los halos oscuros. Debido al (pequeño) momento
angular que adquieren los halos por torcas de marea, el gas que disipa y cae al centro
conservando parcialmente ese momento angular, caerá hasta que entre en equilibrio
centŕıfugo: una galaxia de disco nace entonces. El disco gaseoso es susceptible de in-
estabilidades mismas que disparan la formación estelar (por ejemplo las que describe
el sencillo criterio de Toomre) y/o las interacciones y fusiones aceleran este proceso.
Las estrellas a su vez inyectan enerǵıa y momento al medio interestelar, en especial
cuando éstas explotan como Estallidos de Rayos Gamma y Supernovas (SNs). Este
proceso de retroalimentación frena la formación estelar hasta que el gas disipe nue-
vamente su enerǵıa o en muchos casos, el gas es incluso completamente expulsado
del sistema en los vientos galácticos; esto ocurre más eficientemente mientras menos
masivo es el halo (desde halos de masa < 1012 M⊙). El gas también puede ser ex-
pelido por efectos del Núcleo Galáctico Activo (NGA) que desarrollan las galaxias
más masivas. Además, en los halos más masivos el gas dif́ıcilmente llega a la galaxia
central debido a que se calienta por choques a decenas de millones de grados en el
proceso de colapso del halo; por esto y por la retroalimentación de los NGAs es que
hay un ĺımite superior en la masa bariónica de las galaxias.
Una cuestión clave del escenario descrito es la conexión galaxia-halo, misma
que resume gran parte de los procesos f́ısicos y evolutivos de las galaxias que se
CHAPTER 1. INTRODUCTION 3
ensamblan en los halos oscuros en crecimiento. En los últimos años se han hecho
importantes progresos en constreñir esta conexión en todo el espectro de masas de
las galaxias a través de enfoques semi-emṕıricos que hacen uso de las distribuciones
estad́ısticas de las galaxias observadas y de los halos simulados. En la tesis reciente
de Rodŕıguez-Puebla (2014) se describen a fondo estos enfoques y se da una extensa
lista de referencias. La fig. 1.1 es una reproducción de dicha tesis donde se ilustra la
esencia de estos enfoques. Como se puede apreciar, el cociente de masa estelar (M∗)
a masa de halo (Mh) de las galaxias cambia brúscamente con la masa de halo: hacia
Mh ≈ 1012 M⊙ la eficiencia de ensamblaje de masa estelar parece tener su máximo;
hacia masas menores y mayores la masa estelar con relación a la del halo es cada vez
menor lo cual indica que los procesos de retroalimentación y enfriamiento del gas
descritos en el párrafo anterior están en acción, haciendo la eficiencia de ensamblaje
de la masa estelar cada vez menor.
Las determinaciones de la relaciónM∗–Mh se han hecho en promedio para toda la
población de galaxias/halos (ver trabajos recientes Behroozi et al., 2010; Guo et al.,
2011; Behroozi et al., 2013; Moster et al., 2013; Rodŕıguez-Puebla et al., 2013) aśı
como separadas en galaxias centrales y satélites (Rodŕıguez-Puebla et al., 2012,
2013). Más recientemente, Rodriguez-Puebla et al. (2014) han logrado inferir la
relación M∗–Mh de galaxias centrales separadas en azules y rojas. Sorpresivamente
éstas resultaron ser diferentes, siendo, a paridad de masa estelar, la masa de halo de
las rojas siempre mayor a las de las azules. A muy grosso modo se pueden asociar las
galaxias azules y rojas con las galaxias de tipo tard́ıo y temprano respectivamente.
Entonces podemos contar ahora con la conexión galaxia-halo para estos dos tipos
por separado.
Las galaxias están compuestas también por gas neutro en forma atómica (HI) y
molecular (H2); el gas ionizado es generalmente una fracción despreciable y el He
y metales constituyen cerca del 28.5% del gas neutro. Mientras más pequeñas las
galaxias, mayor fracción de gas tienen. La suma del gas (considerando He y metales)
y las estrellas constituyen la masa bariónica de las galaxias, Mbar. Recientemente,
en la tesis de R. Calette (2014, en prep.), en base a una extensa compilación de
la literatura y a la conexión entre la función de masa estelar de galaxias rojas y
azules y las relaciones de masa de gas–M∗ para ambas poblaciones, se ha logrado
determinar la función de masa bariónica de las galaxias rojas y azules. A través
de la técnica de correspondencia de abundancias (abundance matching), se logró
CHAPTER 1. INTRODUCTION 4
Figure 1.1: Conexión semiemṕırica entre el halo y la galaxia: la población de halos de
materia oscura fŕıa obtenida en simulaciones numéricas cosmológicas es confrontada
estad́ısticamente con la población observada de galaxias bajo la suposición de que
en cada halo/subhalo existe una galaxia central/satélite. Por ejemplo, empatando
las funciones de masa acumulativas de galaxias y halos se infiere la masa de halo
Mh que le corresponde a una dada masa estelar de galaxia M∗ y se puede calcular el
cociente M∗/Mh (eficiencia de crecimiento de la masa estelar) en función de la masa
del halo; como se aprecia en la figura, este cociente a z ∼ 0 tiene un pico de máxima
eficiencia alrededor de Mh ∼ 1012 M⊙ e incluso en el pico el valor es mucho menor
que la fracción bariónica universal, que es 15–17%. La conexión galaxia–halo se
puede afinar más introduciendo modelos de ocupación de satélites en los halos y de
función condicional de M∗, para lo cual se requiere más restrictores observacionales,
como seŕıa la función de correlación de dos puntos (ver por ej. Rodŕıguez-Puebla
et al., 2013).
CHAPTER 1. INTRODUCTION 5
entonces calcular la relación Mbar–Mh de las galaxias azules y rojas por primera vez.
El cociente Mbar/Mh se conoce en la literatura como la fracción bariónica, fbar. Del
trabajo de Calette et al. (2014, en prep.) contamos ahora con la fracción bariónica
en función de Mh, tanto para galaxias azules como rojas. Incluso al pico, el valor
de fbar es más de un factor dos menor al de la fracción bariónica universal, es decir
el cociente entre densidades de fondo de bariones y materia total, Ωb/Ωm. Esto nos
muestra que hay una gran cantidad de bariones en el Universo que en realidad no
están en las galaxias.
La fracción bariónica es un parámero importante en modelos de galaxias; varias
de las propiedades de las galaxias dependen de la cantidad de bariones asignados a
la galaxia que habita el centro de un dado halo. Este tipo de estudios se ha hecho
principalmente para galaxias de disco, tanto con modelos evolutivos de tipo semi-
numérico (v. gr. Firmani and Avila-Reese, 2000; Avila-Reese and Firmani, 2000;
van den Bosch, 2000a) como modelos estáticos donde un disco es cargado dentro de
un halo ΛCDM (v.gr. Mo et al., 1998; Gnedin et al., 2007; Dutton et al., 2007). En
todos estos y otros trabajos se usaban aproximaciones burdas para fbar en función
de Mh, como ser valores constantes o que dependen de Mh como una simple ley de
potencias. Los parámetros de entrada que generalmente tienen estos modelos son:
1. la masa del halo Mh,
2. la fracción bariónica fbar,
3. el parámetro de giro λ que cuantifica el momento angular del halo y
4. la concentración c del halo cuyo perfil es descrito por la función Navarro-
Frenk-White (o la historia de agregación del halo en el caso de los modelos
evolutivos).
En el caso de los modelos estáticos, se define una época (p. ej. la actual, z =
0) para generar una población de galaxias a esa época, entonces para diferentes
Mh’s se asigna el λ y c a cada halo (tomados de las distribuciones probabiĺısticas
de estos parámetros que se miden en las simulaciones cosmológicas) y se asigna
fbar. El valor de este último parámetro (y su dispersión) depende en realidad de
los procesos astrof́ısicos de cáıda y enfriado del gas en los halos aśı como de los
procesos de eyección del gas conducido por los mecanismos de retroalimentación
CHAPTER 1. INTRODUCTION 6
de la galaxia. En simulaciones cosmológicas o en modelos semi-anaĺıticos, fbar es
calculado deductivamente (ab initio) pero no está libre de grandes incertidumbres
relacionadas con los esquemas sub-malla o las recetas usadas, respectivamente. En
el enfoque semi-emṕırico, en cambio, fbar es constreñido (en función de la masa)
sin importar los procesos f́ısicos que la producen. Y el uso de esta inferencia en los
modelos estáticos nos permite explorar los efectos del fbar(Mh) encontrados sobre
las distribuciones de masa internas del sistema galaxia-halo en equilibrio virial y
centŕıfugo.
Los trabajos con el modelo estático de Dutton et al. (2007), Gnedin et al. (2007)
y otros exploraron a fondo los efectos en las relaciones de escala (y sus dispersiones)
de galaxias de disco por parte de parámetros como λ y C, aśı como de procesos
intermedios de la formación de galaxias de disco como ser la contracción/expansión
del halo interno por la presencia del disco. El escenario cosmológico en estos trabajos
es siempre el ΛCDM. Entre las principales conclusiones de estos trabajos están que:
• La relación de Tully-Fisher es reproducida de manera robusta en el sentido
de que variaciones de estos parámetros poco la afectan; no obstante, para los
valores explorados y dependencia con Mh de fbar el punto cero de esta relación
está desfasado en el sentido de que a paridad de masa o luminosidad, la máxima
velocidad de rotación, Vmax, es mayor en promedio a lo que se observa; Dutton
et al. (2007) propusieron entonces que el aparente problema se resuelve si el
interior de halo se expande en vez de contraerse por la formación de disco.
• La relación radio-masa estelar (o luminosidad) de las galaxias modeladas re-
produce aproximadamente las observaciones pero se requieren valores de λ
algo menores al promedio medido en las simulaciones, aśı como valores muy
bajos de fbar.
• La dispersión de la relación radio-masa estelar de las galaxias modeladas es
significativamente mayor a la que se observa para la población de galaxias de
disco.
Ahora contamos con inferencias más precisas de fbar(Mh) y su dispersión, de
tal forma que, a diferencia de previamente, son una entrada bien constreñida para
los modelos. Es de interés explorar el efecto de estas recientes inferencias sobre las
galaxias de disco modeladas. Al mismo tiempo, entonces responder preguntas que
CHAPTER 1. INTRODUCTION 7
los trabajos previos desencadenaron, como ser: ¿El ΛCDM es consistente con las
relaciones de TF estelar y bariónica, en particular con sus puntos ceros, sin introducir
una expansión interna del halo? ¿Existe un problema de radio en las galaxias de
disco formadas en halos ΛCDM? ¿Es el λ original de los halos consistente con el
tamaño de las galaxias observadas? ¿Existe un problema a nivel de las dispersiones
de las relaciones de TF y masa-radio?
Además, siguiendo un enfoque heuŕıstico, se pueden usar los modelos estáticos
para hacer inferencias sobre el rol de los procesos astrof́ısicos intermedios entre
formación de halos y formación de galaxias. Por ejemplo, un problema estudiado en
la literatura es el de si los radios de escala de las galaxias de disco son consistentes
con el parámetro de giro λ medido en las simulaciones cosmológicas (e.g., Cervantes-
Cota et al., 2007; Berta et al., 2008). En estos estudios, los autores infieren para
muestras nutridas de galaxias de catastros como el SDSS, cuál es el λbar que tuvo
el halo de estas galaxias. Encuentran que λbar decrece con la masa o luminosidad
de la galaxia, lo cual podŕıa estar asociado a procesos de transporte de momento
angular durante la formación de la galaxia masiva, principalmente por fusiones que
son comunes en las galaxias más masivas (e.g., Zavala et al., 2008).
1.2 Objetivos y Metodoloǵıa de la Tesis
El principal objetivo de esta tesis es explorar los efectos de la relación fbar–Mh
determinada emṕıricamente para galaxias tard́ıas sobre las relaciones de escala y
correlaciones diversas de la población de galaxias de disco (incluyendo bulbo) a la
luz de un modelo donde las galaxias son insertadas en halos de ΛCDM para formar
un sistema en equilibrio centŕıfugo, tomando en cuenta la deformación central del
halo por el jaloneo gravitacional de la galaxia. Alrededor de este objetivo principal,
se plantearon varios objetivos espećıficos:
• Constatar si las propiedades de los halos ΛCDM (perfil de densidad, concen-
tración, parámetro de giro λ) proveen las condiciones necesarias para que las
galaxias de disco ensambladas en su interior reproduzcan las relaciones de es-
cala y sus dispersiones observadas; en caso de que se obtengan desviaciones,
explorar los cambios que se requieren introducir aśı como los procesos f́ısicos
que explicaŕıan dichos cambios.
CHAPTER 1. INTRODUCTION 8
• Estudiar las condiciones de formación de bulbos y explorar el efecto de los
bulbos sobre la curvas de velocidad circular en función de propiedades globales.
• Mapear la relación global Mbar–Mh y su dispersión en la distribución interna
de masas oscura, de disco y de bulbo de las galaxias. Estudiar las predicciones
obtenidas, como ser el cociente de velocidades máximas de disco-bulbo a total o
de masa bariónica a oscura a 2.2 radios de escala, en función de las propiedades
globales de las galaxias. Comparar con observaciones disponibles y sondear
si los halos ΛCDM enfrentan o no problemas a este nivel de correlaciones de
distribuciones internas con propiedades globales.
• Estudiar el comportamiento de la parte externa de las curvas de velocidad
circular de las galaxias formadas en los halo ΛCDM en función de propiedades
globales de las galaxias y confrontar con evidencias observacionales. ¿Hay un
potencial problema?
Para realizar el estudio y abordar los objetivos planteados, se usará el modelo
estático basado en el enfoque de Mo et al. (1998) pero extendido aqúı para incluir
(a) la formación de un bulbo, (b) la generalización del invariante adiabático usado
para el jaloneo gravitacional del halo por la galaxia y (c) la transformación de gas
en estrellas (formación estelar de acuerdo al criterio de Toomre). El código fue
escrito por Aldo Rodŕıguez-Puebla, quien amablemente lo proporcionó para realizar
modificaciones, como ser la inclusión de un bulbo y su adaptación para la estrategia
heuŕıstica (semiemṕırica) que se seguirá en esta tesis.
La estrategia a seguir trata siempre de estar apegada a un enfoque semi-emṕırico.
Inicialmente, como ya lo hicieron autores previos, se busca generar una población de
galaxias de disco usando las propiedades tal cual de los halos de ΛCDM (condiciones
iniciales “cosmológicas”) y bajo la suposición en el modelo de que las galaxias evolu-
cionan aisladamente. Posteriormente se confrontan correlaciones globales obtenidas
con las observaciones; en caso de haber diferencias, se explora qué suposiciones ini-
ciales son sensatas de relajar, como p. ej. el que las galaxias evolucionan aisladas
y sólo secularmente (las evidencias muestran que una fracción de las galaxias, en
particular las más masivas, sufrieron en el pasado fusiones mayores). Se opera sobre
los parámetros pertinentes para“emular” la acción de fusiones/interacciones sobre
las propiedades finales de la galaxia de tal manera que haya consistencia con ciertas
CHAPTER 1. INTRODUCTION 9
correlaciones observacionales relacionadas con el efecto de las fusiones/interacciones.
Lograda esta consistencia, quedan constreñidos entonces los parámetros en cuestión.
De esta manera el modelo hace predicciones de otras correlaciones y de las distribu-
ciones internas de masa de las galaxias. En este sentido, el modelo se convierte
en un paso intermedio para mapear ciertas correlaciones globales (como la Mbar–
Mh y la masa-radio) sobre otras correlaciones (como la de Tully-Fisher) y sobre las
distribuciones internas de materia oscura, disco y bulbo.
1.3 Contenido
El contenido de esta tesis está dividido en caṕıtulos.
• En el Caṕıtulo 2 se presenta la relación fbar(Mh) para galaxias tard́ıas a ser
usada aqúı, aśı como una recopilación de la literatura observacional de las
relaciones de escala y correlaciones pertinentes a las galaxias de disco.
• En el Caṕıtulo 3 se describe a fondo el modelo estático de galaxia-halo usado en
esta tesis para obtener poblaciones de galaxias de disco. Primero se describe el
modelo originalmente presentado en Mo et al. (1998) y Dutton et al. (2007) y
luego se presentan las actualizaciones que se hicieron sobre el código numérico
de Aldo Rodŕıguez a fin de tomar en cuenta la formación secular de bulbos.
Se describe también la manera en que se toma en cuenta la posibilidad de las
fusiones en la evolución de las galaxias, especialmente las masivas. Se presenta
la estrategia a seguir a fin de generar un catálogo sintético de galaxias de disco.
• En el Caṕıtulo 4 se presentan las relaciones de escala que se obtienen del
catálogo sintético generado con las condiciones iniciales de los halos oscuros
tal cual como se obtienen en simulaciones cosmológicas y bajo la suposición de
sólo evolución secular, sin presencia de fusiones. Se muestra que las relaciones
de escala predichas concuerdan con las observadas hasta masas estelares del
orden de M∗ = 3−5×1010 M⊙. Para galaxias más masivas, los radios efectivos
y las Vmax a paridad de M∗ son sistemáticamente mayores en los modelos que
en las observaciones.
• Mientras que en el Caṕıtulo 5, mismo que junto al Caṕıtulo 6 se presentan
los principales resultados de la tesis. En este Caṕıtulo se introduce el efecto
CHAPTER 1. INTRODUCTION 10
de las fusiones en las galaxias masivas de tal manera que se reproduzcan las
relaciones observadas de radio–M∗, B/T–M∗ y cociente de gas a masa vs. M∗.
Se estudia a fondo las relaciones de Tully-Fisher estelar y bariónica predichas
y sus dispersiones. Se infiere la dependencia del parámetro de giro inicial del
gas del que se forman las galaxias de disco con la masa del halo y de la M∗ de
la galaxia. Se explora también: (1) el rol que juegan las condiciones iniciales
sobre las dispersiones de las relaciones de escala, (2) la posible correlación de los
residuales de estas relaciones con ciertas propiedades galácticas, (3) la posible
correlación de las relaciones Mbar–Mh y M∗–Mh con propiedades galácticas,
y (4) las correlaciones entre los residuales de las relaciones de TF y radio–
masa. Todos estos estudios permiten sondear la viabilidad de la formación de
galaxias de disco en halos oscuros en el contexto de las cosmoloǵıa ΛCDM.
• En el Caṕıtulo 6 se presentan las distribuciones espaciales internas de masa de
las diferentes componentes de los sistemas generados en el catálogo sintético:
bulbo, disco y halo; las fracciones de masa luminosa a masa total a difer-
entes radios son estudiados en función de la masa estelar, Vmax y la densidad
superficial estelar de las galaxias. Se presentan también varios cocientes de ve-
locidades a 2.2RD o al máximo de los mismos en función de estas propiedades
globales con el fin de comparar con las escasas inferencias observacionales que
hay al respecto. Se presenta también la dependencia de la pendiente externa
de las curvas de rotación en función de estas propiedades globales.
• Finalmente en el Caṕıtulo 7 se discuten los principales resultados y conclu-
siones obtenidos en la tesis.
Chapter 2
Observational correlations of disk
galaxies
Disk galaxies follow several correlations among their global properties (luminous,
stellar and baryonic). The most known are those called ”scaling relations”, as the
Tully-Fisher (TF) and the radius–mass relations (e.g., Gnedin et al., 2007; Courteau
et al., 2007; Avila-Reese et al., 2008, and more references therein). These relations
enclose key aspects of the formation, evolution and dynamics of disk galaxies, as
well as of the properties of dark matter. An interesting question is whether these
relations are directly related (or at least are consistent) to the initial conditions
of the cosmological scenario, in particular the ΛCDM scenario, which is the most
popular one (see the Introduction).
On the other hand, if we have a model for the structure and dynamics of the
population of disk galaxies, this model allows us to ”map” some correlations into
other ones, including the scatters. For example, given that the model reproduces
the radius–mass relation of observed galaxies, we can explore how is, within the
context of this model, the TF relation and compare with observations. We can
also explore what are the effects of the global Mbar–Mh relation (semi-empirically
inferred) on these and other relations. As discussed in the Introduction, this relation
resumes the galaxy–halo connection, which includes several key aspects of the galaxy
assembly process. Moreover, the disk galaxy model to be used here (see Chapter 3)
allows to constrain the mass distribution of the bulge, disk, and halo. Therefore,
the global galaxy–halo connection can be mapped into the inner structural and
11
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 12
dynamical properties of disk galaxies, which are related to empirical quantities as
the maximum circular velocity or the circular velocities of the bulge, disk and halo at
a given radius like 2.2 scale radii; therefore, one can pass from the Mbar–Mh or M∗–
Mh relations to, for instance, the Vmax–Mbar, Vmax–M∗, (VD/VT)2.2−M∗ and other
galaxy correlations. In the same way, once generated a population of disk galaxies
through our model, we can explore whether the galaxy stellar mass function or the
Vmax function are consistent or not with, e.g., the TF and radius–mass relations.
This is a very instructive mapping of galaxy correlations to abundance distributions
and viceversa (see e.g., Blanton et al., 2008).
Following, we describe the relevant empirical or semi-empirical correlations that
will be used in our work, either for constraining the model or for comparing with its
predictions.
2.1 The (semi-emprical) baryonic-to-halo mass re-
lation
The galaxy Mbar–Mh relation will be used as input in our static model of disk
galaxies. As mentioned in the Introduction, this relation summarizes most of the
relevant galaxy evolutionary processes. The change of the baryonic fraction fbar ≡
Mbar/Mh with Mh tell us about the ability of the halo of mass Mh to capture and
cool gas, to transform the cold gas into stars, as well as about the role that play the
stellar (mainly through SNe) and AGN feedbacks and the processes of gas loss from
the galaxy. The determination of the fbar–Mh (or Mbar–Mh) relation is not an easy
task mainly due to the difficulty in measuring directly the mass at the virial radius
(see the Introduction for a discussion). Besides the few and yet limited direct studies,
the galaxy-halo connection has been stablished through statistical approaches that
combine the observed galaxy distribution functions and spatial correlation with the
halo/subhalo distributions measured in large cosmological simulations.
In recent years, the new generation of galaxy surveys have provided us with uni-
form and precise measures of properties for a large number of galaxies, complete in
magnitude or volume. Simultaneously, the impressive progress made in cosmological
N-body simulations, allows us to have precise predictions for the evolution, prop-
erties, and distributions of the ΛCDM halos (see a recent review in Knebe et al.,
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 13
2013). In the light of these advances, several statistical approaches have recently
emerged for connecting galaxy properties, such as luminosity, M∗, Mg, Mbar, etc.,
to their dark matter halos in a semi-empirical way. Among them, the abundance
matching technique (AMT) is one the most popular (see the Introduction and ref-
erences therein). In its most simple form, the observed cumulative galaxy number
density at a given property is matched against the theoretical halo plus subhalo
cumulative number density (see Fig. 1.1). As a result, one obtains the relation be-
tween M∗ and the halo mass, without requiring knowledge of the underlying physics
of galaxy evolution. Instead, the results of this kind of connections allow us to infer
semi-empirically which are the key physical processes of galaxy evolution.
In its most basic form, however, the AMT assumes that galaxy properties are
determined only by halo mass, Mh, in such a way that exists a univocal relation
between e.g., M∗ (or Mbar) and Mh. In reality, the galaxy properties, like M∗ and
Mbar, are expected not being determined by only Mh but other halo properties and
evolutionary features also play a role. For example, as it was stated in the pioneering
works of Mo et al. (1998); Avila-Reese et al. (1998); Firmani and Avila-Reese (2000),
the scaling relations of galaxies are actually coined by the mass, spin parameter,
and concentration of the halos (the latter on its own is determined by the halo
mass aggregation history). The mass is probably the most relevant factor because
it establish the gravitational potential and the scaling properties of the galaxy.
However, for a fixed mass, there are other halo properties that are also relevant
for the galaxy. For instance, the halo spin parameter, λ, regulates the galaxy size,
and it plays some role also on the values of M∗, the gas fraction, and Mbar. The
halo mass aggregation history, quantified mainly by the halo concentration, C, is
expected to influence on intensive properties as the galaxy color and gas fraction
but also partially on M∗ and Mbar.
As the result of the effects of different halo parameters on the galaxy properties,
the connection between M∗ or Mbar and Mh is expected to have some intrinsic
scatter, i.e. the galaxy stellar or baryonic masses do not depend only on Mh but
they can be affected also by λ, concentration, mass aggregation history, etc. This
has led to different authors to take into account an intrinsic scatter around the M∗–
Mh (or Mbar–Mh) relation. Commonly, previous studies have found that the width
of this scatter is small and essentially constant with halo mass, but see Rodriguez-
Puebla et al. (2014). An interesting question is whether the scatter around the
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 14
galaxy-halo relation correlates or not with a given galaxy property. Recent studies,
from both direct and indirect probes of halo masses, have found that on average red
galaxies reside in more massive halos than blue galaxies. Potentially, this difference
is showing that galaxy and halo properties are correlated in a non-trivial way. This
is an important point in studies focused to study a particular sample of galaxies,
like late-type galaxies, as it is the case in this Thesis.
Since theM∗–Mh relation of blue galaxies segregates from the average (Rodriguez-
Puebla et al., 2014), then it is important to determine also the Mbar–Mh relation
of blue galaxies in order to correctly study the structural and dynamical correlations
of disk galaxies, which tend to have namely bluer colors. This has been done in
Calette (2014), who used a modification of the abundance matching technique to
include information on gas masses and able to separate the galaxy population into
two groups: blue, disk-dominated and red, bulge-dominated galaxies; the method
was called Multi-Abundance matching Technique Constraints in Halos, MATCH.
By using MATCH, a population of 3 × 106 galaxies that follow the observed
galaxy stellar mass function (GSMF) down to M∗,lim = 107M⊙ at z ∼ 0.1 and agree
with the empirical atomic gas-M∗ and molecular gas-M∗ relations was generated.
The GSMF was constructed from joining two observational determinations, both
based on the NYU-VAGC SDSS catalog: one by Baldry et al. (2008) from the
DR4, corrected to be approximately complete in volume down to M∗ ∼ 107 M⊙,
and other by Rodriguez-Puebla et al. (2014), from DR4 for a larger volume and
complete for masses above M∗ ∼ 109. Rodriguez-Puebla et al. (2014) corrected their
stellar masses in order to take into account that the galaxy surface brightness profiles
extends further away than the commonly used aperture limits in the SDSS, specially
for central massive galaxies, see e.g., Bernardi et al. (2013); Kravtsov et al. (2014).
This implies larger stellar masses, specially for the most massive (mostly central)
galaxies. Baldry et al. (2008) applied some corrections to volume completeness at
low masses, this is why their GSMF could be extended down to M∗∼ 107 M⊙.
However, at the high-mass end the Rodriguez-Puebla et al. (2014) GSMF is more
suitable due to the larger volume of the updated SDSS data release they have used.
Galaxies in MATCH were divided into two populations, blue/late-type and
red/early-type galaxies, respectively. The idea in Calette (2014, as well as in
Rodriguez-Puebla et al. 2015) was to have an approximate division of the over-
all galaxy population into two groups, which roughly can be associated to disk-
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 15
dominated and bulge-dominated galaxies. Because for the construction of the GSMF,
an updated version of the NYU-VAGC SDSS catalog (Blanton et al., 2005b) was
used, where colors are reported for almost all the galaxies, then the color should be
used as the property to separate galaxies into two groups. On the other hand, the
empirical correlations of gas-to-stellar mass were inferred from a diverse compilation
from the literature, where in some cases galaxies are characterized by their colors
and in other cases, by their morphology or star formation activity. It is well known
that color correlates with the galaxy morphology and star formation activity. How-
ever, it is also known that there is a fraction of galaxies that do not follow these
correlations and, according to the criteria used to divide them into two groups, they
can be disk-dominated but red and passive (e.g., Maller et al., 2009; Masters et al.,
2010, the fraction of these galaxies is not larger than ∼ 10% of the late-type ones
and it is more common among massive galaxies), or bulge-dominated but blue and
star forming (e.g., Schawinski et al., 2009, these galaxies are less than 6% of early-
type ones and are not found among massive galaxies). The M∗–Mh and Mbar–Mh
relations that Rodriguez-Puebla et al. (2014) and Calette (2014) attempted to infer
are not for a fine and continuous separation of galaxies by color or morphological
type, but only for two rough galaxy populations. Therefore, the “contaminations”
produced by the above mentioned (few) deviations from the general correlations are
not so relevant as to change these mass relations when separated just in two groups.
In order to separate the GSMF into blue and red galaxies, Calette (2014) used
a fit to the fraction of red (blue) galaxies from the NYU-VAGC SDSS DR7 catalog
used in Rodriguez-Puebla et al. (2014):
fred =
1
1 + (M∗/Mchar)γ
, (2.1)
where Mchar is the mass where fred = fblue = 0.5. The fitting to the data provides
that log(Mchar/M⊙) = 10.1 and γ = −0.40; the blue fraction is fblue = 1 − fred.
Colors in Rodriguez-Puebla et al. (2014) were K-corrected to z = 0.1 and separated
into blue and red galaxies based on the color-magnitude criteria of Li et al. (2006).
The latter authors used a bi-Gaussian fitting model to separate blue and red galaxies
in several absolute magnitude bins. They have used the 0.1(g − r) color and a
definition of the magnitudes at z = 0.1 Calette (2014) uses these fractions in order
to compute the GSMFs of blue and red galaxies as: φg,i = fi × φg, where subscript
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 16
‘i’ refers either to blue or red galaxies. Note that for masses smaller than ∼ 5× 108
M⊙, the red/blue fractions are just extrapolations of the determinations mentioned
above. In any case, observations suggest that most of dwarfs are indeed blue, star-
forming galaxies, unless they are satellites. However, the fraction of satellite to
central galaxies is small, specially at low masses.
The strategy followed in Calette (2014) consisted in generating a Monte Carlo
mock catalog of 3 × 106 galaxies. Each galaxy is extracted with a stellar mass
M∗ from the overall GSMF described above, taking into account their error bars.
For a given M∗, whether the extracted object is blue or red is randomly assigned
from the probability given by eq. (2.1). Then, to each galaxy are assigned HI
and H2 masses, MHI and MH2
, respectively, by using the MHI-to-M∗ and MH2
-to-
M∗ correlations recently constrained for blue/late- and red/early-type galaxies in
Calette (2014); these correlations have intrinsic scatters, which are assumed to be
lognormally distributed, and were taken into account in the Monte Carlo trials.
These empirical correlations are based on a compilation and homogenization of
observations ofMHI andMH2
for local galaxies with known stellar masses and galaxy
type and/or color. The correlations inferred in Calette (2014), are presented in
Appendix A. Thus, galaxies in the MATCH mock catalog reproduce by construction
the observational correlations of MHI and MH2
with M∗ along with their scatters
for both blue/late-type and red/early-type galaxies. As a direct consequence of the
above, galaxies in the MATCH catalog have also defined the total cold gas mass,
Mg = (MHI + MH2
) + M(He+Z) = 1.4(MHI + MH2
), and the total baryonic mass,
Mbar = Mg +M∗. The factor 1.4 is for accounting for He and metals.
The MATCH mock galaxy sample, by construction, is volume-complete in stellar
mass down to ∼ 107 M⊙. By applying the abundance matching technique for blue
and red galaxies, Calette (2014) inferred the M∗–Mh for the blue and red galaxy
populations. For this, it is necessary to know the probability of a subhalo/halo to
host a blue or red galaxy. In the case of central galaxies (associated to halos), such
a probability as a function of Mh has been inferred semi-empirically in Rodriguez-
Puebla et al. (2014). In the case of satellite galaxies (associated to sub halos), Calette
(2014) introduced an approximation, based on Peng et al. (2012), to calculate the
probability of a subhalo to host a blue or red satellite, which assumes that this
probability is similar to the case of central galaxies but corrected by a factor that
depends on the effects of satellite quenching due to the halo environment. In any
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 17
case, have in mind that the overall population of galaxies is dominated by centrals,
so that any uncertainty in the proposed approximation for satellites/subhalos, has
a minor effect on the overall probability as a function of Mh.
From the abundance matching applied to the blue and red total GSMF’s, the
M∗–Mh relations of blue/red galaxies, including their scatters, are found. Then, a
halo mass is assigned to each blue or red galaxy of mass M∗ of the mock catalog
by using the corresponding M∗–Mh relation and its lognormally distributed scatter.
For the given M∗ and color type, as mentioned above the empirical MHI-to-M∗ and
MH2
-to-M∗ correlations are used to assign MHI and MH2
to the galaxies, taking
into account the lognormal scatter distributions of these correlations. Therefore, it
is possible to construct the galaxy mass functions for HI, H2, cold gas, and baryons,
separately for blue and red galaxies, which are associated to late- and early-type
galaxies, respectively. The obtained total HI and H2 mass functions agree well with
direct determinations of them using surveys in radio as ALFALFA, HIPASS, and
others (Calette, 2014). Note that we need many Monte Carlos trials for seeding
galaxies in the halos in order to sample well the halo mass function as well as the
scatter distributions of the GSMFs, the blue/red galaxy fractions, the blue/red halo
mass fractions, the M∗-to-Mh relations, and the empirical MHI–M∗ and MH2
–M∗
relations. Finally, the obtained relation between Mbar and Mh (or fbar=Mbar/Mh
and Mh) and its scatter for blue/late-type galaxies will be used in our static model
of disk galaxies as one of the basic input parameters. We will use actually the best
Behroozi et al. (2013) function fit to these results:
log10(Mbar/Mh) = log10(ǫM1) + f(x)− f(0), (2.2)
where
f(x) = −log10(10
−αx + 1) +
δ[log10(1 + exp(x))]γ
1 + exp(10−x)
, (2.3)
and x = log10(Mh/M1). The recent inferred parameters are: log10M1 = 11.69,
ǫ = 0.0668, α = 1.5, δ = 2.6993, γ = 0.8275.
In Fig. 2.1 the Mbar–Mh relation of blue galaxies (left panel) as obtained in
Calette (2014) is plotted along with its intrinsic scatter in dex (lower panel). For
comparison, the relation for red galaxies and the average relation of both populations
are also plotted. The corresponding fbar–Mh relations are plotted in right panel.
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 18
Figure 2.1: The Mbar–Mh relations of blue and red galaxies (blue and red lines)
and the average relation (black line) as obtained in Calette (2014) by the MATCH
semi-empirical approach. The intrinsic scatters around these relations are shown in
the lower panel. The corresponding fbar–Mh relations are plotted in the right panel.
In Fig. 2.2, we plot the best Behroozi function fit for blue/late-type galaxies
with its respective intrinsic scatter. The fbar–Mh relation and its scatter are inputs
of our static model. For the intrinsic scatter, we have fitted two linear functions
in the log-log plots to the scatter reported in Calette (2014) (blue line in the lower
panel of Fig. 2.1).
12 -1 
~ 11 0 
:::;: 
~ 
10 
• • ~ 
,Q :::;:"' -2 :::;: 9 
'-. 
Oll • • o B ,Q 
~ :::;: 
~ 
7 Oll 
-3 0.6 o 
~ 
1 
'" 0.4 
~ 
.9 
b 0.2 
O -4 
10 11 12 13 14 10 11 12 13 14 
log Mh [M0 l log Mh [M0 l 
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 19
Figure 2.2: The mean with its scatter (gray shaded area) are plotted for the fbar–Mh
relation for the blue galaxies.The black solid line is the Behroozi-like function that
best fits the MATCH result for the blue Mbar-Mh relation.
2.2 The size-stellar mass relation
One of the most important scaling relation is the one between size and mass of a
galaxy. The size of the galaxies is related to the overall scale of the system (given
for instance by the mass) and to the angular momentum. Therefore, it is expected
that the scatter around the mean size–mass relation is associated mainly to the spin
parameter λ: for a given mass, the larger the λ, the more extended is expected to be
the disk. The size–mass relation is a key observational constrain in our approach.
Since the size of galaxies for a given mass depends mainly on the spin parameter,
this relation will be important for constraining namely the spin parameter.
For late-type disk galaxies, their surface brightness (density) profiles are typically
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 20
well fitted by an exponential distribution, characterized by the scale radius, RD. For
a perfect exponential disk, the radius where half of the luminosity (mass) is contained
is given by Re=1.68RD. However, the presence of a bulge, which is typically more
concentrated than a exponential disk, can alter this equality for real galaxies, more
as larger is the bulge-to-disk ratio. Because the bulge-to-disk decomposition is not
an easy task, specially when applied to large surveys, the most common reported
characteristic radius of galaxies in these surveys is Re.
Following, we report and discuss several Re–M∗ relations for blue or late-type
galaxies obtained from large surveys with different methods. These relations will be
compared to our model results and, eventually, they will help to constrain the model.
• Shen et al. (2003, S03): From the DR1 SDSS survey, these authors separated
galaxies into late- and early-type by using the photometric concentration index C
and index Sérsic n. The former were assigned to those with C < 2.86 and n < 2.5;
the latter to those with C > 2.86 and n > 2.5. The stellar masses are obtained from
the Max-Planck-Institute for Astrophysics–Johns Hopkins University (MPA-JHU)
catalog (Kauffmann et al. (2003b); Salim et al. (2007)). The Re–M∗ relations of
both groups are different, with low-concentration galaxies being larger for a given
M∗ than high-concentration ones. The low-concentration galaxies follow the mean
and intrinsic scatter relations:
Re[kpc] = γ
(
M∗
M⊙
)α (
1 +
M∗
M0
)β−α
(2.4)
and
σlnRe
= σ2 +
(σ1 − σ2)
1 + (M∗/M0)
, (2.5)
respectively, where α, β, γ and M0 are free fitting parameters. α and β represent the
slopes of the relation, and M0, the characteristic mass, which determines the stellar
mass at which the slope of the relation change. The values of these parameters are:
α = 0.14, β = 0.39, γ = 0.10, M0 = 3.98× 1010 M⊙ , σ1 = 0.47 and σ2 = 0.34. The
corresponding relation with its scatter, are plotted in Fig. 2.3 (black solid line with
the dotted lines for the scatter); note that the scatter in the figure is in dex.
• Mosleh et al. (2013): These authors present their size-stellar mass relation
from the New York University Value-Added Galaxy Catalog (NYU-VAGC; Blanton
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 21
et al. (2005c)) updated with the DR7 data from the SDSS (Abazajian et al. (2009)).
The stellar masses are also obtained from MPA-JHU catalog updated to the DR7.
The galaxies are classified into different sub-samples based on their color, morphol-
ogy and specific star-formation rate (sSFR). We use the sub-sample based in color.
Following Blanton et al. (2005c), the criterion to separate galaxies into red and blue
objects is: (g−r) = 0.68−0.032(Mr+20). To measure the effective radii, they used
two methods: (a) ”parametric”, where the half-light radius of galaxies is calculated
by using best-fit, two-dimensional analytic models, and (b) ”non-parametric”, where
the observed one-dimensional light profiles and the total flux are used. They use
the same functional form employed for late-type (low-concentration) galaxies in S03
(see eq. 2.4 and 2.5 above) for their Re–M∗ and intrinsic scatter relations. The cor-
responding fitting parameters for the sample selected by color are α = 0.185±0.081,
β = 0.329±0.164, log(γ) = −1.406±0.750, M0 = 10.325±0.299, σ1 = 0.574±0.059
and σ2 = 0.056 ± 0.122. The mean relation for this parameters are plotted in Fig.
2.3 (magenta dots connected with solid line).
• Cebrain & Trujillo (2013). These authors considered the effect of envi-
ronment where galaxies are. As in Mosleh et al. (2013), the sample is taken from
SDSS-DR7. The estimates of stellar mass are obtained from (Blanton and Roweis
(2007)) using a Chabrier (2003) Initial Mass Function (IMF) with population syn-
thesis model from Bruzual and Charlot (2003). The size of the galaxies were deter-
mined using a Sersic (1968) fit to the radial intensity profile obtained using circular
apertures (Blanton et al. (2005a)). This relation is studied in different large scale
environments, considering whether the galaxies are in a high-density or a low-density
region. The galaxies are segregated by their morphology (using the Sérsic index as
we describe above). Consequently, it is applied a ”Maximum Likelihood estimator”
to obtain the most likely values for the mean size and the scatter of the distribution,
for a given mass bin. Thus, using this method, it can be estimated the more likely
mean size R(M), dispersion σlnR(M) and their 1σ-errors. We use the data given
in tables A1 and A2 of Cebrian and Trujillo (2014) for the Re–M∗ relation and its
dispersion, which is displayed in Fig.2.3 (green dots connected with solid line and
dashed lines for the corresponding scatter).
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 22
Figure 2.3: Stellar mass-size relation of observed galaxies in different samples. In left panel,
are plotted the relations for total effective radius. The relation inferred by Shen et al. (2003) are
plotted with black solid and dashed lines for the mean and its intrinsic scatter. The same relation
and scatter but with parameters inferred for blue galaxies in Mosleh et al. (2013) are plotted with
magenta dots connected with solid line. The green dots connected with solid line corresponding to
the observations of Cebrian and Trujillo (2014). The sample studied in Bernardi et al. (2012) are
represented with brown dots connected with solid line. The blue dots with error bars correspond
to observations from Baldry et al. (2012). In right panel are plotted with black solid and dashed
lines, the relations for disk effective radii, assuming that Re is 1.68RD, from Dutton et al. (2011).
• Bernardi et al. (2012): These authors obtain their stellar mass-size relation
using a sample from SDSS DR7 (as both previous authors). The estimated stellar
masses come from Bernardi et al. (2010), assuming a Chabrier IMF. To quantify
stellar mass-size is performed fits to the Main Galaxy Sample images using the
PyMorph package. This alghoritm is used to fit single component deVaucouleurs
and Sérsic profiles, and two component exponential + deVaucouleurs (deVExp) and
exponential + Sérsic (SerExp) profiles to each image. As a result of their analysis,
they find that the reductions of the SerExp model to return the least biased estimates
1.5 1.5 
• Eern..,.<l.i (2014) 
• Cebr"in-Trujillo (2014) 
_ Dutton et ,,[. (2011) 
• llo . 1eh et al. (2013) 
1 • Ea ldry (20 11) 1 
_ Sh"n "t ,!l, (2003) 
0.5 0.5 
O I O 
-0.5 -0.5 
7 8 9 10 11 12 7 8 9 10 11 12 
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 23
of R and M∗, and hence the R−M∗ relation, finding the following relation:
〈log10Re/[kpc]|O〉 = p0 + p1O + p2O
2, (2.6)
where O = log10(M∗/M⊙) and p0, p1 and p2 are the fitting parameters. We use
the values of parameter for late-type galaxies presented in the tables 3 and 4 from
Bernardi et al. (2012): p0 = 11.2699, p1 = −2.3026 and p2 = 0.1227. In Fig. 2.3 the
corresponding relation is plotted with brown dots connected with solid line.
All the relations presented above are for the total effective radius, Re, i.e., they
do not take into account any bulge-disk decomposition. If the bulge-disk decompo-
sition is applied, then in principle it can be obtained the scale radius, RD, of the
exponential disk. In this case, Re=1.68RD. The next authors present the size–mass
relations inferred for only the disks.
• Dutton et al. (2011) present Re obtained as 1.68RD for the SDSS DR7 sam-
ple, where RD was measured from 2D bulge+disk decompositions, using GIM2MD
simultaneously on g and r images. For this sample, the stellar masses are calculated
using the relation between the r-band mass-to-light ratio and (g−r) color from Bell
et al. (2003) with an offset of −0.1 dex to correspond to a Chabrier (2003) IMF.
They select blue-cloud galaxies for this study. The criterion to select blue galaxies is
by requiring that (g − r) < 0.71− 0.067(log10 M∗ − 10). The median of the relation
between Re=1.68RD and M∗ found in Dutton et al. (2011) is given by:
Re = R0
(
M∗
M0
)α [
1
2
+
1
2
(
M∗
M0
)γ](β−α)/γ
, (2.7)
where α is the slope at low massesM∗ ≪ M0, β is the slope at high massesM∗ ≫ M0,
γ control the sharpness of the transition between the two slopes, M0 is the transition
mass, and R0 is the value of the R at this M0. They find that log10(M0/[M⊙]) =
10.44, log10(R0/[kpc]) = 0.72, α = 0.18, β = 0.52 and γ = 1.8 provide a good fit to
the data. The scatter around the mean relation is assumed as lognormal distributed
with s = σlnR|M , given by following relation:
s = s2 + (s1 − s2)/[1 + (M∗/M0)
γ]. (2.8)
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 24
Here s1 is the scatter at low masses, s2 is the scatter a high masses, M0 is the
transition mass and γ control the sharpness of the transition. They find that s1 =
0.47, s2 = 0.27, log10(M0/[M⊙]) = 10.30 and γ = 2.2 give a good fit to the data.
In the right panel of Fig. 2.3, we plot the mean relation and its scatter as given
in Dutton et al. (2011) for blue galaxies. Recall that, though it is given for Re,
it was actually obtained for RD, by using disk-bulge decomposition, and Re was
just assumed to be 1.68RD. Therefore, in this case Re is actually the effective or
half-mass radius of the disk and not of the total (bulge+disk) galaxy.
2.3 The Tully-Fisher relation
The relation between the maximum rotation velocity, Vmax, and the luminosity
(or stellar mass) of disk galaxies is called the Tully-Fisher (TF) relation. This is
probably the tightest correlation in Extragalactic Astronomy and it was subject of
hundreds of observational studies since it was stablished (Tully and Fisher, 1977).
The stellar and baryonic TF relations are predicted in the static model of disk
galaxies. Therefore, we need observational TF relations (both stellar and baryonic)
in order to compare with the predictions of our static model.
In Avila-Reese et al. (2008), the authors compiled a sample of disk galaxies
larger than M∗ ∼ 109 M⊙ with the observational information necessary to obtain
the stellar masses, Vmax, radii, colors, gas masses, etc. including the uncertainties.
The sample is not pruned to minimize the scatter in the TF relation, as usually
was done in the literature in order to use this relation as a distant estimator. The
intrinsic scatter of the TF relation contains valuable information related to galaxy
formation and evolution and we will also compare it with our model predictions.
Since the compiled sample of galaxies by Avila-Reese et al. (2008) contains in-
formation on stellar mass and cold gas masses, they determined both the stellar and
baryonic TF relation with their respective intrinsic scatters (the latter, after resting
the observational uncertainties). The compiled sample is presented in Fig. 2.4 for
both the stellar and baryonic cases. The authors carried out several line fittings
to the log-log data. We reproduce in Fig. 2.4 their orthogonal fits including the
intrinsic scatters. These are the fits that we will use to compare with our model
predictions.
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 25
The determination of the TF relation for dwarf galaxies is difficult due to obser-
vational limitations. It seems that the intrinsic scatter strongly increases as smaller
are the masses. There are some pieces of evidence that the main relation bends
slightly at M∗ ∼ 3− 5× 109 M⊙ in the sense that at a given Vmax smaller galaxies
have lower stellar masses than the corresponding to the extrapolation of the TF
relation stablished at larger masses. This has been also found in cosmological N-
body+Hydrodynamics simulations (e.g., de Rossi et al., 2010, see for observational
references therein). In Fig. 2.4 we plot individual estimates of Vmax and M∗ for
dwarf galaxies as reported in McGaugh (2005) and Geha et al. (2006).
Figure 2.4: Observational dependence of Vmax as a function of stellar mass (left panel) and
baryonic mass (right panel) for several samples. The blue dots with error bars and the orthogonal
fits with 1σ intrinsic scatter (red solid and dotted lines) in both panels represent a compiled sample
of disk galaxies by Avila-Reese et al. (2008). The brown dots and its error bars, are the the observed
dwarf galaxies from the compiled sample by Geha et al. (2006). The observed dwarf galaxies from
McGaugh (2011) are the green dots with corresponding error bars.
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 26
2.4 The bulge-to-total mass ratio vs M∗
The bulges contain important clues to galaxy formation and evolution. It is believed
that the bulges are the result of dynamical processes. Those called classical seem
to have formed through violent processes, such as hierarchical clustering via major
mergers. Those formed through longer time-scales, via disks instabilities and secular
evolution processes are called pseudo-bulges (e.g., Wyse et al., 1997; Kormendy,
2013).
• Gadotti (2009).- The data used for the bulge-to-total mass ratio by this
author are taken from the SDSS. He performed 2D bulge/bar/disk decom-
positions using g, r and i-band images of a representative sample of 1000
galaxies. He finds that the Petrossian concentration index is a better proxy
for the bulge-to-total ratio than the global Sérsic index. He also finds that
the pseudo-bulges can be distinguished from classical bulges as outliers in the
Kormendy relation. The sample is composed by all objets spectroscopically
classified as galaxies in the SDSS DR2, applying the following criterion. First,
all galaxies with redshift in the range 0.02 ≤ z ≤ 0.07 are selected. Secondly,
all galaxies with stellar masses below 1010M⊙ are excluded. Galaxy stellar
masses were obtained from Kauffmann et al. (2003a), having then a volume-
limited sample, i.e., a sample which includes all galaxies more massives than
1010M⊙ in the volume defined by redshifts cuts and the DR2 footprint. Fi-
nally, galaxies with an axial ratio b/a ≥ 0.9 are selected, where a and b are
the semi-major and semi-minor axes of the galaxy, respectively, taken from
the SDSS data base (these are measured at the 25 mag arcsec−2 isophote in
the g band).
• Fisher and Drory (2011).- These authors present an inventory of non-
edge-on galaxies, from elliptical galaxies to galaxies with classical and pseudo-
bulges and bulgeless galaxies, in a volume-limited sample within the local 11
Mpc sphere complete down to MB < −15.2, using Spitzer 3.6 µm and Hubble
Space Telescope (HST) data. They studied the abundance of pseudo-bulges
and classic bulges. The sample is taken from the Kennicutt et al. (2008) local
survey, requiring Galactic latitudes of |b| > 20◦. The final sample contains 320
galaxies. The full sample and measured quantities are listed in their Table 1.
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 27
Their galaxies are decomposed into a Sérsic bulge and an exponential disk,
where the decompositions are taken from Fisher and Drory (2010). The Sérsic
index, n, is used to diagnose bulges into pseudo- (n < 2) and classical (n > 2)
Fisher and Drory (see 2008). For those bulges with n ∼ 2, is supplemented
bulge identification with nuclear morphology from (HST) images. Ellipticals
are assigned B/T = 1. Galaxies in which the decomposition yields B/T < 0.01
are assigned B/T = 0 and are called ”bulgeless”. The total luminosity is de-
terminate by integrating the near-IR SB profile and convert to stellar mass
using RC3 B-V color as described in Fisher et al. (2009) following Bell and de
Jong (2001).
• Weinzirl et al. (2009).- The data set analyzed by these authors are derived
from the 182 H -band images from the public data release of the Ohio State
University Bright Spiral Galaxy Survey (OSUBSGS; Eskridge et al. (2002)).
OSUBSGS is a magnitude-limited survey with objects whose distance range
up to ∼ 60 Mpc. The sample is reasonably complete for bright (MB ≤ −19.3
or LB > 0.33L∗) galaxies. To derive global stellar masses for most of the
OSUBSGS sample galaxies is used the relation between stellar mass and rest-
frame B−V color from Bell et al. (2003). Using population synthesis models,
the latter study calculates the stellar M/L ratio as a function of color using
functions of the form log10(M/L) = aλ + bλ ×Color+C, where aλ and bλ are
bandpass dependent constants and C is a constant that depends on the stellar
initial mass function (IMF). For the V band, Bell et al. (2003) find aλ = −0.628
and bλ = 1.305; assuming a Kroupa et al. (1993), they find C = −0.10.
The Weinzirl et al. (2009) sample is limited to only disk galaxies, which they
assume to be those with B/T< 0.5. The bulge-to-disk decomposition is carried
out using several alghoritms for two-dimensional luminosity decomposition,
including GIM2D (Simard et al. (2002)), GALFIT (Peng et al. (2002)) and
BUDDA (deSouza et al. (2004)).
• Cibinel et al. (2013).- The data and B/T ratios presented by these authors
are taken from the Zurich Environmental Study (ZENS; Carollo et al. (2013),
which is based on a sample of 1484 galaxies, members of 141 galaxy groups
extracted from the 2-degrees Field Galaxy Redshift Survey (2dFGRS; Colless
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 28
et al. (2001), Colless et al. (2003)), and specifically from the Percolation-
Inferred Galaxy Group (2PIGG) catalog Eke et al. (2004). The group halo
mass, the distance from the center of the group, the rank within the group
(i.e., whether the galaxy is the central or a satellite) and the location on the
large-scale structure (LSS) are given for each galaxy. The bulge-disk decompo-
sition is carried out by using the Galaxy IMage 2D (GIM2D) software package
(Marleau and Simard (1998); Simard et al. (2002)).
Figure 2.5: The observationally inferred B/T mass ratio as a function of stellar mass. The blue
filled circles with error bars are from the groups sample of Cibinel et al. (2013). The red filled
squares with error bars are from Weinzirl et al. (2009) and refer only to disk-dominated galaxies
(B/T< 0.5). The green dots with error bars are from Gadotti (2009). The black filled squares
with error bars are from the local 11-Mpc volume study of Fisher and Drory (2011).
2.5 The gas-to-stellar mass ratio vs. stellar mass
The determination of the gas fractions for large samples of galaxies is not an easy
task since it requires of dedicated campaigns of observations in radio, both for the
atomic HI and molecular H2 gas contents. For the latter, a tracer of H2 should be
CHAPTER 2. OBSERVATIONAL CORRELATIONS OF DISK GALAXIES 29
used for massive studies, for example the CO emission lines. Besides, in order to
infer the gas fractions or the gas-to-stellar mass ratios, observations in the optic that
provide the data to infer M∗ are necessary.
A recent compilation and homogenization of data from the literature of samples
that contain information on M∗ and MHI and/or MH2
was presented in Calette
(2014). The total cold gas mass is calculated as Mg = (MHI +MH2
) +M(He+Z) =
1.4(MHI +MH2
), where the factor 1.4 accounts for helium and metals. In Calette
(2014), blue/late-type and red/early-type galaxies were treated separately because
the gas fractions in both populations are quite different. The author determined the
MHI-M∗ and MH2
–M∗ correlations, and from them inferred the Mg–M∗ correlation,
all the time by separate for blue/late-type and red/early-type galaxies.
In Fig. 2.6, the Mg/M∗–M∗ correlation for blue/late-type galaxies as inferred
from observations in Calette (2014) is plotted. The dashed lines correspond to the
1σ scatter.
Figure 2.6: Cold gas-to-stellar mass ratio of blue/late-type galaxies as a function of M∗ as
inferred in Calette (2014). The dashed lines are the 1σ scatter.
Chapter 3
The Semi-empirical Method
3.1 The static model
3.1.1 A disk-halo model
As already mentioned above, the ”static” model introduced by Mo et al. (1998)
(hereafter MMW98) became a powerful tool for probing the correlations of disk
galaxies formed inside CDM halos. We call ”static” this model in order to differen-
tiate it from those where the evolution of the galaxy is followed (e.g., Firmani and
Avila-Reese, 2000; Avila-Reese and Firmani, 2000; van den Bosch, 2000b; Dutton
and van den Bosch, 2009). In the static model, a disk is loaded instantaneously into
the gravitational potential of a ΛCDM halo, and the dynamics is solved in order the
disk settled into centrifugal equilibrium, taking into account its drag onto the inner
halo. The halo is assumed to be spherical and with an initial NFW density profile
(Navarro+1997):
ρ =
4ρs
(r/rs)(1 + r/rs)2
, (3.1)
where rs is the characteristic radius at which the logarithmic slope of the density
distribution dlnρ/dlnr = −2 and ρs = ρ(rs). We define the virial radius, Rvir, as
the radius that encloses a mean density equal to ∆vir times the mean density of
the universe, where ∆vir is obtained from the spherical top-hat collapse model. The
30
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 31
halo mass profile obtained by integrating equation (3.1) is:
M(r) = 8πρsr
3
s
[
1
1 + Cx
− 1 + ln(1 + Cx)
]
, (3.2)
where x ≡ r/Rvir and
C =
Rvir
rs
(3.3)
is the NFW concentration parameter. We also define the virial velocity Vvir as the
circular velocity at the virial radius, V 2
vir = GMvir/Rvir, where G is the gravitational
constant, Mvir is the virial mass and Rvir is the virial radius.
The angular momentum of the halo is expressed through the so-called spin pa-
rameter λ:
λ =
Jvir |E|
1
2
GM
5
2
vir
=
Jvir/Mvir√
2RvirVvir
f
1
2
c , (3.4)
where Jvir is the total angular momentum at the virial radius, E is the halo energy
and
fC =
C
2
1− 1/(1 + C)2 − 2ln(1 + C)/(1 + C)
[C/(1 + C)− ln(1 + C)]2
, (3.5)
which measures the deviation of E from that of a singular isothermal sphere with
the same mass (see MMW98). The spin parameter is a measure of the centrifugal
rotational support of a structure against the potential energy. For λ = 1, the
structure is fully supported by rotation, while for λ << 1, rotation is not important
for the virial equilibrium.
Due to the gravitational effects produced by forming galaxies, the halo contracts
its inner regions. This happens when baryons cool and concentrate in the center of
the spheroid, modifying the shape of the gravitational potential. Since this process is
slow with respect to the dynamical time of the system, the halo contracts to conserve
its adiabatic invariants taking into account that its spherical symmetry remains.
This way, the angular momentum of individual dark matter particles, which moves
on circular orbits, is conserved during contraction. Therefore, a particle that is
initially (before the formation of the disk) at radius ri, ends up at radius rf , where
GMf (rf )rf = GMi(ri)ri. (3.6)
Here Mi(ri) is given by equation (3.2) (NFW profile) and Mf (rf ) is the total final
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 32
mass within rf . The final mass is the sum of the dark matter mass inside the initial
radius ri and the mass of an exponential disk (Blumenthal et al. (1986)):
Mf (rf ) = Md(rf ) +Mh,f (rf ) = Md(rf ) + (1− fbar)Mi(ri), (3.7)
where fbar is the fraction of the total mass locked in the baryonic disk of mass Md,
that is, fbar ≡Md/Mh, and
Md(r) = Md
[
1−
(
1 +
r
RD
)
e−r/RD
]
. (3.8)
Thus, from equations (3.6) and (3.7), we can find iteratively a contraction factor
Γ(ri) ≡ rf
ri
, which allows us to calculate the mass distribution of the contracted dark
matter halo. The relation between ri and rf can be generalized as
rf = Γνri, (3.9)
where ν is a free parameter and varies according the type of contraction. For ex-
ample, ν = 1 is the standard contraction for radial orbits from Blumenthal et al.
(1986), ν = 0 is for non contraction, and ν < 0 corresponds to models of expansion
of the inner dark matter halo instead of contraction. In Dutton et al. (2007) it was
proposed that most of halos actually should expand due to galaxy formation, be-
cause of the strong effects of SN-driven outflows. In a more standard context, based
on numerical cosmological hydrodynamical simulations, Gnedin et al. (2004) have
found that the Blumenthal et al. (1986) adiabatic contraction formalism should be
slightly corrected (the particle orbits deviate from the pure circular motion). The
corrections suggested by their results is such that instead of the standard value
ν = 1, one should use ν = 0.8. This is the value that we will use in our model.
In the static model of disk galaxies in centrifugal equilibrium loaded inside
ΛCDM halos, it is assumed that the baryons initially had the same density profile
and angular momentum distribution as the dark matter. Therefore, if we assume
detailed angular momentum conservation (i.e., the specific angular momentum of
the disk, Jgal/Md remains the same as that one of the halo, Jvir/Mh), then the spin
parameter of the baryons that will form the disk is the same than the halo one,
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 33
λbar = λh. If this assumption is relaxed, then
λbar ≡
(
jgal
fbar
)
λh, (3.10)
where
jgal = Jgal/Jvir. (3.11)
The total angular momentum for a disk with a surface density distribution Σ(R)
and a circular velocity profile V (R) is
Jgal = 2π
∫ Rvir
0
Σ(R)RVT(R)RdR. (3.12)
We assume that the disk has an exponential surface density distribution Σ(R) =
Σ0 exp(−R/RD), so that
Jgal = MdRDVvirfV , (3.13)
where
fV =
1
2
∫ Rvir/RD
0
e−uu2V (uRD)
Vvir
du, (3.14)
a factor that takes into account the gravitational effects of the disk. Assuming that
the angular momentum of the disk is a fraction of the total angular momentum of
the halo (i.e. Jgal = jgalJvir), we can combine equations (3.4) and (3.12) to obtain:
RD =
1√
2
(
jgal
fbar
)
λbarRvirf
−1/2
c f−1
V , (3.15)
with λbar=λh when detailed angular momentum conservation is assumed.
Once calculated the scale radius RD, we can calculate the (total) circular velocity
VT(R), which is the sum in quadrature of contributions from disk and halo:
VT
2(R) = VD
2(R) + VH
2(R), (3.16)
where
VH
2(R) = G[Mf (R)−Md(R)]/R, (3.17)
and
VD
2(R) =
GMd
RD
2y2[I0(y)K0(y)− I1(y)K1(y)] (3.18)
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 34
is the circular velocity produced by an exponential disk, where y = r/(2RD), and In
and Kn are modified Bessel functions (Freeman, 1970). However, in order to define
RD, according to eq. (3.15), we should know the total circular velocity profile (see
eq. 3.14). This circularity leads us to use an iterative approach in order to calculate
the final disk scale radius RD of the disk in centrifugal equilibrium with the potential
of the cosmological halo contracted on its own by this disk.
3.1.2 A disk-bulge-halo model
In this Section, we describe the inclusion of a bulge proposed to be formed by
internal instabilities of the disk, following Dutton et al. (2007, hereafter D+2007).
The baryonic mass of the galaxy is now the sum of a disk and a bulge:
Mbar ≡ fbarMvir = Md +MB. (3.19)
The bulge-to-disk mass ratio is defined as:
θ ≡ MB
Md
(3.20)
and the disk mass and the bulge mass are given in terms of this ratio, so that
Md =
1
1 + θ
fbarMvir, (3.21)
and
MB =
θ
1 + θ
fbarMvir. (3.22)
According to eq. (3.11), the fractional angular momentum of the baryons is
given by jgal = Jgal/Jvir. Let Jb = jbJvir be the original angular momentum of
the baryonic material out of which the bulge forms. However, during the bulge
formation process this angular momentum is transferred to the disk plus the halo,
and therefore the final angular momentum of the bulge is zero (D+2007). If ft is
the fraction of Jb that is transferred to the disk, then the final angular momentum
of the disk is
Jgal = [jgal − (1− ft)jb]Jvir, (3.23)
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 35
and the specific angular momentum of the final disk is
Jgal
Md
= (1 + θ)(1− flost)
(
jgal
mgal
)
Jvir
Mvir
, (3.24)
where the parameter flost is the fraction of the total angular momentum of the
material out of which the bulge plus disk forms and which has been lost to the halo:
flost = (1− ft)
(
jb
jgal
)
. (3.25)
We define other important parameter:
fx ≡ flost
(
1 + θ
θ
)
= (1− ft)
(
Jb/MB
Jgal/Mgal
)
, (3.26)
which expresses the ratio of specific angular momentum that has been lost to the
halo due to bulge formation to the total specific angular momentum of the material
out of which the disk and bulge have formed. This parameter should be less than
unity if the bulge is formed out of material with low specific angular momentum
(van den Bosch et al., 2002).
We can obtain a new relation for the scale length of the disk after a bulge formed
from the disk by dynamical instabilities by combining equations (3.4), (3.12), (3.11).
One obtains:
RD =
1
fV
√
2fc
[1 + (1− fx)θ]λbarRvir, (3.27)
where λbar is now the effective initial spin parameter of the material out of which
the bulge and disk form, and it is given by
λbar ≡
(
jgal
fbar
)
λ, (3.28)
where fbar is the baryonic fraction. As we can see in equation (3.25), the bulge
formation affects the final scale length of the disk through the parameters θ and
fx. Following D+2007, it is assumed that the material that forms the bulge has the
same specific angular momentum as the disk. Here, we assume fx = 0.5 (see Shen
et al., 2003).
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 36
The new total circular velocity V (R) is given by the equation:
V 2
T (R) = V 2
D(R) + V 2
B(R) + V 2
H(R), (3.29)
the component VB is the circular velocity of the bulge, which is derived from the mass
distribution of the bulge. In principle, the instabilities are instrinsic (isolated disk),
giving rise to secular evolution that ends in a pseudo-bulge (see e.g. Kormendy,
2013). Observations show that pseudo-bulges are well described by a Sérsic surface
brightness (surface density) profile with index n between 0.5 and 3. A Sérsic surface
density profile (Sérsic, 1963; Sersic, 1968), is given by
I(R) = Ie,Bexp
{
−bn
[
(
R
Re,B
)1/n
− 1
]}
. (3.30)
where Ie,B is the surface brightness at the effective radius Re,B, which is defined
as the projected radius that encloses one half of the total luminosity (mass), by
adjusting the value of bn. The conditions to sets the value of bn is (see Graham and
Driver, 2005; Ciotti, 1991):
Γ(2n) = 2γ(2n, bn), (3.31)
where Γ and γ are the Gamma and lower Incomplete Gamma Functions, respectively.
Capaccioli (1989) introduced the following approximation for bn, valid for 0.5 < n <
10:
bn ≈ 2n− (1/3). (3.32)
Assuming a spherical density distribution, the surface density profile given by equa-
tion (3.30) can be translated into a volumetric density distribution.
According to the formation mechanism, the mass distribution of bulges are ex-
pected to be different. Bulges formed by secular evolution of the disk (our start-
ing scenario) have typically the properties of pseudo-bulges (e.g., Kormendy and
Kennicutt, 2004; Avila-Reese et al., 2005), with Sérsic profiles shallower than a de
Vacouleours profile, that is with Sérsic index n < 4. Here, we assume an average
value of n = 2 for the secular bulges (see e.g., Gadotti, 2009), in which case the
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 37
mass profile is
ν(R) = MB
{
1− exp(−
√
α2η)
[
1 +
√
α2η +
(α2η)
2
+
(α2η)3/2
6
+
(α2η)2
24
+
(α2η)5/2
120
]}
,
(3.33)
where MB is the total mass bulge, α = 5.67 and η ≡ R/Re,B. The corresponding
circular velocity profile is:
VB(R) =
√
Gν
R
, (3.34)
where G is the gravitational constant.
As it will be discussed below, in the case of massive galaxies, interactions and
mergers are actually common. Therefore, the massive disks are expected to undergo
strong induced instabilities that end commonly with the formation of large bulges
with a de Vacouleours profile or n = 4 Sérsic profile. The corresponding density
profile in this case is given by the Hernquist function (Hernquist, 1990):
ρb =
MB
2π
rb
R(R +Rb)3
, (3.35)
where Rb = Re,B/1.815 is a scale length. The corresponding circular velocity is:
VB(R) =
√
GMBR
(R +Re,B)
. (3.36)
Thus, for galaxies smaller than a given mass, we will assume Sérsic distribution
with n = 2, while for massive galaxies, n = 4. The corresponding circular velocities
are given by equations (3.34) and (3.36). In both cases, the relevant parameters
are MB and Re,b. Since our model is not evolutionary, we are not in conditions of
calculating Re,B for our bulges. Instead, we use an empirical relation between Re,B
and MB, as given in Gadotti (2009):
logRe,B[kpc] =





−2.062 + 0.2log(MB/M⊙), log(MB/M⊙) < 10
−3.057 + 0.3log(MB/M⊙), log(MB/M⊙) > 10.
(3.37)
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 38
Finally, we include the bulge component to the final mass budget and eq. (3.7)
is then rewritten as:
Mf (rf ) = Md(rf ) +MB(rf ) +MDM,f (rf ) = Md(rf ) +MB(rf ) + (1− fbar)Mi(ri).
(3.38)
Thus, we calculate the adiabatic invariant dark matter halo contraction by using
this new mass distribution.
Bulge mass assignment
Let us first consider the case of internal dynamical instabilities produced by
the secular evolution of a self-gravitating disk. For this, the critical instability
parameter let be β(R) = VD(R)/VT (R). According to the numerical simulations of
Christodoulou et al. (1995) the disk is stable as long as β(R) < βcrit, where the
value of the parameter βcrit falls roughly in the range 0.52 ≤ βcrit ≤ 0.70. Following
D+2007, we calculate the mass of the disk contained within the radius where β(R)
attains a maximum and check if βmax > βcrit:
βmax = max
0≤R≤Rvir
β(R) < βcrit. (3.39)
If this is the case, then all this mass is assigned to a bulge. We use βcrit = 0.62 for
the case of an isolated disk.
However, the secular bulge formation mechanism is not suitable in the case of
galaxy interactions and mergers. The mergers transfer the secondary galaxy to the
bulge of the primary, and induce disk instabilities in the primary (Hopkins et al.,
2010, see more references therein),. In order to take into account these effects, yet
within our scheme of βmax, we propose that the bulge formation by mergers can
be emulated by lowering the value of βcrit. It is well known that mergers become
important in the mass assembly of only massive galaxies (M∗ ≥ 2 − 3 × 1010M⊙,
Mh ≥ 3 − 5 × 1011M⊙), being more frequent as more massive they are (see e.g.,
Gottlöber et al., 2001; Zavala et al., 2012). Therefore, one expects that βcrit will
decrease on average with Mh. This dependence will be fixed in order our model
could reproduce the observational trend of the bulge-to-disk ratio with M∗.
Note that in our scheme the bulge is formed only from stars of the primary disk.
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 39
This might not be the case when the mergers are majors because in this case most
of the stars in the bulge come from the accreted secondary galaxy rather from the
disk. This is why our scheme becomes somewhat uncertain for galaxies dominated
by bulges.
3.1.3 Stellar and gas fractions
So far in our static model of disk galaxies in centrifugal equilibrium loaded inside
ΛCDM halos, we did not make any difference between stellar and gas components in
the galaxies. Since the static model is not an evolutionary model, we can not follow
a star formation history as well as we can not follow the secular or major merger
processes. However, as was done in the case of bulge formation, the structural and
dynamical internal distributions of our ”instantaneous” galaxies can be used for
estimating when the conditions of large-scale star formation due to disk instabilities
are fulfilled. The simple Toomre criterion (Toomre, 1964) will be used for this
purpose.
First, we consider that if a fraction of the disk is converted into bulge, according
to the criterion described in the previous subsection, then the bulge is composed
only by stars. Second, for the remaining baryonic disk, we estimate the fraction of
it that could have been transformed into stars. According to the Toomre criterion,
instabilities in a gaseous disk able to drive star formation happen when Σcrit(R) <
Σd(r), where
Σcrit(R) =
σgas(R)κ(R)
3.36QG
. (3.40)
Here, Σd(r) is the baryonic (assumed gaseous initially) disk surface density profile,
κ(R) and σgas are the epicycle frequency and velocity dispersion radial distributions,
and Q is the Toomre stability parameter. The two first distributions are given by the
solution of our static model. Regarding σgas, observational studies of neutral, atomic
(HI) gas for spiral galaxies, including our galaxy, show that it lies in a range of 5−15
km/s (Tamburro et al., 2009; Petric and Rupen, 2007; van Zee and Bryant, 1999)
without any clear trend with radius. We will assume a constant value of σgas = 7
km/s.
The Q stability parameter originally given in Toomre (1964) for an infinitesimal
thin disk has the value of 1. However, in numerical simulations of more realistic
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 40
disks, the value found for Q is typically 1.5− 2.5. In the early stays of the gaseous
disk, the formation of massive clumps make the disk more unstable and the value
of Q can be larger (Dekel and Birnboim, 2008; Perez et al., 2013). Besides, the
Toomre criterion changes a little when taking into account the coexistence of two
fluids, stars and gas (Shen and Lou, 2003)). The value of Q is affected by other
factors, such as the presence of a bulge; a bulge provides more dynamical support
against instabilities, making the Q parameter larger. Other factors affecting Q are
the height of disk, which is derived from vertical equilibrium, as well as the gas
dissipation which extends the instability to small scales, producing infinitesimally
thin disks unstable for all Toomre-Q values and reasonable thick disks stable at high
Q, primarily because of thickness effects (Elmegreen, 2011; Romeo and Wiegert,
2011). The turbulences of gas also change Q; the perturbations in the molecular
and atomic gas have a significant effect on both the inner and outer regions of
the disk. This can drive the inner gas disk to a regime of transition between two
stability phases such that the outer disk is more prone to star-dominated instabilities
(Hoffmann and Romeo, 2012; Shadmehri and Khajenabi, 2012).
There are also observational studies that attempted to constrain the value of
Q along the disks of local galaxies. For example, Bottema (1993) conclude that Q
along the disks of their analyzed spiral galaxies is around 2 and 2.5.
We will take a constant value of Q = 2.5 for the “secular” disks, i.e. those that
are not expected to have suffered mergers. This applies on average for low-mass
galaxies, let say those smaller than M∗ ≈ 3×1010 M⊙ (Mh ≈ 5×1011 M⊙. However,
as already discussed in the previous subsection, for galaxies formed in more massive
halos, the major mergers become more frequent in such a way that the central galaxy
likely suffers also mergers and interactions, which trigger strong disk instabilities
able to produce bursts of star formation. Since our model is not evolutionary, we
are not able to calculate these processes, but if we keep ourselves in the scheme of
the Toomre instability criterion, such an enhanced star formation can be estimated
someway by increasing artificially the value of Q. As in the case of bulge formation
in massive galaxies, we again will proceed empirically. The “artificial” value of Q as
a function of Mh for the massive halos will be set in such a way that the gas fraction,
Rgas, vs stellar mass empirical relation is reproduced. The observations show that
as larger are the galaxies, the smaller gas fractions have (or larger stellar fractions
have). In the next Section, it is described in more detail our procedure.
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 41
Following the Toomre instability criterion, the stellar mass contained up to the
radius r along the disk is calculated as:
Md,∗(r) = 2π
∫ r
0
[Σs(r)]r
′
dr
′
(3.41)
where Σs(r) = Σbar(r) − Σcrit(r) is the stellar surface mass density in the re-
gions where the local criterion for instability is fulfilled. Md,∗(r) is computed up
to the radius where Σb(r) = Σcrit(r). The total baryonic surface density is then
Σbar(r) = Σd(r) + Σg(r). Since by construction Σbar(r) is an exponential function,
the stellar surface mass density is expected then to be deviate from an exponential
distribution.
Summary
The initial parameters of our static model are the halo mass Mh (its initial distri-
bution is assumed to follow the NFW density profile), the halo λ and C parameters,
and fbar. By means of iterations, a baryonic exponential disk in centrifugal equilib-
rium is constructed, taking into account the gravitational drag of the halo within
the adiabatic invariance approach. During the iterations, the regions of the disk
that obey our criterion of dynamical instability are converted into a Sérsic bulge,
which has its own contribution to the gravitational potential. As the result, a bulge-
disk-halo system in equilibrium is obtained; the main criterion of convergence of the
iterations is when RD changes less than a given ǫ. For the remaining disk in centrifu-
gal equilibrium, the Toomre gas stability criterion allows us to estimate the amount
of stars formed along the disk. The bulge is assumed to be made only of stars.
The main equations solved in the iterations are (3.38), (3.8), (3.20), (3.27) and
(3.29). Following MMW98 and D+2007, we start with a guess for RD by setting
fV = 1 in equation (3.27). Then, we obtain Md from equation (3.8) and MB from
equation (3.20); by replacing this into equation (3.38) and using equation (3.6),
we find ri as a function of r to obtain Mf from equation (3.38). Thus, we have
V 2
T (R) and the bulge mass MB for the given RD, and we can calculate the bulge
circular velocity and the new disk circular velocity from equations (3.17) and (3.29).
Replacing this into eq. (3.14) and using (3.27), finally we obtain a new value for
RD and MB and check for convergence. In Fig. 3.1 we show the total circular
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 42
velocity profile and its decomposition into bulge, disk, and halo for models of four
halo masses and with the average values of λbar, fbar and C corresponding to these
masses. For the average λbar values, we use those ones constrained and discussed
in Chapter 5. The radii are normalized to the stellar effective radius, Re, of each
model, which are indicated inside each panel. From panel (a) to panel (d), the
masses are log(Mh/M⊙) = 10.5, 11.5, 12.0 and 13.5.
In the case of the lowest halo mass model, stars nor bulge are not formed (the
respective instability criteria are not obeyed at any radius); therefore, the baryonic
component corresponds only to a gaseous disk. The halo component completely
dominates in the circular velocity profile at all radii, whose maximum is attained at
≈ 4Re; at inner radii, Vc keeps increasing with radius. For the log(Mh/M⊙)=11.5
model, stars and a bulge form. However, the circular velocity continues being dom-
inated by the the halo component, excepting at the very inner regions, where the
bulge component dominates, though not too significantly as to affect the maximum
of Vc; Vmax is attained at ≈ 2.7Re. For the log(Mh/M⊙)=12.0 model, the compact
bulge component is already significant enough as to produce a bumped maximum
in Vc at the very center of the galaxy; at slightly larger radii, the Vc profile follows
the decreasing behavior of the bulge component and at ∼ 1Re it flattens and then
gently decreases with radius. In the case of the log(Mh/M⊙)=13.5 model, the bulge
also produces an inner bump in the Vc profile, but at larger radii, Vc again rises
with radius, following the halo component which is dominant, as in the case of the
low-mass systems; this is because fbar is very small for this model, following the
semi-empirical fbar–Mh relation. The Vmax is attained at a very external radius.
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 43
Figure 3.1: The circular velocity profiles and their decompositions into bulge, disk,
and bulge (solid black, dotted green, dashed red and long-dashed blue lines, respec-
tively). Each panel corresponds to a halo mass, indicated inside the panel. The
radii are normalized to the Re of each model. The λbar, C and fbar input parameters
of each model were taken as the average ones corresponding to the given Mh.
3.2 The procedure
In this Section, we describe the steps to generate a mock catalog of disk galaxies,
which is our ultimate goal. We set up an uniform halo mass distribution in bins of
0.1 dex in log10 Mh in the mass range 1010 to 1014M⊙. Note that the ΛCDM halo
mass function (HMF) is not uniform, but in order to obtain scaling relations, we
60 
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..>: 
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:>-" 50 
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" :>-
50 
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:>-" 200 
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CHAPTER 3. THE SEMI-EMPIRICAL METHOD 44
need to sample all masses with good statics. The ΛCDM HMF strongly decreases
with mass and it is described well, for example, by the Sheth-Thormen function
(Sheth and Tormen, 1999). In order that our mock catalog predicted statistical
distributions of the galaxy properties, we can convolve it with this function. The
mock catalog is generated as follows:
For each halo mass bin of 0.1 dex, we assign the input parameters by taking
them randomly from their corresponding distributions. These distributions are as
follow:
⋆ For the spin parameter, it is a lognormal function:
P (log λ) =
1
σλ
√
2π
exp
[
− log2(λ/λ0)
2σλ
]
, (3.42)
where λ0 is the mean and σλ is the width. Cosmological N-body simulations
show that both λ0 and σλ do not depend on Mh. We use the values measured
in the Millennium Simulations for unperturbed halos by Bett et al. (2007):
λ0 = 0.03687± 0.000016 and σλ = 0.2216± 0.00012.
⋆ For the NFW concentration parameter, C, the N-body cosmological simula-
tions show that it follows a lognormal distribution at a fixed halo mass, with
the mean of this distribution, C̄, slightly decreasing with Mh, while the width,
σlgC , remains roughly constant. For the mean, we use here the C̄−Mh relation
reported for the Bolshoi Simulation in Muñoz-Cuartas et al. (2011) as
log10C̄ = a(z)log10(Mh/[h
−1M⊙]) + b(z), (3.43)
where a(z) = wz−m and b(z) = [(α/z+γ)+ (β/(z+γ)2)] are the coefficients
given at any redshift and the additionals fitting parameters have been set equal
to w = 0.029, m = 0.097, α = −110.001, β = 2469.720 and γ = 16.885. For
the width, we adopt σlgC = 0.11 dex
⋆ Regarding fbar, the semi-empirical dependence of fbar on Mh for blue/late-
type galaxies presented and discussed in Chapter 2 is used. According to
it, for a given Mh, fbar follows a lognormal distribution with constant width
σ
lgfbar = 0.12 dex. The mean of this distribution depends on Mh. The results
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 45
obtained in Calette (2014), show that the mean Mbar–Mh relation of blue/late-
type galaxies is well fitted by a Behroozi et al. (2013) function:
log10(Mbar/Mh) = log10(ǫM1) + f(x)− f(0), (3.44)
where
f(x) = −log10(10
−αx + 1) +
δ[log10(1 + exp(x))]γ
1 + exp(10−x)
, (3.45)
and x = log10(Mh/M1). The fit of this function to the Calette (2014) results
give the next values for these parameters: log10M1 = 11.69, ǫ = 0.0668, α =
1.5, δ = 2.6993, γ = 0.8275.
For eachMh and the set of initial conditions taken from the distributions above
mentioned, we apply the static model of disk galaxies in centrifugal equilibrium
inside ΛCDM halos described in Section 3.1, and generate the global properties
and internal mass distributions of the given galaxy. As discussed above, in the
case the disks evolved only secularly, without mergers and interactions, the
dynamical criterion for bulge formation in isolated disks (§§3.1.2) and the
Toomre criterion for disk star formation (§§3.1.3) apply normally. In both
cases, the criteria are given by one parameter, βcrit and Q, respectively. The
values we assume for these parameters are βcrit = 0.62 and Q = 2.5 (see for
details the mentioned subsections).
For each halo mass bin, we perform 1000 random extractions from the dis-
tributions for λ, C and fbar presented above.1 We have checked that 1000
extractions are enough to sample well each one of these distributions. Since
we cover the halo mass range from 1010 to 1014 M⊙ in bins of 0.1 dex, we
have 40 mass bins. For each bin, we perform 1000 realizations of our three
parameters, for a total of 40,000 models. In Chapter 5, we will show how
1We make use of the fact that the integrals of two probability distributions are equal,
∫ x
0
h(x′)dx′ =
∫ y
0
g(y′)dy′. In the case of a uniform distribution between 0 and 1,
∫ x
0
U(x′)dx′ = x.
Therefore, a random number y from the distribution g(y) can be obtained by resolving the equation
x =
∫ y
0
g(y′)dy′, where x is a random number from the [0,1] interval. In all the three cases of the
parameters λ, C, and fbar, they are log normally distributed, or, that is the same, the logarithms of
these parameters, are Gaussian distributed. Therefore, we solve the above equation for a Gaussian
function.
CHAPTER 3. THE SEMI-EMPIRICAL METHOD 46
well sampled are the given distributions of the parameters with our statistical
realizations.
Chapter 4
Disk galaxy correlations: the
case of isolated galaxies
In the previous Section, we have described our stationary model for generating
a population of disk galaxies inside ΛCDM halos. This model has two sets of
input parameters:
1. Initial conditions parameters, which define the halo-galaxy global
properties. They are the halo mass, Mh, halo spin parameter, λ, halo
concentration, C, and the galaxy baryon fraction, fbar=Mbar/Mh. The
three first parameters are related to the “cosmological” dark matter halos.
The baryon fraction fbar encloses the complex astrophysical processes of
gas accretion and cooling into the galaxy, as well as the stellar- and AGN-
driven outflows. The relationship of fbar on Mh and its scatter for disk
galaxies (actually, blue/late-type galaxies) has been recently determined
in Calette (2014) and we use it here. The initial spin parameter out of
which the galaxy forms, λbar, actually could deviate from the one of the
halo; by the moment we assume detailed angular momentum conservation
in which case λbar = λ.
2. Disk instability parameters, which are related to the criteria that
define the amount of disk mass transformed into bulge (βcrit), and the
amount of disk gas mass transformed into stars (the Q Toomre parame-
ter).
47
CHAPTER 4. DISK GALAXY CORRELATIONS... 48
All these parameters are in principle thought for distinct (“isolated”) halos,
inside which formed one central disk that evolves secularly, i.e., the disk does
not suffer mergers and external perturbations, and it is formed under detailed
angular momentum conservation. In this Chapter, these will be our initial
(naive) assumptions. We proceed by generating a mock disk galaxy catalog as
described in subsection 3.2.
Before presenting the main results from the generated catalog, we note that
there is a (small) fraction of model galaxies with some peculiarities:
– Galaxies with B/T ratios larger than 0.5. Elliptical galaxies are typically
defined as those with B/T>0.5. Therefore, we exclude from our statistics
models with B/T>0.5. They are actually few and mostly in the 1010−1011
M⊙ stellar mass range.
– Galaxies with no stellar masses, i.e., those for which the Toomre criterion
never was attained. These are purely gaseous galaxies. In the lowest halo
mass bins (Mh . 3× 1010 M⊙), they are more than 50% of the cases. At
higher masses, the gaseous galaxies are very rare.
– Models that did not converge. The overall fraction of not converged
models is 1.8%. There is not a trend of not converged models with Mh,
fbar and λ; instead, many of them happen for low values of C.
4.1 Results
In Fig. 4.1, the stellar and baryonic effective radius–mass relations are plotted
(left and right upper panels, respectively). The effective radius is defined as
the one inside which half of the (bulge+disk) stellar mass (Re) or (bulge+disk)
baryonic mass (Re,bar) is contained. The solid line and shaded area correspond
to the mean and standard deviation from our model results in M∗ bins of
0.3 dex. The blue dashed line and the dotted lines are the mean and the
1σ scatter from observations corresponding to Shen et al. (2003), while the
pink and red dots connected by long-dashed lines correspond to the studies
by Mosleh et al. (2013) and Bernardi et al. (2012), respectively. See Section
2.2 for details on these observational inferences as well as on other ones; we
CHAPTER 4. DISK GALAXY CORRELATIONS... 49
consider that the showed observational determinations are representative of
the Re–M∗ observational relation and its scatter.
CHAPTER 4. DISK GALAXY CORRELATIONS... 50
Figure 4.1: Different radius–mass relations from the models (B/T≤ 0.5) compared to some
observational determinations (indicated inside the panels). The black solid lines and the shaded
areas are the model means and standard deviations obtained in small M∗ bins. Upper left panel:
Stellar effective radius vs. M∗. Upper left lower panel: Disk stellar effective radius vs. M∗. Note
that the observational correlation by Dutton et al. (2011) is actually for the exponential disk scale
length; for this case, Re,D=1.68RD. Upper right panel: Baryonic stellar effective radius vs. Mbar.
Lower right panel: Disk baryonic scale radius vs. Mbar.
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log lo(M*/Mc) loglo(Mb "jMe) 
CHAPTER 4. DISK GALAXY CORRELATIONS... 51
In the left lower panel, we plot the effective radius of only the stellar disk com-
ponent, Re,D, versus the total galaxy stellar mass (black solid line and shaded
area). This is compared to the fit to observations presented in Dutton et al.
(2011, blue dashed and dotted lines, for the mean and 1σ scatter, respectively).
These authors used bulge-disk decompositions to estimate the exponential disk
scale radius; the Re,D they report is then Re,D= 1.68RD (see Section 2.2 for
details). In the right lower panel, we plot the Rbar–Mbar relation, where Rbar is
the baryonic disk scale radius, which by definition is exponential. We compare
these results with the respective observational data from a compilation of disk
galaxies presented in Avila-Reese et al. (2008, blue circles with error bars).
The model results agree well with the Re–M∗ relation from observations up
to log(M∗/M⊙) ≈ 10.7; at larger masses, the trend of increasing Re with M∗
is steeper in the models than in the observations. The scatter around the
relations are also similar in models but at log(M∗/M⊙) & 10.7, the scatter
is larger in the former. The recent results from Bernardi et al. (2012) show
that the Re–M∗ relation can be steeper at large masses than previous works.
However, even in this case, the model Re–M∗ relation is steeper at the high-
mass. A similar situation is seen for the Re,D–M∗ relation, though model
and observations agree better than in the case of the Re–M∗ relation. This
suggests that the potential problem is mainly at the level of the bulge component.
Regarding the baryonic disk radius–baryonic mass relation (lower right panel),
the model results are also consistent with the observations.
As previous studies have shown, the size–mass relation of disk galaxies depends
mainly on the spin parameter λ and the baryon fraction fbar, and the scatter
around this relation is given by the scatter in the λ distribution (see e.g., Mo
et al., 1998; Firmani and Avila-Reese, 2000; Dutton et al., 2007; Avila-Reese
et al., 2008). The combination of the recently determined fbar–Mh relation for
blue/late-type galaxies and halo λ distribution measured in large cosmological
simulations, produces galaxies that at masses lower than log(M∗/M⊙) ≈ 10.7
have mean sizes for their masses in good agreement with observations; the
agreement is also remarkable at the level of the scatter around the mean re-
lation. However, the models disagree with observations in the size–mass rela-
tions at larger masses in the direction of being the sizes and the dispersions too
CHAPTER 4. DISK GALAXY CORRELATIONS... 52
large. This suggest mainly the presence of some evolutionary mechanism(s)
able to lower the angular momentum of the baryons that ultimately form the
bulge-disk galaxy. In figure 4.2, we show the stellar and baryonic TF relations
(Vmax vs. M∗ and Vmax vs. Mbar, respectively). Here, Vmax is the maximum
of the total (bulge+disk+contracted halo) circular velocity profile. The solid
lines and shaded areas correspond to the means and standard deviations from
our model results in M∗ bins of 0.3 dex. The red solid line in each panel is the
best orthogonal fit to observations reported in Avila-Reese et al. (2008, see
Section 2.3); the dotted lines enclose the 1σ intrinsic scatters given by these
authors. The data points with error bars correspond to individual observations
of dwarf galaxies by Geha et al. (2006, red dots) and McGaugh (2011, green
dots)(see Fig. 2.4).
CHAPTER 4. DISK GALAXY CORRELATIONS... 53
Figure 4.2: Predicted dependence of Vmax as a function of stellar mass (left panel) and baryonic
mass (right panel) using initial cosmological parameters and the fbar–Mh relation inferred by
Calette (2014). The average and 1σ of the model population distribution are plotted with solid
lines and shaded areas. The fits and 1σ intrinsic scatter in both panels to a compiled sample
of normal disk galaxies by Avila-Reese et al. (2008) are plotted with red solid and dotted lines,
respectively. The red dots are for observed dwarf galaxies by Geha et al. (2006), while the green
dots are from McGaugh (2011).
For intermediate masses, the model results are in rough agreement with the
observations both in the stellar and baryonic TF relations; if any, the models
have slightly larger values of Vmax than the observational fits. However, for
masses larger than log(M∗/M⊙) ≈ 10.7 (log(Mbar/M⊙) ≈ 11), the models start
to significantly deviate from the observations; as larger is the mass, the larger
is Vmax in the models with respecto to observations. At the low–mass side,
CHAPTER 4. DISK GALAXY CORRELATIONS... 54
the models show a bending in the TF relations, though less pronounced in the
case of the baryonic relation. The observations at low masses show a large
scatter; a bending seems evident from the McGaugh (2011) data, while for the
Geha et al. (2006) the scatter is too large. The sample of the later authors
contain actually a large fraction of satellite galaxies. On the other hand, some
observational studies for samples containing massive and dwarf galaxies show
that indeed dwarf galaxies lie on average above the extrapolation of the stellar
TF relation of massive galaxies (e.g., De Rijcke et al., 2007). For the baryonic
TF relation, the observations seem to show that there is not a bending towards
low masses (e.g., De Rijcke et al., 2007). The intrinsic scatters around model
TF relations are significantly larger than those inferred for the observations,
in particular at large masses.
The upper panel of Fig. 4.3 shows the B/T ratio vs. M∗. The mean B/T ratio
and the standard deviation from our model results for M∗ bins of 0.3 dex are
plotted with the solid line and shaded area; recall that we excluded models
with B/T>0.5, since they are closer to elliptical galaxies. The observational
data from Cibinel et al. (2013) and Weinzirl et al. (2009) are also reproduced
in this plot; for more details and other observational reports see Section 2.4.
The scatter in the B/T–M∗ relation is large in both the observations and
models. Most of the models at low masses are bulgeless (B/T<0.1). The B/T
ratio increases with M∗ but at masses larger than ≈ 5 × 1010 M⊙, the B/T
ratio decreases on average with M∗. The observations show that the B/T ratio
increases on average with M∗. The sample of Weinzirl et al. (2009) includes
only spiral galaxies; this is why their average B/T ratios are small and the
statistics at large masses is poor (most of massive galaxies are of early type).
In the case of models, galaxies that likely would be classified as S0 are actually
taken into account. Therefore, we should compare with something in between
the data from Weinzirl et al. (2009) and Cibinel et al. (2013). We conclude
that, within the large scatter, our models present a trend of B/T with M∗
in agreement with observations but at large masses, the model galaxies show
B/T ratios systematically smaller as larger is M∗, in poor agreement with the
observational trends.
Finally, in the lower panel of Fig. 4.3, the gas-to-stellar mass ratio, Rgas ≡
CHAPTER 4. DISK GALAXY CORRELATIONS... 55
Mg/M∗, vs. M∗ is plotted. The means and the standard deviations from our
model results in M∗ bins of 0.3 dex are plotted with the black solid line and
the shaded areas. The blue solid line is the mean relation inferred from a
compilation of observations in Calette (2014, see Section 2); the blue dashed
lines enclose the estimated 1σ scatter in the observations. The agreement
between models and observations is remarkable until log(M∗/M⊙) ≈ 10.5;
at larger masses, the models have too large gas mass ratios as compared to
observations.
CHAPTER 4. DISK GALAXY CORRELATIONS... 56
Figure 4.3: Distibutions of the B/T and gas mass ratios, respectively, as a function of stellar
mass. The mean values and the standard deviations of our models are plotted with the solid black
lines and shaded areas, respectively. Some observational results are also plotted (see the labels
inside the panels for the sources as well as Section 2).
CHAPTER 4. DISK GALAXY CORRELATIONS... 57
4.2 Summary and discussion
We have obtained different galaxy correlations with the static disk galaxy
model implemented here using the “cosmological” initial conditions, the semi-
empirical fbar–Mh relation, and assuming that the galaxies are isolated. The
results are in general in good agreement with observations up to stellar masses
≈ 3 − 5 × 1010 M⊙. At larger masses, the models significantly deviate from
observations:
– the size–mass relations become steeper, i.e., as larger is M∗, the larger
are the model galaxy sizes with respect to observations;
– the stellar and baryonic TF relations become steeper, i.e., as larger is M∗,
the larger are the model galaxy Vmax with respect to observations;
– the B/T ratio decreases on average with M∗, opposite to what observa-
tions show;
– the gas mass ratio, Rgas, remains constant on average while for the ob-
served massive galaxies, this ratio decreases with M∗.
It is encouraging that the “cosmological” halo initial conditions, the simple
disk “secular” instability criteria for isolated galaxies, and the recently con-
strained fbar–Mh relation for blue/late-type galaxies are consistent with the
main scaling correlations of the local disk galaxy population up to M∗ ≈
3− 5× 1010 M⊙. In particular, the zero-points of the model TF and size–mass
relations at intermediate masses are in agreement with observations. Recall
that in our model, the inner halo suffers adiabatic contraction due to galaxy
formation. Thus, at difference of Dutton et al. (2007), we find that it is not
necessary to lower the spin parameter from halo to disk galaxy formation nor
to invoke halo expansion instead of the standard adiabatic halo contraction.
We attribute this difference with that and other studies, mainly to the new
fbar–Mh relation and its scatter used here. It is important to remark that this
relation, by construction, is in agreement with the observed local galaxy stellar
and baryonic mass functions! Note also that we use the λ distribution from
the Millennium Simulation reported for unperturbed halos (Bett et al., 2007),
CHAPTER 4. DISK GALAXY CORRELATIONS... 58
which has the mean and scatter slightly lower than those used in Dutton et al.
(2007).
Nevertheless, that the model results disagree with observations at large masses,
it means that we are omitting the effects of astrophysical and evolutionary
processes relevant for the properties of massive galaxies. Indeed, this is the
case, since in our model it is not taken into account the role of mergers in
the formation of the disk-bulge systems, the angular momentum transport,
and the interaction-induced star formation. Namely for galaxies formed in
halos larger than 1012 M⊙, the effects of mergers seem to be relevant in their
evolution, more as more massive is the halo (see e.g., Zavala et al., 2012, and
more references therein).
The mergers contribute directly with stars to the bulge, and make the disk
more unstable to both processes of bulge growth and disk star formation.
Therefore, we can not already use the values of the βcrit and Q parameters
that apply for isolated disks. The effects of mergers during galaxy evolution
affect also the behavior of the angular momentum distribution of baryons.
The mergers seem to work as a mechanism of angular momentum transference
from baryons to dark matter (e.g., Zavala et al., 2008). Hence, one expects
that the more massive the halo, the lower will be the spin parameter of the
baryons out of which the disk forms with respect to the spin parameter of the
host halo, λbar < λ.
It is interesting that the mentioned above effects of mergers are expected to
work just in the direction of improving the failures of the static model at large
masses. In the next Chapter we will explore this possibility.
Chapter 5
A semi-empirical picture of
local disk galaxy correlations
In the previous Chapter, we have generated a catalog of disk-bulge-halo sys-
tems by means of the static model of disk galaxies in centrifugal equilibrium
loaded inside ΛCDM halos described previously in Chapter 3. As a first ex-
ploration, we have used the “cosmological” halo initial conditions and the disk
“secular” instability criteria that apply for systems growing in isolation. Be-
sides, for the parameter that contains the information about the astrophysical
processes, we have used the recently constrained fbar–Mh relation (and its scat-
ter) for blue/late-type galaxies. The static model predictions at this level are
encouraging: several structural, dynamical and star formation-related correla-
tions are in good agreement with observations from1 M∗ ≈ 109 to ≈ 3−5×1010
M⊙. However, at larger masses the model galaxies deviate systematically from
observations, suggesting that we have missed an important factor in the evo-
lution of massive galaxies: the mergers.
Since the static model is not an evolutionary approach, we are not in position
to follow the merger history of the halo-galaxy systems and, consequently,
to estimate the merger-driven bulge growth and star formation, as well as the
1We have set a lower limit in the masses of the halos, Mh = 1010 M⊙, which corresponds on
average to M∗ ≈ 7 − 10 × 108 M⊙. The empirical constraining of the fbar–Mh relation at lower
masses is uncertain and it seems that the scatter turns on very large at lower masses in such a way
that the methods to infer this relation are expected to fail.
59
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 60
merger-driven angular momentum transport processes. However, as mentioned
in Chapter 3, we can attempt to emulate the effects of mergers, yet within our
scheme, by artificially lowering the instability parameter βcrit and the spin
parameter λ, and by increasing the star formation Toomre parameter Q. In
order to constrain the change of these parameters with Mh, we may proceed
in a heuristic way, looking for reproducing the observed scaling correlations at
high masses that mostly depend on these parameters, as we will show below.
From our models and previous works, we know that the size–mass relation de-
pends strongly on the spin parameter λ. Regarding the B/T–M∗ and Rgas–M∗
correlations, they depend mostly on the parameters βcrit and Q, respectively.
For low masses, we leave the models to be generated in our mock catalog with
the same parameters as in the previous Chapter, while for larger masses, the
parameters are varied as a function of Mh in such a way these observational
correlations are reproduced at high masses (at lower masses our model already
reproduced them). We introduce dependences of these parameters with halo
mass from log(Mh/M⊙)≈ 3 − 5 × 1011 M⊙ by using, first, a subset of models
with the mean parameter values in λ, C and fbar. From these experiments, we
establish the functionalities that describe how the parameters λ, βcrit and Q
should depend on Mh in order to improve the model predictions in the size–
mass, B/T–M∗ and Rgas–M∗ correlations at high masses. Then, we calibrate
with better accuracy the parameters of these functionalities in order the whole
generated mock catalog reproduced the high-mass ends of the mentioned above
observational correlations.
At this point, it should be said that our approach turned on to be less deduc-
tive (ab initio) and more heuristic or semi-empirical. The approach is aimed
in general to generate ΛCDM-based models that are consistent with several
observational scaling correlations of disk galaxies. This allows us then:
– to predict other global correlations of disk galaxies and compare them
with observations in order to test the overall consistency of the model as
well as of the underlying cosmological background;
– to probe the effects of the recent determinations of the Mbar–Mh and
M∗–Mh relations (galaxy-halo connection) on the structural/dynamical
properties of galaxies accesible to direct observations;
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 61
– to explore and constrain how much the initial parameters (e.g., the spin
parameter) and the key assumptions (e.g., adiabatic invariant halo con-
traction) of the model should be modified in order to generate a disk
galaxy population in agreement with observations;
– to use these constraints for speculating about the physical and evolu-
tionary processes relevant to disk galaxy evolution at different scales, for
example the role of mergers;
– to explore the inner structure and dynamics of the generated bulge-disk-
halo systems as a function of global properties.
Below, we present the results obtained from our mock catalog generated as
described in Section 3.2 but with a modification in some parameters for halos
more massive than 3− 5× 1011 M⊙, following the strategy mentioned above.
In Section 5.1, the scaling correlations are presented and discussed. In Sec-
tions 5.2 and 5.3 we explore the origin of the scatters in the correlations, as
well as wether the scatter around the Mbar–Mh and M∗–Mh correlations are
segregated or not by some galaxy properties; Section 5.4, we explore the cor-
relations among the residuals of the scaling relations. In Chapter 6, the inner
mass distributions of the different components and the implied dynamics are
presented as a function of several global galaxy properties.
5.1 Disk galaxy correlations
5.1.1 The size–mass relation and inference of the spin
parameter of galaxies
Similar to Fig. 4.1, in Fig. 5.1 we plot the Re–M∗ and Re,bar–Mbar rela-
tions (left and right upper panels, respectively), as well as the Re,D–M∗ and
Rbar–Mbar relations (left and right lower panel). While the first two radii are
calculated for the whole (disk + bulge) stellar or baryonic galaxy, the latter
radii refer only to the disk stellar or baryonic component. The black solid lines
and gray shaded areas are the mean and standard deviations in bins of 0.1 dex
in M∗ for galaxies with B/T≤ 0.5. The means and standard deviations for the
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 62
few galaxies with B/T> 0.5, are plotted with green dots and error bars, re-
spectively. The same observations as in Fig. 4.1 are shown in Fig. 5.1. While
at masses lower than M∗ ≈ 3×1010 M⊙, the model results are similar to those
in Fig. 4.1 (Chapter 4), at higher masses, the too steep increasing of the radii
with M∗ is now shallower and in good agreement with the observations.
The agreement is good also at the level of the scatters around the size–mass
relations, even at large masses, where the scatter was too large previously (see
Fig. 5.1). In Fig. 5.2, we compare the one sigma scatter of the Re-M∗and
Re,D-M∗ relations (showed in Fig. 5.1) from our models (black solid lines)
with those of the corresponding observations (blue dashed lines). As seen
both scatters are similar, in special for the Re,D-M∗ relation. Note that in
both cases the scatters decrease with mass.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 63
Figure 5.1: Different radius–mass relations from the semi-empirical models compared to some
observational determinations (indicated inside the panels). The black solid lines and the shaded
areas are the model means and standard deviations (B/T≤ 0.5) obtained in small M∗ bins. The
green dots with error bars are the means and standard deviations for models with B/T> 0.5. Upper
left panel: Stellar effective radius vs. M∗. Upper left lower panel: Disk stellar effective radius vs.
M∗. Note that the observational correlation by Dutton et al. (2011) is actually for the exponential
disk scale length; for this case, Re,D=1.68RD. Upper right panel: Baryonic stellar effective radius
vs. Mbar. Lower right panel: Disk baryonic scale radius vs. Mbar.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 64
Figure 5.2: Intrinsic scatters of the Re-M∗ (upper panel) and Re,D-M∗ (lower panel) relations
showed in fig 5.1. Solid black and dashed blue lines are for the models and the observations,
respectively.
As mentioned above, the parameter that most affects the size–mass relations
is λ. The semi-empirical galaxies presented in Fig. 5.1 are with the same
“cosmological” mean λ distributions as in Chapter 4 but with the mean of λ
decreasing with Mh since Mh = 3× 1011 M⊙ as:
λ0 → λ0,bar = 0.145− 0.0095× log(Mh/M⊙). (5.1)
Therefore, our static model of disk galaxies in centrifugal equilibrium inside
ΛCDM halos allows us to probe the initial spin parameter of the baryons
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 65
out of which disk galaxies formed. Different observational size–mass relations
and their scatters are well reproduced if this spin parameter is log-normally
distributed following the results from N-body cosmological simulations (see
eq. 3.42) with σλ = 0.22 dex and a mean that is equal to that one of the
simulations up to Mh = 3 × 1011 M⊙ (λ0,bar = 0.037), and for larger masses
with a mean that decreases with Mh as given in eq. (5.1).
Thus, our results show that the ratio of the mean initial spin parameter of
the baryons out of which the galaxy is assembled to the one the halos had
originally, ℓ ≡ λ0,bar/λ0, is equal to one for halos smaller than Mh = 3× 1011
M⊙, and it decreases with Mh from this mass as:
ℓ = 3.955− 0.257× log(Mh/M⊙), Mh > 3× 1011M⊙. (5.2)
For instance, ℓ = 0.5 for halos of mass logMh = 13.44 M⊙. An implication of
this result is that galaxies formed in halos less massive than Mh ≈ 3 × 1011
M⊙, on average, conserve in detail the angular momentum acquired by the
ΛCDM halo, while in more massive halos, the baryons out of which galaxies
form transported a fraction of their angular momentum to substructures and
dark matter (see e.g., Zavala et al., 2008). This process is more efficient as
more massive is the halo (eq. 5.2), consistent with the fact that more massive
halos assemble larger fractions of their masses in major mergers than the less
massive ones.
In the literature, there were some observational attempts to infer the spin
parameters of the baryons out of which galaxies formed (e.g., Cervantes-Sodi
et al., 2008; Berta et al., 2008). These inferences were done for SDSS galaxies
by using very simplified approaches to estimate λbar from only the optical
information. It is interesting that these studies agree in that λbar decreases on
average with the luminosity or stellar mass of galaxies, with little dependence
on environment. Our models of disk galaxies in centrifugal equilibrium inside
ΛCDM halos offer a sophisticated way to constrain λbar and its scatter, taking
into account even the effect of the halo contraction. The mock galaxy catalog
we use reproduces several observational correlations of galaxies and it also
agrees with the GSMF through the used fbar–Mh relation. Therefore, the λbar
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 66
values of the disk galaxy models are expected to reflect well the λbar values real
galaxies should have if they were formed in ΛCDM halos.
The galaxies in our mock catalog have on average λ0,bar ≈ λ0 = 0.037 up
to M∗ ≈ 3 × 1010 M⊙ and for larger masses it decreases down to λ ≈ 0.02
at M∗ = 5 × 1011 M⊙ (see the upper left panel of Fig. 5.6 to be presented
below). The scatter is basically equal to the scatter of the halo spin parameter
used as input in the model, σλ = 0.22 dex (compare both shaded areas in
the upper panels of Fig. 5.6). Interesting enough, the scatter around both
the Re–M∗ and Re,D–M∗ observed relations was reproduced namely with this
“cosmological” scatter in λbar, as shown in Fig. 5.2. Therefore, the dispersion
in the spin parameter values of the relaxed ΛCDM halos explains the dispersion
in the size–mass relations of observed local disk galaxies
5.1.2 The B/T–M∗, Rgas–M∗ and Σe–M∗ correlations
At large masses, the parameters related to the bulge formation (βcrit) and disk
gas transformation into stars (Q), need to be calibrated in order to take into
account the effects of the mergers. We perform this calibration by making
that the predicted B/T–M∗ and Rgas–M∗ correlations were in agreement with
the observations. The upper panels of Fig. 5.3 show the B/T– and Rgas–M∗
correlations (solid lines surrounded by shaded areas) from the models that
include a correction in the instability parameters βcrit and Q at large masses
to take into account the effects of mergers; the correction to λ as a function
of Mh discussed in the previous subsection remains the same. The green solid
circles with error bars correspond to the galaxies with B/T> 0.5. The same
observational data as in Figs. 4.3 are plotted in these figures.
As in the case of the size–mass relations, the semi-empirical models are prac-
tically the same as in the previous Chapter for M∗ . 3 − 5 × 1010 M⊙. The
corrections introduced in βcrit and Q (and λ) for halos larger thanMh= 3×1011
M⊙, worked namely in the direction of improving the comparison of models
with observations at stellar masses larger than M∗ ≈ 3 − 5 × 1010 M⊙. As
can be seen from Fig. 5.3, the agreement with observations is now good at all
masses:
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 67
– The B/T ratio increases on average with M∗; the increasing is weak for
masses up to M∗≈ 1010 M⊙ and stronger for larger masses; most of the
low-mass mass galaxies are actually bulgeless, B/T< 0.1.
– The model galaxies with B/T> 0.5 are all massive galaxies. The fraction
of them increases with M∗; for M∗ > 1011 M⊙ it accounts for 10.6%.
If we add to this fraction, those models that did not converge (1.8%,
they are mostly too unstable), the fraction is then 12.4% . This can be
considered as the fraction of early-type galaxies predicted by our semi-
empirical approach.
– The (cold) gas-to-stellar mass ratio, Rgas=Mg/M∗, is on average 10 at
M∗ ≈ 108 M⊙ (or 91% in fraction, Mg/(Mg+M∗)) and it decreases with
M∗; for a Milky Way-sized galaxy, Rgas ≈ 0.15 (0.13 in fraction). At
larger masses, the mean Rgas decreases fast.
– The 1σ scatter around the Rgas–M∗ relation increases slightly as M∗ is
smaller; in general, the intrinsic scatter of the model Rgas–M∗ relation is
slightly larger than that one of the observational inference, but it should
be taken into account that the latter is very uncertain due to sample
selection effects, observational errors, and non-detections. It should be
said that at low masses the fraction of pure gaseous galaxies is significant;
for M∗< 109 M⊙, this fraction is 17.8%.
In order to attain a good agreement with observations of the B/T ratios at large
masses, we have artificially lowered the value of βcrit in the criterion of bulge
formation (see subsection 3.1.2 and eq. 3.39 therein) for log(Mh/M⊙) > 11.5.
Lowering βcrit it implies that the disk is more unstable and larger bulges are
formed; the source of instability is not already intrinsic, but external, produced
by mergers. This parameter was lowered as:
βcrit = −0.0217[log10(Mh/M⊙)]
2 + 0.4319 log10(Mh/M⊙)− 1.4932 (5.3)
for log(Mh/M⊙) > 11.5. Note, however, that in our scheme, the bulge is all
the time formed from stars of the disk, producing this that as larger is the
formed bulge, the lower is the surface density of the disk. This might not be
the case when the mergers are major because in this case most of the stars in
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 68
the bulge come from the accreted galaxy rather from the disk, and the latter
remains the same or even more dense due to the merger-induced instabilities.
This is why our scheme is valid only for galaxies with non-dominant bulges,
let say B/T. 0.5.
Similarly, in order to agree with the gas-to-stellar mass ratios of massive
blue/late-type galaxies, we have increased artificially the value of the Toomre
parameter Q (see subsection 3.1.3) for log(Mh/M⊙) > 11.5. For the largest
masses, the increasing should be very large, to values up to Q ∼ 200. This is
because in these cases, a prominent (stellar) bulge forms; then, as mentioned
above, the remaining disk is less dense and more extended than the original
one, therefore it is stable against axisymmetric perturbations and it does not
form stars. However, the same merger(s) that produced the bulge, likely pro-
duce strong bursts of induced star formation that we emulate by increasing
Q.
Finally, the lower panel of Fig. 5.3 shows the effective stellar surface density,
Σe, vs. M∗ for the the models with B/T≤ 0.5 (solid line surrounded by the
gray area). This effective surface density refers to half of the sum of the bulge
and disk components. The semi-empirical disk galaxies show a significant cor-
relation between Σe and M∗. Up to ∼ 2×1010 M⊙, the slope of the correlation
is ∼ 0.6 and for larger masses it systematically flattens. The flattening of the
Σe–M∗ correlation at large masses is associated to the steepening seen in the
Re–M∗ correlation at these masses (Fig. 5.1).
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 69
Figure 5.3: Upper panel: Bulge-to-total mass ratio vs. M∗ for the semi-empirical models (black
solid line surrounded by a gray area) compared to some observational determinations (indicated
inside the panels). The gray area and the error bars are the standard deviations at each M∗
bin. The semi-empirical models and the observations by Weinzirl et al. refer to disk galaxies
(B/T≤ 0.5). The green solid circles with error bars are the averages and standard deviations
corresponding to the models with B/T> 0.5. Medium panel: Gas-to-stellar mass ratio vs. M∗ for
the semi-empirical models (black solid line surrounded by a gray area) compared to the empirical
correlation inferred in Calette et al. (2014; blue solid line surrounded by the blue dashed lines).The
green solid circles with error bars are the averages and standard deviations corresponding to the
models with B/T> 0.5. Lower panel: Effective stellar surface density, Σe, vs. M∗ for the disk
(B/T≤ 0.5) semi-empirical models. The blue line indicates a slope of 1/3 predicted for disk
galaxies when fbar and λ are constant.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 70
Our results regarding the Σe–M∗ or (Re–M∗) correlation show that the shape of
the semi-empirical fbar–Mh relation has an imprint on this correlation. As pre-
vious authors have already shown, for pure disks and assuming that M∗=Md,
then Σe scales as M∗
1/3 when fbar and λ are constant (e.g., Mo et al., 1998;
Avila-Reese et al., 1998, this dependence is indicated with a thin blue solid
line in the bottom panel of Fig. 5.1). This is evident from the following
dependences for exponential disks:
Σe =
(M∗/2)
2π(1.68RD)2
∝ fbarMh
(λRh)2
∝ f
2/3
barM∗
1/3
λ2
, (5.4)
where we have made use of RD = g(C)fbarMhλ (see Chapter 2; g(C) is a
function that almost does not depend on mass), and Rh ∝ Mh
1/3 for the
CDM halos. So, if fbar and λ do not depend on mass, Σe ∝ M∗
1/3. However,
our semi-empirical galaxies were constructed with a baryon fraction and spin
parameter that depend on mass. At low masses, roughly fbar ∝ Mh
0.4 (see
Fig. 2.2), and since Mh=M∗/fbar, then fbar ∝ M∗
0.4/1.4 = M∗
0.29; λ does not
depend on Mh. Therefore, using eq. (5.4) we obtain that at the low-mass end
Σe ∝ M∗
0.53, close to the slope seen in the lower panel of Fig. 5.3.
At the high-mass end, roughly fbar ∝ Mh
−0.56, then fbar ∝ M∗
−0.56/0.44 =
M∗
−1.27, and using eq. (5.4), Σe ∝ M∗
−0.53. However, at high masses, λ
decreases with Mh according to our semi-empirical approach: very roughly
λ ∝ Mh
−0.22 or ∝ M∗
−0.28. Therefore, according to eq. (5.4), Σe ∝ M∗
0.03,
that is the Σe–M∗ correlation flattens in agreement with what is seen in Fig.
5.3. Note that these approximate calculations refer to an exponential stellar
disk (no bulge, no gas). The results presented in Fig. 5.3 are for a disk that
can contain gas and is not perfectly exponential plus a stellar bulge. In any
case, this approximation and our results agree very well.
The conclusion is that the Σe–M∗ correlation of disk galaxies results with a
slope steeper than 1/3 at low masses because the baryonic fraction decreases
as Mh (or M∗) is smaller, and it flattens at the largest masses because the
baryonic fraction decreases as Mh (or M∗) is larger; actually, the slope of
the Σe–M∗ relation can be even negative at the high-mass end if λ keeps
constant, however, our semi-empirical model shows that the spin parameter
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 71
out of which massive galaxies form should decrease with mass (see above),
which compensates the effect of the low values of fbar, making the slope of the
Σe–M∗ relation at large masses almost flat. A large survey of galaxies with
measured Σe values and stellar masses would provide valuable information
in the light of the predictions of our semi-empirical approach. The surface
density of galaxies is notably affected by the astrophysical processes that coin
the fbar–Mh and λbar −Mh relations.
5.1.3 The Tully-Fisher relations
The predicted stellar and baryonic TF relations from our (semi-empirical)
mock catalog are plotted in Fig. 5.4 (black dashed line surrounded by a gray
area) similar as in Fig. 4.2 in Section 4.1. By comparing both figures, we see
that the strong steepening of the TF relations at high masses ameliorated a
little now. This is mainly because after the formation of larger bulges at high
masses here with respect to the models in the previous Chapter (compare Figs.
4.3 and 5.3), the overall mass distribution of the baryons strongly concentrates
to the center (the bulge is compact), being the peak of the baryon circular
velocity component shifted to the very inner radii. Since the total circular
velocity is the sum in quadratures of the halo and the baryon contributions,
at the radius where the baryon component has now its maximum, the halo
contribution is low in such a way that the quadratic sum of both is yet low. The
maximum velocity, Vmax, is attained at larger radii, where the disk dominates
over the bulge, but since the disk is now less massive and more extended, this
Vmax results lower than the one when the bulge was smaller. The net result
is that the Vmax–Mbar and Vmax–M∗ relations flatten at large masses. In the
case of the Vmax–M∗ relation, note that M∗ also increased slightly as larger is
the mass with respect to the models in the previous Chapter; this is because
massive galaxies are now more efficient in transforming gas into stars (compare
Figs. 4.3 and 5.3).
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 72
Figure 5.4: Stellar and baryonic TF relations from our (semi-empirical) mock catalog compared
to observations (indicated inside the panels). The dashed line and the gray area are the means and
standard deviations calculated in small M∗ bins. Only galaxies with B/T≤ 0.5 are considered.
It is quite encouraging that the predicted TF relations are in good agreement
with the observations, confirming this that the ΛCDM cosmology offers a
natural explanation for the origin of these relations (see extensive discussions
in Avila-Reese et al., 1998, 2008, and more references therein). Note that
there is not a problem in the zero-point of both the stellar and baryonic TF
relations as it was suggested in the literature2 (e.g., Dutton et al., 2007),
and, by construction, our initial conditions are in agreement with the galaxy
stellar and baryonic mass functions. Recall that the fbar–Mh relation used as
input here, was constrained by using the observed baryonic mass function. In
Fig. 5.5, we plot the fbar–Mh relation and its scatter from our semi-empirical
2The problem was enunciated originally for the semi-analytic models: if their parameters are
tuned to fit the observed GSMF, then the modeled galaxies used to have too large Vmax for their
masses (or luminosities) as compared to observations.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 73
models and compare them with the semi-empirical fbar–Mh relation taken from
Calette (2014); the latter relation has been used actually as input to generate
our mock catalog. Therefore, this plot is just to show that we have sampled
well the input fbar–Mh relation (see Fig. 2.2 in Chapter 2). In Fig. 5.5, it
is also plotted the Fs-Mh relation (Fs ≡ M∗/Mh) and its scatter from our
models, and it is compared with the semi-empirical inference of this relation
for blue galaxies given in Calette (2014); this relation is the one that map
the ΛCDM halo mass function to the GSMF of blue galaxies. The overall
agreement between our prediction and the semi-empirical inference implies
that our semi-empirical model would map well the observed GSMF. 3
3However, according to Fig. 5.5, at low halo masses, our galaxies have systematically slightly
higher Fs values than those in Calette (2014) and the scatter around the Fs-Mh relation signif-
icantly increases. This is because in halos smaller than 1011 M⊙, a fraction of the disk galaxy
models are quite stable to the Toomre criterion for gas transformation into stars and form small
amounts of stars or not form stars at all. The fraction of purely gaseous galaxies increases as Mh
decrease. Some fraction of these pure-gas model galaxies should actually have small stellar masses,
which would contribute to lower the average values of Fs at these masses.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 74
Figure 5.5: Galaxy baryonic and stellar mass fractions as a function of Mh from the (semi-
empirical) mock catalog. The means and standard deviations are plotted with solid lines and
shaded areas, respectively. The blue lines are the means from the semi-empirical inferences by
Calette (2014) for blue/late-type galaxies. Our catalog was generated actually by using as input
the fbar–Mh from these authors, see Fig. 2.2. Only galaxies with B/T≤ 0.5 are considered.
The bending of the stellar TF relations at low masses was already discussed
in Section 4.1. It is less pronounced for the baryonic TF relation than for
the stellar one, in good agreement with observations. Regarding the bending
at high masses, it is difficult to say whether observations exhibit also such
a systematic bending since very massive disk galaxies are rare, and the fit
to observations is just an extrapolation at the largest masses. It would be
interesting to have more observations of massive disk galaxies in order to test
this prediction.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 75
5.1.3.1 The intrinsic scatter around the TF relations
The standard deviations in mass bins of our semi-empirical models in the TF
relations (associated to the 1σ intrinsic scatter of these relations), are in gen-
eral larger than the intrinsic scatter estimated from the observations.4 The
lower panels of Fig. 5.4 shows the logarithmic standard deviations of the
Vmax–M∗ and Vmax–Mbar relations as a function of mass. The semi-empirical
models show a minimal scatter at stellar and baryonic masses in between ap-
proximately 5×109 and 1.5×1010 M⊙. At lower and larger masses, the scatter
increases, specially for the stellar TF relation. Even in the mentioned mass
range, the model scatter is larger than the estimated from the observations.
The dashed horizontal lines in both panels of Fig. 5.4 show the intrinsic scat-
ters inferred from the observations in Avila-Reese et al. (2008). The scatter
around the TF relation is a long standing problem of the models in the context
of the CDM cosmologies stated originally in Eisenstein and Loeb (1996) and
Avila-Reese et al. (1998).
In the next subsection, we will explore the role that the initial parameters
play on the scatter of the scaling TF and size–mass relations. We will see
that in the case of the TF relations, the scatter is mainly produced by the
dispersion in the halo concentration parameter C. At this point, we should
mention that the mock catalog was generated by sampling the distributions of
the parameters C, λbar and fbar in the assumption that they are independent
among them. If these parameters are correlated among them, then the scatter
in the TF relations could reduce (or it could increase). We leave for a future
work the exploration in detail of these possibilities.
4In Avila-Reese et al. (2008), the compiled and homogenized galaxy sample was not pruned to
minimize the scatter as other samples do it (the main goal of these samples was to use them for
calibrating the TF relation as distance indicator). Avila-Reese et al. (2008) have also subtracted
the observational errors in order to estimate the intrinsic scatter of the TF relations.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 76
5.2 The source of scatter in the disk scaling
relations
The predicted size–mass relations have a large scatter, while the TF relations
are much tighter, i.e., they have smaller scatters. We explore further which
parameters from the initial conditions are related to the scatters around these
disk galaxy (B/T≤ 0.5) scaling relations. First, we present how the initial
condition parameters, λbar, fbar, and C, correlate with M∗ (left panels of Fig.
5.6). The correlations are actually a result of the input dependences of λbar,
fbar, and C on Mh, shown for completeness in the right panels of Fig. 5.6.
However, since our semi-empirical model results agree very well with several
of the observed scaling correlations, we can say also that our models constrain
in an empirical fashion the values and how the λbar, fbar, and C parameters
correlate with M∗ or Mh. This was discussed already for the case of the spin
parameter in §§5.1.1.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 77
Figure 5.6: Correlations of the model input parameters λbar, fbar, and C with M∗ (left panels)
and Mh (right panels) from our (semi-empirical) mock catalog. The dashed lines are the means
and the gray shaded areas represent the standard deviations. The correlations of these parameters
with Mh are actually a sampling of the assumed input relations and their scatters. Only galaxies
with B/T≤ 0.5 are considered.
In Fig. 5.7, we plot the residuals of the Re–M∗ and Vmax–M∗ scaling relations
vs. the residuals of the dependences of λbar, C, and fbar on M∗ (see Fig.
5.6). This plot shows us how much the “input” parameters affect the intrinsic
scatter of the scaling relations of disk galaxies (B/T≤ 0.5). The dispersion
around the Re–M∗ relation is large and it is produced actually by the three
parameters (the residuals of these relations correlate with the residuals of the
three parameters). The main source of scatter, at a given M∗, is λbar: the
larger the λbar, the larger is the galaxy size; the slope of the correlation ∆lgRe
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 78
vs. ∆lgλbar is higher than 1. The second source of scatter is fbar; the higher
the fbar, the smaller is the galaxy size, with a correlation slope around 1. The
residuals of the Re–M∗ relation correlate only weakly with the residuals of
the C − M∗ relation (the scatter is large), in the sense that, for a given M∗,
more concentrated halos tend to produce galaxies with smaller Re, i.e., more
concentrated in their stellar mass distributions.
The dispersion around the Vmax–M∗ relation, which is small, is produced
mainly by the C parameter. For a given M∗, the higher the halo concen-
tration C, the larger is the galaxy Vmax. The correlation among the residuals
is tight with a slope around 1. The ∆lgVmax residuals anticorrelate weakly with
the ∆lgλbar and ∆lgfbar residuals. Thus, the scatter around the TF relation is
small because its source is practically only the scatter of the C parameter.
We have also studied the correlations among the residuals of the baryonic
size–mass and TF relations and the residuals of the λbar–Mbar, C–Mbar, and
fbar–Mbar relations. They are actually similar to those of the stellar case
discussed above, though more scattered in the case of the size-mass relation
and less scattered in the case of the TF relation.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 79
Figure 5.7: Residuals of the stellar TF and Re–M∗ relations vs. the residuals of the λbar−, C−,
and fbar–M∗ correlations. The correlations among these residuals show how much the origin of
the scatters around the scaling relations depend on the “initial” parameters. Only galaxies with
B/T≤ 0.5 are considered.
We would like to know whether the scatter around the disk galaxy (B/T≤
0.5) scaling relations is segregated by a given galaxy property. In the upper
panels of Fig. 5.8, we present the correlations among the residuals of the Re–
M∗ relation and the global galaxy properties Rgas, B/T, and effective surface
density (bulge + disk), Σe. The means (dashed lines) an standard deviations
(shaded areas) are shown. As seen, there is not an evident segregation of
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 80
the the Re–M∗ relation by the gas mass ratio Rgas, while there is a weak but
systematical trend with the B/T ratio in the sense that those galaxies that
for a given M∗ are smaller (more concentrated), have also higher B/T ratios
on average. Regarding Σe, for the stellar low-surface density galaxies there is
not a segregation in the Re–M∗ relation due to Σe, but for the high-surface
density ones, those that deviate to smaller radii at a given M∗, also are those
that have the highest effective surface densities. Note that the low-surface
density galaxies are gas dominated (and a significant fraction did not even
appear in the plot because they are completely gaseous).
In the lower panels of Fig. 5.8, the correlations among the residuals of the
Vmax–M∗ relation and the same global galaxy properties than above are pre-
sented. For low-gas content galaxies (which are mostly massive galaxies), there
is a trend with in the sense that those that deviate in the stellar TF relation
more to the high-Vmax side have on average smaller gas contents. The op-
posite happens for the high-gas content galaxies (which are mostly low-mass
galaxies); those that more deviate to the the high-Vmax side have on average
higher gas contents, though this trend is very noisy.
The residuals of the stellar TF relation almost do not correlate with the B/T
ratio. At intermediate B/T ratios there is a very weak correlation of the residu-
als with B/T but at larger B/T ratios disappears at all. For intermediate-mass
galaxies, the bulge component is not too big but it can contribute significantly
to Vmax; therefore, those galaxies that have larger B/T ratios, can be also
those with larger deviations to high velocities in the TF relation. However,
more massive galaxies, in spite that can have larger B/T ratios, their bulges
already do not contribute significantly to Vmax because they tend to be dark
matter dominated (see below); therefore, there is not anymore a correlation of
the TF relation residual with B/T.
Finally, the TF relation residuals do not correlate with Σe for the low-surface
density disk galaxies, while for the high-surface density ones, the residuals
strongly correlate with Σe. Note that the low-surface density galaxies are gas
dominated (and a significant fraction did not even appear in the plot because
they are completely gaseous). For the high-surface density, it is expected that
the more concentrated the bulge-disk system, the more it can contribute to
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 81
increase Vmax.
B/T
Figure 5.8: Residuals of the stellar TF (upper panels) and Re–M∗ (lower panels) relations vs.
disk galaxy properties, namely the gas mass and B/T ratios, and the central stellar surface density.
Only galaxies with B/T≤ 0.5 are considered.
I I I I 1 • I I I I 
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CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 82
5.3 On the scatter of the Mbar–Mh and M∗–Mh
relations
The results from our semi-empirical mock catalog give a possibility to study
the interesting question about the scatter of theMbar–Mh andM∗–Mh relations
of disk galaxies. As discussed in Chapter 2, these relations have been recently
inferred from direct and semi-empirical studies. They are relatively tight but
some scatter around them is present; see Fig. 5.5. In Fig. 5.9 we explore how
the residuals from both relations do correlate with internal galaxy properties.
Our semi-empirical disk galaxies show that the residuals of the fbar–Mh relation
are in some degree segregated by the galaxy properties studied here, mainly by
the gas mass ratio Rgas ≡ Mg/M∗ and the effective stellar surface density Σe:
as higher is Rgas and Σe, the more the galaxies deviate on average to the low-
fbar and high-fbar sides of the fbar–Mh relation, respectively. This means, that
those galaxies that in the fbar–Mh plane are below the mean relation tend also
to be those with higher gas mass ratios and lower effective surfaces densities.
Regarding the B/T ratio, there is a noisy correlation among the residuals of
the fbar–Mh relation and this ratio: those galaxies with high fbar values with
respect to the average relation tend to have roughly large B/T ratios.
In the case of the Fs–Mh relation, similar trends as for the fbar–Mh relation
are seen, though in the detail there are some differences. The residuals get
larger in this relation than in the fbar–Mh relation, specially at high values of
Rgas and low values of Σe and B/T, that is, when the galaxies are less massive
(see also Fig. 5.5).
The results presented here are clear predictions that in future empirical and
semi-empirical studies will be likely able to test.
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 83
B/T
Figure 5.9: Correlations among the residuals of the fbar–Mh (upper panels) and Fs–Mh (lower
panels) relations with some disk galaxy internal properties. The means and standard deviations
of the correlations are plotted with dashed lines and blue shaded areas, respectively.
5.4 Correlations among the residuals of the
scaling relations
In Fig. 5.10, the correlation among the residuals of the Re–M∗ and Vmax–M∗
relations (left panel), and among the residuals of the Re,bar–Mbar and Vmax–
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 84
Mbar relations (right panel) are shown. The dashed lines and shaded blue areas
correspond to the means and standard deviations. For exponential disks with-
out dark matter, the logarithmic residuals of the radius–mass and Vmax–mass
relations anti-correlate with a slope of −0.5 (Courteau and Rix, 1999, blue
line in the panels). This is because for a given mass, the smaller the disk scale
radius (the more compact the disk), the higher is Vmax (the more peaked is the
circular velocity rotation). The presence of a dark matter halo, softens this
dependence and in the case of dark halo dominion, the dependence may com-
pletely disappear: Vmax is determined by the halo internal mass distribution
and it does not depend on the disk component.
Our results show that there is a weak and scattered anti-correlation among
the residuals both for the baryonic and stellar scaling relations, implying that
the semi-empirical disk galaxies are mostly dark matter dominated already at
radii around the one where Vmax is attained. However, a weak anti-correlation
is seen, means that there is a little trend: those galaxies that at a given mass
deviate to larger Vmax values tend to have smaller effective radii. The weak
anti-correlations among the residuals of the scaling relations of model galaxies
are similar to those inferred from observational samples (e.g., Dutton et al.,
2007; Courteau et al., 2007; Avila-Reese et al., 2008). If any, the latter authors
have found that the correlation in the baryonic case is slightly stronger and
steeper than in the stellar case. These authors interpreted this difference as an
compensation effect in the TF and size–mass relations when passing from the
baryonic to the stellar quantities (see Firmani and Avila-Reese, 2000). Our
results are surprisingly similar to these empirical inferences; the green lines in
Fig. 5.10 are the orthogonal linear fits to their data presented in Avila-Reese
et al. (2008).
CHAPTER 5. A SEMI-EMPIRICAL PICTURE... 85
Figure 5.10: Residuals of the stellar and baryonic TF relations vs. the residuals of the respective
stellar and baryonic effective radius–mass relations. The residuals are in dex. The blue lines
correspond to a slope of −0.5, valid for exponential disk galaxies without dark halo and bulge.
The absence of correlation is indicative of dark matter dominion at radii where Vmax is attained.
The green lines are the orthogonal linear fits to the observations presented in Avila-Reese et al.
(2008). Their slopes are indicated inside the panels.
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Chapter 6
The inner mass distributions of
the semi-empirical disk galaxies
We have found in Section 5.4 of the previous Chapter a weak and shallow anti-
correlation among the residuals of the baryonic and stellar scaling relations,
which suggests nearly dark matter dominion at radii where the maximum
circular velocity is attained (see Fig. 5.10). Our goal in this Section is to
study in more detail the inner mass distributions and dynamics of the galaxy
disk-bulge-halo systems build up inside ΛCDM halos. The fbar–Mh relation
used as the ”astrophysical” input in our model, and the closely related Fs–Mh
relation (Fig. 5.5, recall that Fs ≡ M∗/Mh) summarize the key aspects of
galaxy formation and evolution of galaxies within the ΛCDM halos. We will
explore now how these relations do project into the inner mass distributions
of the disk galaxy systems, within optical radii of the galaxies instead of the
hypothetical virial radii.
6.1 Galaxy-to-total maximum velocity ratios
The ratio of disk (or disk + bulge) to total circular velocity is a quantity
related to the ”luminous”-to-dark matter ratio. The velocities can be defined
(and measured) at a given radius or estimated at their respective maxima (the
total one, Vmax, comes directly from observations while the disk or disk+bulge
86
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 87
ones are inferred from the observed luminosity mass distributions or from
vertical disk dynamics studies).
In Fig. 6.1, we plot three circular velocity ratios vs. some global properties:
Vmax, the effective stellar (bulge + disk) surface density Σe, and M∗. The
model results for the disk galaxies (B/T≤ 0.5) are shown as the averages and
standard deviations calculated in small bins of these properties (dashed lines
and shaded areas, respectively). We also show the corresponding means and
standard deviations for the “early-type” subsample, i.e. those models with
B/T> 0.5 (green dots with error bars, respectively). Recall that a significant
fraction of the low-mass model galaxies are gaseous, so they can not be plot-
ted in the panels where stellar quantities appear (central and right columns).
However, in the first column, they can be included and are plotted with red
dots and error bars. The upper panels are for the ratio of disk circular ve-
locity to total circular velocity at 2.2RD, where RD is the baryonic disk scale
length (exponential by definition). The medium panels are for the ratio of the
maximum circular velocity of the disk component, VD,max, to Vmax while the
lower panels are for the ratio of the baryonic (bulge + disk) maximum circular
velocity, Vbar to Vmax note that the disk or disk+bulge maximum circular ve-
locity may occur at radii different to the one where the total circular velocity
has its maximum.
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 88
Figure 6.1: Different “luminous”-to-total velocity ratios as a function of Vmax, Σe and M∗ for
the semi-empirical mock disk galaxy catalog (B/T≤ 0.5); the dashed lines and shaded areas are
the means and standard deviations, respectively. Green dots with error bars correspond to the
sample of galaxies with B/T> 0.5. The red dots with error bars correspond to the sample of purely
gas disk galaxies; they can not be plotted in the medium and right panels because they do not
have stars. Upper panels: Disk-to-total velocity ratio measured at 2.2 scale radii of the baryonic
disk. Medium panels: Maximum disk-to-maximum total velocity ratio. Lower panels: Maximum
disk+bulge-to-total velocity ratio.
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CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 89
The VD,2.2/V2.2 and VD,max/Vmax ratios of the semi-empirical model galaxies
are almost always lower than 0.6, far from the disk-dominated case which has
values of 0.75–0.85 (Sackett, 1997) (not taking into account a bulge). At low
Vmax, Σe, and M∗ the ratios slightly decrease as lower are these quantities. At
Vmax ≈ 150 km/s, Σe ≈ 200 M⊙/pc
2, and M∗ ≈ 2 × 1010 M⊙, these ratios
have a shallow maximum, and for larger values the ratios decrease. Deter-
minations of the VD,2.2/V2.2 ratio (which is close to the VD,max/Vmax ratio for
small-bulge galaxies) with vertical-disk dynamics studies using spectroscopical
data were carried out by Bershady et al. (2011). For the small sample these
authors present, they infer a correlation of the VD,2.2/V2.2 with Vmax, which
we reproduce with a solid line (the dotted lines show the scatter of their cor-
relation). The low-velocity semi-empirical model galaxies have larger velocity
ratios than those of the small observational sample by Bershady et al. (2011).
If we include the significant fraction of gaseous model galaxies (red dots and
error bars), then the agreement would be better. It is probably that our algo-
rithm for gas transformation into stars based on a simple Toomre instability
criterion underpredicts the amount of stars formed at our lowest halo masses.
On the other hand, the Bershady et al. (2011) velocity ratios seem to be too
low; previous studies with similar techniques showed higher ratios, around 0.6
(e.g., Bottema, 1993). The sample of this author was small, only 12 galaxies,
and he did not find a dependence of VD,2.2/V2.2 on luminosity or Vmax.
At maximum velocities around 200 km/s, the velocity ratios of the Bershady
et al. (2011) and (Bottema, 1993) observations are of the order of our pre-
dictions. However, for the observational sample from the former authors, it
seems that the VD,2.2/V2.2 ratio continues increasing with Vmax, while in our
mock sample, the ratio at these values decreases with Vmax. Unfortunately,
there are not dynamical observational determinations for galaxies with Vmax
values larger than ∼ 250 km/s.
Zavala et al. (2003) provided estimates of the VD,max/Vmax ratio for 79 disk
galaxies, whose properties were compiled from the literature (it is the same
sample used in Avila-Reese et al., 2008). Their data are plotted in Fig. 6.1
(color dots). The data do not show a significant correlation with Vmax nor
with M∗, but a trend of increasing VD,max/Vmax with Σe is seen. At low
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 90
values of Vmax, Σe, and M∗ the semi-empirical galaxies are in rough agreement
with the observational inferences, but at large values of these quantities, the
semi-empirical galaxies decrease their VD,max/Vmax ratios with respect to the
observational inferences. Recall that that the disk velocity contribution in the
massive models becomes smaller in favor of a bulge component that increases
the total maximum circular velocity, Vmax. So, it could be that when the
bulge is significant, too much of the inner disk has been reduced. On the other
hand, it should be said that the observational inferences could be contaminated
partially by a bulge component at the time of estimating VD,max.
In the lower panels of Fig. 6.1, the disk+bulge velocity component is used. As
expected, this component is now higher than in the only disk case, specially
for the larger and higher surface density model galaxies. From our results, we
see that in general the inclusion of the bulge increases significantly the max-
imum circular velocities ratios; this is even more notorious for the B/T> 0.5
galaxies (green dots with error bars). The circular velocity decompositions
into bulge, disk and halo from observations are actually not trivial. On the
other hand, the total velocity maxima can be at very different radii than those
where the disk or disk+bulge maxima are attained. Therefore, these ratios are
not reflecting already the luminous-to-dark matter contents at a given radius,
specially for the bulge-dominated or dark-matter dominated galaxies. Un-
fortunately, observational samples with structural and dynamical information
are scarce and do not allow for a full disk-bulge-halo decomposition in such a
way that we can get the maximum circular velocities for each component and
compare with our predictions.
For all the maximum velocity ratios presented in Fig. 6.1, there is a general
trend: these ratios as a function of Vmax, Σe andM∗ follow a bell-shaped depen-
dence; after a maximum, they decrease as smaller/less dense and larger/more
dense are the galaxies. The latter behavior could seem unexpected, but, as we
will see below, its related to the dependence of fbar on Mh used as an input in
our static model of disk galaxies.
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 91
6.2 The inner baryon and dark matter mass
fractions
In Fig. 6.2, we present the direct measures of the baryon (bulge+disk) mass
fraction (i.e., the amount of baryon matter with respect to the total one)
in the semi-empirical galaxies as a function of Vmax, Σe and M∗. The solid
lines and shaded areas are the means and standard deviations of this fraction,
Mbar/Mtot, measured up to one stellar effective radius, Re, for our disk (B/T≤
0.5) nongaseous galaxies. The green dots and error bars correspond to the
galaxies with B/T> 0.5. The dashed lines are the means of the measured
mass fractions at 0.5, 1.5 and 2.0Re, from top to bottom, respectively. Note
that the corresponding dark matter mass fractions are 1−Mbar/Mtot.
Figure 6.2: Baryon-to-to total mass ratio at a given radius, Mbar/Mtot, of the semi-empirical
galaxies as a function of Vmax, Σe, and M∗. The solid lines and shaded areas are for measures of
this ratio at 1 Re (the means and standard deviations, respectively). The dashed lines, from top
to bottom, are the means of the Mbar/Mtot ratio measured at 0.5, 1.5, and 2.0 Re, respectively.
The green dots with error bars are the means and standard deviations of this ratio at 1 Re for the
sample of B/T> 0.5 galaxies.
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 92
From Fig. 6.2, we learn that:
– The baryon mass fraction at 1Re has a maximum on average for galaxies
with Vmax ≈ 200 km/s, Σe ≈ 500 M⊙/pc
2, and M∗ ≈ 5×1010 M⊙. These
maximal baryon fractions are slightly above 0.5, i.e. the mentioned above
galaxies have a slight dominion of baryon matter over dark matter at
1Re. At values lower and higher than these, the mass ratios decrease.
The decreasing is more pronounced for the dependence on M∗. Such a
behavior of the inner baryon mass fraction is partially a consequence of
the global fbar–Mh relation of disk galaxies used as input in our stationary
model. This implies that the fbar–Mh relation (or the closely related
Fs–Mh relation) has a clear imprint in the inner galaxy and halo mass
distributions.
– As expected, the baryon (dark matter) mass fraction increases (decreases)
as the radius is smaller, but for small, low-density galaxies, the differences
between 0.5 and 2.0Re are on average very small (the galaxies are already
dark matter dominated since ∼ 0.5Re), while for larger, higher-density
galaxies, the differences are more notorious, being many giant galaxies
baryon-matter dominated at 0.5Re but dark-matter dominated already
at 2Re. It is interesting that our Milky-Way sized galaxies, contain on
average 50% of baryons and 50% of dark matter at 1Re; at 2Re, the
proportion change on average to 40% and 60%, respectively.
– The baryon mass fraction at 1Re depends more tightly on Σe than on M∗
and Vmax. This can be seen from the small scatter in the medium panel
with respect to the left and right ones. The inner baryon mass fraction
in the ΛCDM-based semi-empirical galaxies is determined by both the
galaxy surface density (concentration) and the mass, but it dominates
the first one.
– For the subsample of bulge-dominated semi-empirical galaxies (green
dots with error bars), which are mostly high-Vmax, massive galaxies, the
baryon mass fractions at 1Re are high, being most of them actually baryon
matter-dominated at this radius.
Our results clearly show how are expected to be the “luminous” and dark
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 93
matter distributions in disk galaxies formed in ΛCDM halos. An interesting
prediction is that very massive, high-surface density galaxies become dark
matter dominated already at 1Re. The trend of increasing the baryon matter
dominion as galaxies are more massive and of higher surface density is broken,
and the reason is that the halo baryon fraction, fbar, decreases significantly
with Mh for massive halos. Note that massive galaxies could be even more
dark matter dominated than we show in Fig. inner-mass if the halo spin
parameter λ would have remained the same for the baryons (detailed angular
momentum conservation). After introducing here a decreasing of the initial
spin parameter out of which galaxies form as Mh is larger (see Chapter 5),
the galaxies are more concentrated and hence, increase their contribution to
the total mass in the inner regions. But not enough as to dominate over dark
matter at radii ∼ 1Re.
6.3 The outer slope of the circular velocity
curves
A debated question in the literature is the one related to the rotation velocity
shape of disk galaxies (e.g., Persic and Salucci, 1997; Catinella et al., 2006).
Are the rotation curves described by an universal profile? Do the parameters
of this profile depend more on luminosity (or M∗) or in surface brightness or
any other galaxy property? The answers to these questions are tightly related
to how are distributed the luminous and dark matter mass components.
Here, we focus only on the outer slope of the circular velocity profiles. By
outer we understand galaxy radii further away the radius where the luminous
(bulge + disk) component attained its maximum. For example, in the case
of an exponential disk-dominated system, where the maximum is attained at
≈ 2.2RD, or equivalently, ≈ 1.3Re, the outer slope could be measured let say
at ∼ 3RD. Because of our galaxies are composed of bulge, disk and halo,
it is not so obvious where is the maximum. If the halo dominates, then the
maximum happens typically at radii larger than the one of the disk component.
The opposite happens if the bulge dominates. In order to assure a region far
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 94
away the maximum of the bulge+disk component, we measure the slope of the
circular velocity profile in between 3.0 and 3.5RD, αout. This can be considered
as an outer radius in the sense of the luminous galaxy. We could measure the
slope at even outer radii, but in the hope to compare in the future our results
with observational determinations, it is better to keep the measure not too
further away. The observational measures of the rotation velocities do not
extend typically to much outer radii.
In Fig. 6.3, we plot the dependence of the outer slope of the total circular
velocity curves (αout) of our semi-empirical disk galaxies as as function of Vmax,
Σe, and M∗. According to the figure, αout correlates with these three galaxy
properties, but more with Σe (the tightest correlation is with this property),
implying that outer shape of the circular velocity is mainly determined by
the galaxy surface density. In the case of this correlation, we have actually
obtained the means and standard deviations for the mock galaxies smaller
than M∗ = 5×1010 M⊙ separate from those larger than this mass. Recall that
an important result from our semi-empirical mock galaxy catalog is that the
Σe–M∗ correlation flattens at the high-mass end (see subsection 5.1.1 in the
previous Chapter) in such a way that increasing M∗, Σe almost already does
not increase. This behavior in combination with that α strongly increases with
M∗ at high masses, make that in the αout–Σe plane appeared a clear bimodality
driven by mass. In order to take into account such a bimodality at the time of
obtaining means and standard deviations, we have separated our mock sample
into two groups by the mass criterion mentioned above.
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 95
Figure 6.3: Outer slope of the circular velocity of our semi-empirical galaxies as a function of
Vmax, Σe, and M∗. The solid lines and shaded areas are the means and standard deviations. In
the case of Σe (medium panel), we have calculated these means and standard deviations separated
into galaxies smaller and larger than M∗ = 5 × 1010M⊙ because a strong bimodality driven by
mass appears in the αout–Σe plane. The green dots with error bars are the means and standard
deviations for the B/T> 0.5 galaxies.
Small, low-surface density galaxies have rising circular velocities (αout > 0)
at 3–3.5RD, but as the surface density and galaxy size increase, the slope
becomes flatter and even decreasing (αout < 0). However, for M∗ & 5 × 1010
M⊙ (Vmax & 200 km/s) the slope again increases on average; the velocity
profiles are rising for the largest disk galaxies. In other words, the baryonic
component is sub-dominant for these galaxies and the total circular velocity
curve is mainly the one of the dark matter halo at these radii. This is in
agreement with the above reported luminous-to-total mass contents at 1.5
and 2Re, see Fig. 6.2. The circular velocity profiles of large galaxies are rising
at large radii mainly because these galaxies become dark matter dominated,
as the low-mass ones. It should be said that the B/T> 0.5 galaxies follow the
same trends (green dots with error bars).
What about the dependence of αout on Σe for high-surface density galaxies?
0.2 
• B /T>O,~ 
0.1 
* " O J, 
O~ 
-0.1 
-0.2 ~~~LLLU~~LLLLUL~~~~LULU~UL~LU~~LU~~LU~ 
1.8 2 2.2 2.4 2.60.5 1 1.5 2 2.5 3 8 9 10 11 12 
loglO(V moJ[kms-'l loglO(Eo) [Mo/pc'l 
CHAPTER 6. THE INNER MASS DISTRIBUTIONS... 96
According to Fig. 6.3, on average αout would continue decreasing up to the
largest possible values of Σe. When separated by mass, many of the most
massive galaxies can have lower values of Σe and positive values of αout. A
clear bimodality in the αout–Σe plane appears. This is why we plot the means
and standard deviations in this plane separated by mass.
Our results show that a way to probe the luminous-to-total content of galaxies
yet within optical radii, is by the outer shape of the rotation curves. With the
advent of large galaxy surveys, large samples of galaxies with rotation curves
measured up to 3–3.5RD (1.5-2Re) will be available, and it will be possible then
to test whether their outer slopes have the dependences that we have found
in our ΛCDM-based semi-empirical disk galaxies. The secondary sample of
the MaNGA/SDSS-IV survey (Bundy et al., 2014) is designed to attain 2.5Re.
Therefore, we will have optical rotation curves at least up to this radius for
∼ 1500 galaxies, for which most of the optical properties will be also available.
Chapter 7
Summary and conclusions
El escenario cosmológico ΛCDM proporciona el contexto más adecuado para
interpretar la naturaleza, origen y evolución de las galaxias, en particular aque-
llas que dominan en el Universo local: las galaxias de disco (ver p. ej. Mo et al.
1998; Avila-Reese et al. 1998,2000,2008; Firmani & Avila-Reese 2000; Gnedin
et al. 2007; Dutton et al. 2007). En esta tesis nos propusimos explorar la con-
sistencia de dicho escenario con las principales correlaciones observacionales
de la población de galaxias de disco locales, tomando en cuenta las recientes
inferencias semi-emṕıricas de la fracción bariónica, fbar=Mbar/Mh, en función
de la masa de halo, Mh. Esta relación, fbar–Mh (o Mbar–Mh), contiene la in-
formación de los principales procesos astrof́ısicos que operaron en la evolución
de las galaxias en función de la masa como ser la cáıda y enfriamiento del
gas hacia los halos y galaxias, la formación estelar, la retroalimentación de
este proceso hacia el medio interestelar aśı como el de los núcleos galácticos
activos, etc. Las relaciones M∗/Mh–Mh y Mbar/Mh–Mh, incluyendo su dis-
persión intŕınseca, fue inferida por primera vez por separado para galaxias
tard́ıas y tempranas locales (Rodriguez-Puebla et al., 2014; Calette, 2014),
siendo éstas diferentes en realidad. En esta tesis se hizo uso de tales inferen-
cias correspondientes a las galaxias azules/tard́ıas (mayormente de disco).
Se implementó como herramienta de trabajo un modelo estático (no evolutivo)
de población de galaxias de disco que consiste en cargar discos en equilibrio
centŕıfugo en los halos de ΛCDM, tomando en cuenta la contracción adiabática
97
CHAPTER 7. SUMMARY AND CONCLUSIONS 98
que sufre el halo internamente por la galaxia (Mo et al., 1998). Se implementó
la posibilidad de calcular la formación de bulbos por inestabilidades intŕınsecas
del disco aśı como de transformación de gas en estrellas por inestabilidades tipo
Toomre (Dutton et al., 2007). Las condiciones iniciales del modelo estático son
las propiedades “cosmológicas” del halo (masa virial Mh, parámetro de giro λ
y concentración C) y la relación fbar–Mh que resume los procesos astrof́ısicos
de incorporación y pérdida de bariones por parte de la galaxia. Cada uno de
estos parámetros sigue en realidad una distribución estad́ıstica que tomamos
en cuenta.
Armados de este modelo semi-emṕırico, se procedió a generar un catálogo
sintético de decenas de miles de galaxias de disco locales cubriendo el intervalo
de masa de halo de 1010 a 1014 M⊙. Para ser consistentes en las comparaciones
con las observaciones, tomamos en cuenta como galaxias de disco sólo aquellas
con el cociente B/T menor o igual a 0.5. La fracción de galaxias con B/T>
0.5 es baja (≈ 11%) y aplica sólo para galaxias masivas.1 Un repaso de las
principales conclusiones relacionadas a las relaciones de escala y correlaciones
de propiedades galácticas globales con la masa (Caṕıtulos 4 y 5) es como sigue:
– Partiendo de las distribuciones que las simulaciones cosmológicas de N
cuerpos dan para los parámetros λ y C, y suponiendo conservación detal-
lada de momento angular, la población de galaxias de disco generada re-
produce muy bien las relaciones de escala observadas (radio–masa y TF,
tanto estelares como bariónicas) aśı como correlaciones de propiedades
intensivas (B/T, cociente de masa de gas a estelar –Rgas = Mg / M∗–,
densidad estelar superficial efectiva, Σe) con la masas estelar M∗. No
obstante este acuerdo es válido sólo hasta masas M∗ ≈ 3× 1010 M⊙ que
corresponde a halos de Mh ≈ 3 − 5 × 1011 M⊙. A masas mayores, las
galaxias tienen radios y Vmax mayores a los observados, B/T demasiado
pequeños y contenidos de gas muy altos. Además las densidades superfi-
ciales de las galaxias masivas decrecen con M∗.
1Nótees que esta fracción no pretende ser una predicción de la fracción de galaxias tempranas
pues desde un principio, se estuvo tratando sólo la población de galaxias azules/tard́ıas. La fracción
puede ser más bien interpretada como galaxias azules que de todos modos presentan un bulbo
dominante.
CHAPTER 7. SUMMARY AND CONCLUSIONS 99
– El desacuerdo de los modelos con las observaciones a altas masas es de
esperarse: en el escenario ΛCDM los halos masivos han ensamblado sus
masas con una fracción importante de fusiones mayores implicando que
las galaxias también sufrieron fusiones e interacciones en su evolución
reciente (e.g., Avila-Reese et al., 2014). En estos casos la formación de
bulbos y la transformación de gas en estrellas en un régimen de inestabil-
idades intŕınsecas (evolución secular) no es suficiente; de igual manera, la
suposición de conservación detallada de momento angular deja de ser del
todo correcta pues las fusiones son mecanismos de transporte de momento
angular, de los bariones del que se forman las galaxias a la materia oscura
(Zavala et al., 2008). En vista de esto, nuestros resultados nos indican el
rol que las fusiones/interacciones juegan en las principales propiedades de
las galaxias de disco actuales: prácticamente ninguno para las de masas
menores a M∗ ≈ 3 × 1010 M⊙ y cada vez mayor a medida que la masa
crece a partir de Mh ≈ 3− 5× 1011 M⊙.
– Operando sobre los parámetros que controlan las inestabilidades del disco
para hacerlos “artificialmente” más inestables mientras más masivos son
sus halos debido a las fusiones y disminuyendo el parámetro de giro del
gas del que se forma la galaxia, logramos que las galaxias de altas masas
estén en acuerdo con la relación radio–masa y las correlaciones B/T–M∗
y Mg–M∗ observadas. Nuestro método puede entenderse a este punto
como una manera (semi)emṕırica sofisticada de determinar el parámero
de giro inicial del que las galaxias de disco observadas se formaron, λbar en
función deM∗ yMh. El resultado es que hasta masas deM∗ ≈ 3−5×1010
M⊙, este parámetro es similar en magnitud y distribución (lognormal) al
de los halos ΛCDM (λ promedio de 0.036 sin una dependencia con Mh),
pero a masas mayores, disminuye con Mh como λbar ∝ −0.0095 logMh o,
en función deM∗, como λbar ∝ −0.0126 logM∗; es decir, hay ”pérdida” de
momento angular a masas mayores. La dispersión de la distribución ”cos-
mológica” original se mantiene; con ella se logra reproducir la dispersión
observada en las relaciones radio-masa.
– Las relaciones Re–M∗ y Re,D–M∗ predichas tienen una dispersión con-
siderable, que aumenta hacia las masas menores; estas dispersiones son
CHAPTER 7. SUMMARY AND CONCLUSIONS 100
similares a la que se infieren de las observaciones. A paridad de M∗, la
dispersión tiene su fuente principalmente en la dispersión de λbar: a mayor
λbar, radio más grande. Sin embargo, encontramos que también es una
fuente de dispersión el parámetro fbar (radio más pequeño en promedio
mientras fbar es mayor) y en mucho menor grado, la concentración C (se
observa una leve tendencia a que el radio es mayor cuando C es mayor).
– La relación de la densidad superficial estelar efectiva, Σe, con M∗ que se
predice es estrecha. La dependencia inicial de fbar con Mh que usamos
tiene su clara huella en esta relación, haciendo que sea más empinada a
bajas masas que el caso esperado Σe ∝ M∗
1/3 (la correlación que medimos
va como M∗
0.6) y que se aplane a las altas masas. De hecho, Σe incluso
decreceŕıa con M∗ a las altas masas (por el hecho de que fbar decrece
con la masa en este régimen) si es que λbar fuese constante. No obstante,
λbar en nuestro modelo decrece con la masa en este régimen como ya se
discutió arriba.
– Una vez que a altas masas se reprodujeron las relaciones radio-masa,
B/T–M∗ y Mg/M∗–M∗, las relaciones predichas de TF estelar y bariónica
están en excelente acuerdo con las observaciones. En particular, no encon-
tramos ningún problema con el punto cero, siendo que por construcción
nuestros modelos están de acuerdo con la función de masa bariónica (y
estelar) inferida de las observaciones. A bajas masas, la TF estelar sufre
un doblez que aparentemente también lo muestran las galaxias enanas; a
nivel de TF bariónica este doblez es mucho menos perceptible. A masas
muy altas, los modelos semi-emṕıricos muestran también un doblez que
será interesante comprobar si en las observaciones se da o no.
– La dispersión en la relaciones de TF predichas es moderada y producida
casi únicamente por la dispersión en la concentración C de los halos. Los
parámetros λbar y fbar no parecen ser fuente de dispersión y por ende las
relaciones de TF son muy robustas a variaciones en estos parámetros. No
obstante, la dispersión intŕınseca en las relaciones de TF de los modelos
semi-emṕıricos son mayores a la que presentan las observaciones. Este
podŕıa ser un potencial problema, aunque hacemos notar que el catálogo
fue construido bajo la suposición de que las dispersiones en λbar, C y fbar
CHAPTER 7. SUMMARY AND CONCLUSIONS 101
son independientes. En caso de que hubiese cierta correlación entre C
y los otros dos parámetros, la dispersión en la TF podŕıa disminuir (o
aumentar!).
– Los residuales de la relación Re–M∗ no correlacionan significativamente
con Rgas, pero śı anticorrelacionan con el cociente B/T (galaxias que se
desv́ıan hacia radios menores para su masa, tienden a tener B/T mayores)
y con Σe (galaxias que se desv́ıan hacia radios menores para su masa,
tienden a ser de mayor densidad superficial, aunque esto no se observa en
el caso de las galaxias con Σe bajos). Los residuales de la relación Vmax–
M∗ (TF) prácticamente no correlacionan con el cociente B/T y lo hacen
débilmente con Rgas: anticorrelacionan con Rgas en caso de galaxias poco
gaseosas (que son las masivas) pero correlacionan con Rgas en caso de
galaxias gaseosas (que son poco masivas). Los residuales de la relación
TF correlacionan con Σe para galaxias de alta densidad superficial pero
esta correlación desaparece para densidades superficiales menores.
– Se exploró también los residuales de las relacionesMbar–Mh yM∗–M∗, en-
contrando que hay cierta segregación en estas relaciones con las propiedades
galácticas estudiadas aqúı: Rgas, B/T y Σe. Será de interés comprobar
en un futuro con las observaciones directas estas segregaciones predichas,
mismas que son más evidentes en el caso bariónico.
– Los residuales de las relaciones Re–M∗ y Vmax–M∗ anticorrelacionan muy
débilmente con una pendiente mucho menor a −0.5 y en acuerdo a cier-
tas inferencias observacionales. Algo similar ocurre con los residuales de
las relaciones de escala bariónicas, mostrando una anticorrelación ligera-
mente mayor. La anticorrelación entre los residuales con una pendiente
tan baja es evidencia de que la componente disco+bulbo en la velocidad
de rotación es poco dominante, es decir, es más importante la componente
del halo oscuro.
En el Caṕıtulo 6 estudiamos más en detalle la cuestión de las contribuciones
a la velocidad circular por parte del disco y del disco + bulbo, cuestión rela-
cionada directamente con las distribuciones de masa interna en las galaxias de
disco. Los principales resultados son:
CHAPTER 7. SUMMARY AND CONCLUSIONS 102
– El cociente de velocidades de disco a total a 2.2RD, VD,2.2/V2.2, es casi
siempre menor a 0.6, lejos del caso de “disco máximo” (0.75-0.85). Este
cociente para las galaxias semi-emṕıricas decrece en promedio con Vmax,
Σe y M∗ en el lado de valores bajos de estas propiedades globales de
las galaxias. Alcanza un máximo suave (alrededor de 0.6) a Vmax ≈
150 km/s, Σe ≈ 200 M⊙/pc
2, and M∗ ≈ 2 × 1010 M⊙, y para valores
mayores, el cociente VD,2.2/V2.2 vuelve a decrecer. En comparación con
estudios espectroscópios de dinámica vertical en galaxias de disco que
infieren estos cocientes para muestras muy pequeñas, nuestros resultados
son someramente consistentes dentro de las incertidumbres.
– El cociente de velocidades VD,max/Vmax (donde los máximos pueden ser a
diferentes radios) muestra un comportamiento con Vmax, Σe y M∗ similar
al mencionado arriba: el cociente crece a medida que estas propiedades
aumentan en valor, llega a un máximo y para galaxias más grandes y den-
sas, vuelve a decrecer. Este comportamiento es una huella de la relación
fbar–Mh usada como condición inicial en los modelos semi-emṕıricos.
Cuando se usa la velocidad al máximo del disco + bulbo, el cociente
VB+D,max/Vmax, se incrementa, en especial para las galaxias más masi-
vas y de altas densidades superficiales. Lamentablemente no hay mu-
chos datos observacionales aún de este tipo como para confrontar con las
predicciones.
– Los cocientes de velocidades son importante porque es la manera de ex-
plorar observacionalmente la distribución dinámica de la masas de difer-
entes componentes (bulbo, disco, halo). No obstante, nuestros modelos
dan la predicción directa de dichas distribuciones. Ellos muestran que la
fracción de masa bariónica (bulbo+ disco) hasta 1Re alcanza su máximo
en promedio para galaxias con Vmax ≈ 200 km/s, Σe ≈ 250 M⊙/pc
2, and
M∗ ≈ 5×1010 M⊙; estas galaxias son ligeramente dominadas por materia
“luminosa”, con fracciones algo superiores a 0.5 en promedio. Para val-
ores de Vmax, Σe, y M∗ menores y mayores a los mencionados, la fracción
de masa “luminosa” decrece, es decir domina la masa oscura dentro de
1Re para ellas. Esto es consecuencia de la relación fbar–Mh usada como
condición inicial.
CHAPTER 7. SUMMARY AND CONCLUSIONS 103
– La fracción de masa “luminosa” aumenta hacia radios más centrales,
aunque para galaxias pequeñas y de baja densidad superficial, el aumento
es mı́nimo, por ejemplo a 0.5 Re, es decir estas galaxias ya son dominadas
por materia oscura a estos radios. Nuestros resultados muestran que una
galaxia similar a la Vı́a Láctea contiene en promedio 50% de bariones y
50% de materia oscura a 1Re; a 2Re la proporción cambia en promedio a
40% y 60% respectivamente.
– Una cantidad que se podŕıa obtener observacionalmente para catastros
de galaxias y que está relacionada con el cociente de masa “luminosa” a
oscura, es la pendiente externa (entre 3 y 3.5RD) de la curva de rotación,
αout. Calculamos αout para nuestras galaxias semi-emṕıricas y encon-
tramos que esta pendiente correlaciona con Vmax, Σe y M∗, pero más con
Σe, implicando esto que es la densidad superficial la que más afecta a la
forma externa de las curvas de rotación.
– Galaxias pequeñas, de baja densidad superficial tienen perfiles externos
de Vc crecientes (αout > 0). A medida que las galaxias son más masivas
y densas, αout decrece en promedio haciéndose negativo (curvas decre-
cientes), pero para galaxias de M∗ >∼ 5×1010 M⊙ (Vmax >∼ 200 km/s),
la pendiente vuelve a crecer mientras más masivas son las galaxias; las
galaxias más masivas tienen curvas crecientes, mismas que están trazando
al halo (la contribución luminosa decrece ya a estos radios); es decir son
galaxias dominadas por materia oscura a 3-3.5 RD. Una vez más, este
comportamiento bimodal de αout con la masa (o Vmax) es debido a la
relación fbar–Mh usada como condición inicial.
Appendix A
The empirical HI-to-M∗ and
H2-to-M∗ correlations for local
late- and early-type galaxies
Calette (2014) have compiled and homogenized from the literatura several
samples of local galaxies with reported measures of stellar mass, HI and H2
masses, as well as morphology type and/or color. By defining the mass ratios
RHI ≡ MHI/M∗ and RH2
≡ MH2
/M∗, in this work, it is found that the
RHI–M∗ and RH2
–M∗ correlations for blue/late-type (LTG) and red/early-
type (ETG) galaxies are better described by a double-power law.
The double power law proposed to fit the RHI vs M∗ correlations is:
log10 (RHI
) = log10(B)− log10
[
(
M∗
M s
)ξ
+
(
M∗
M s
)ρ
]
, (A.1)
where B is the normalization, ξ and ρ are the slopes of the function and M s
is the breakdown mass. In Table A.1, the values of these parameters for both
LTGs and ETGs are given.
For the scatter around the determined RHI-M∗ correlations, they assume that
it is log-normal distributed in agreement with several previous studies (see
Calette 2015 for the references). The lognormal intrinsic scatter is proposed
104
APPENDIX A. The empirical HI-to-M∗ and H2-to-M∗ correlations... 105
Galaxy population Parameters.
ETG
log10(B) ξ ρ log10(M
s)
-0.82 0.20 1.86 9.50
LTG
log10(B) ξ ρ log10(M
s)
0.06 0.20 0.85 9.50
Table A.1: Parameters of the RHI–M∗ two-power law relations
to slightly depend on M∗ as:
σlog10 RHI
= A+ ϕ log10
(
M∗
M s
)
. (A.2)
In Table A.2, the values of A and ϕ for the two galaxy populations are reported.
Galaxy population Parameters
ETG
A ϕ
0.70 0.00
LTG
A ϕ
0.41 -0.04
Table A.2: Parameters for the scatters around the RHI–M∗ relations
The double-power law proposed to fit the RH2
vs M∗ correlations is:
log10 (RH2
) = log10(C)− log10
[
(
M∗
M s
)β
+
(
M∗
M s
)γ
]
, (A.3)
where C is the normalization, β and γ are the slopes of the function and M s
is the breakdown mass. In Table A.3 are given the values of these parameters.
Galaxy population. Parameters.
ETG
log10(C) β γ log10(M
s)
-1.52 0.0 1.50 9.50
LTG
log10(C) β γ log10(M
s)
-0.5 0.34 0.78 9.50
Table A.3: Parameters of the RH2
–M∗ double power-law correlations
Regarding the intrinsic scatter around the RH2
–M∗ relations, again, Calette
(2015) assumes that it is log-normally distributed. The width of this distribu-
APPENDIX A. The empirical HI-to-M∗ and H2-to-M∗ correlations... 106
tion, in the case of LTGs, which are mostly star-forming objects, is expected
to be close to the width of the specific star formation rate (sSFR) vs. M∗
relation, the so-called “main sequence”1. The observations of local galaxies
show that the scatter around this relation is 0.3− 0.5 dex, increasing slightly
to lower masses (e.g., Salim et al., 2007). Based on this reasoning as well as on
the scatter that observations show in the largest compiled samples in Calette
(2015), he finds that a good description for the intrinsic scatter of LTGs is:
σlog10 RH2
= A′ + ϕ′ log10
(
M∗
M s
)
. (A.4)
with A′ = 0.37 and ϕ′ = −0.01.
In the case of ETGs, the intrinsic scatter seems to be larger than for LTGs. The
former are in general passive, devoid of gas reservoirs, but probably a fraction
of them can acquire some gas and trigger star formation during interactions
and mergers. Then, the amount of H2 depends on the kind of merger and on
the conditions to transform the atomic gas to molecular one. The range of
possibilities is huge, hence, the scatter around the RH2
–M∗ relation should be
large. Calette (2015) assumes the width of the log-normal distribution to be
σlog10 RH2
= 0.7 dex and constant with mass.
1The sSFR is a measure of the efficiency of current SFR of a galaxy with respect to its past
average efficiency. Since star formation proceeds in molecular clouds, the fraction of H2 in a galaxy
with respect to its size (M∗), is expected to correlate with the sSFR
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