Dynamical stabilities in isolated and perturbed barred models.

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Dynamical Stabilities in

Isolated and Perturbed Barred

Models

by

Diego Valencia Enríquez

Thesis submitted in partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY IN

ASTROPHYSICS

at the

Instituto Nacional de Astrofísica, Óptica y

Electrónica

March 2018

Tonantzintla, Puebla

Under the supervision of:

Ph.D. Ivânio Puerari

Tenured Researcher INAOE

Ph.D. Irapuan Rodrigues de Oliveira Filho

Tenured Researcher UNIVAP,

c

INAOE 2018

The author hereby grants to INAOE permission

to reproduce and to distribute publicly paper and

electronic copies of this thesis document in

whole or in part.

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To my parents who gave me the first push

to reach for the stars.

To my brother and sister who have always

believed in me.

To my wife who has been an engine to keep

going.

To my little daughter who has been an

inspiration to me.

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"NGC 1042" taken with the Fabry-Perot (OAN-SPM) by Diego Valencia.

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Declaration

This thesis is submitted for the degree of Doctor of Philosophy to the Instituto Nacional

de Astrofísica, Óptica y Electrónica (INAOE). I hereby declare that this dissertation is

not the same as any that I have submitted for a degree, diploma, or any other

qualifica-tion in other universities. Some of the work presented here has also been presented in

the following publications:

• D. Valencia-Enríquez, I. Puerari, L. Chaves-Velasquez,Detecting the growth of

structures in pure stellar disc models, 2017, Rev. Mex. Astron. Astrofis., 53, 257

• D. Valencia-Enríquez, I. Puerari, I. Rodrigues,Lower Limit on Global Stabilities

in Disc Galaxies to Generate a Bar, 2018a, second revision sent.

• D. Valencia-Enríquez, I. Puerari, I. Rodrigues, Evolution of Global Stabilities

in Perturbed Disc Galaxies Models, 2018b in preparation

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Abstract

This doctoral dissertation comprises the study of structures and dynamical stabilities in disc galaxies using N-body simulations. Particularly in the evolution and growth of spirals and bar structures.

From the theoretical point of view; it has been shown that the spirals are explained as density waves traveling through the disc, and the bar structure is the result of angular momentum exchange within the disc and between the disc, bulge, and halo. As well as, it depends on the properties of the model - the mass ratio between disc and halo, the halo concentration and velocity dispersions. The spiral arms can be long-lived quasi-stationary density waves with a constant pattern speed, or they can be evanescent waves that grow from short leading spirals waves to large trailing spirals waves, or they are a superposition of waves as a result of global modes. Bars are triaxial stellar systems with higher pattern speed than that the spirals, and also they can be thought as a density wave. N-body simulations have shown that the spiral arms appear as assembling particles that grow as a transient spiral wave and fade out like high winding while the bar structure emerge as gathering particles in the bar potential, which are losing angular momentum near the corotation. However, it is still unclear how the dynamical stabilities change and evolve as the structures (spirals, and bar) grow. In order to address these concerns, during this thesis, we have developed N-body simulations to understand the nature of spiral and bar structures and how they are affected by a flyby of a perturbation. These are briefly described below.

The first part of this thesis was published in my first paper (Valencia-Enríquez et al., 2017), which is presented in the third chapter. There we use a set of 3D N-body simulations for unbarred and barred models. Then we analyzed the growth of spirals and/or bar structures using 1D and 2D Fourier Transforms FT methods. We show that spectrograms and diagrams of the amplitude of Fourier coefficients as a function of time, radius and pitch angle show that the general morphology of our modeled galaxies is due to the superposition of structures which have different values of pitch angle and number of arms. Also, in barred models we made a geometric classification of orbits from the bar reference frame showing that the bar potential and the Lagrangian points

L4 andL5 catch approximately one-third of the total disc mass.

In the second part of this thesis, which is present in the fourth chapter, is part of my second paper. There we figure out what are the lower limits of global stabilities

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rotation configuration (spin parameter and critical spin parameter) of a model, which characterize the Global Stabilities, determine the evolution of a model. However, the literature does not show measurements of critical limits of Global Stabilities on their models to get a model stable or unstable to bar formation, and how they behave during the evolution of a model. Therefore, we resolve this issue by measuring and analyzing the whole evolution of Global Stabilities and their critical values with galactic models in which we only change the spin parameter to get models from disc dominated to halo dominated ones. We showed that while the Global Stabilities are closer from their stabilities limit, the bar formation is delayed, the spiral structures are stronger, the pattern speed of the bar is slower, and the bar catches more particles at the bar saturation phase than when the bar is formed in a more disc dominated model.

N-body simulations have shown that the bar can be triggered by two processes: (1) by own instabilities in the disc, or (2) by interactions with other galaxies. Both mecha-nisms have been widely studied. However, the literature has not shown measurements of the critical limits of Global Stabilities, GSP; therefore, to complete our previous studies we described before, we perturbed such isolated models to study the evolution of GSP under perturbation, which is part of the fifth chapter of this thesis, and also it is taken from my third paper, which is in preparation. We find that the critical limits of GSP are not much affected in barred models. However, when the bar is triggered by the perturbation, the disc fall in the unstable regimen. We show in our models that the bar triggered by a light perturbation grows into two phases: first, the bar appears as slow rotator, then it evolves toward fast rotator; second, when the perturbation is far from the target galaxy, the bar evolves from fast to slow rotator. Nevertheless, when the bar is triggered by a heavy perturbation, it appears as fast rotator and evolves toward slow rotator like classical bar models. Furthermore, the capture of orbits by the bar potential is different for bar generated in isolated models, and for bar triggered by a perturbation. While isolate models capture more compact orbits than loop or bar orbits at the beginning of the bar formation, all perturbed models caught many loop and bar orbits, but few compact orbits.

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Resumen

La presente tesis doctoral se centra en el estudio de estructuras (espiral y barra) y esta-bilidad dinámica en galaxias de disco usando simulaciones de N-cuerpos.

Desde el punto de vista teórico; se ha mostrado que las espirales se explican como ondas de densidad que viajan a través del disco, y la estructura tipo barra es el resul-tado del intercambio de momento angular dentro del disco y entre disco, bulbo y halo. También, esto depende de las propiedades internas del modelo; principalmente en el cociente entre masas del halo y disco, concentración del halo y de la dispersión de ve-locidad de las componentes de la galaxia. Los brazos espirales pueden ser ondas de densidad casi estacionarias de larga vida con velocidad angular constante, o pueden ser ondas evanescentes que inician como ondas espirales tipo “adelantada” (leading) y terminan como ondas espirales tipo “atrasada” (trailing), o también se consideran como superposición de ondas la cual es el resultado de modos globales en el disco. Así mismo, las barras son sistemas estelares tri-axiales que pueden considerarse como una onda de densidad con una mayor velocidad patrón que las ondas espirales. Por otro lado, simulaciones de N-cuerpos han demostrado que los brazos espirales apare-cen ensamblando partículas, que creapare-cen como ondas de densidad espiral transitorias y se desvanecen con un alto grado de enrollamiento, mientras que la estructura de barra emerge acumulando partículas en el potencial, las cuales están perdiendo momento angular cerca de la corotación. Sin embargo, aún no está claro cómo los límites de estabilidad dinámica cambian y evolucionan a medida que las estructuras (espiral o barra) crecen. Con el fin de abordar este tema, para esta tesis, hemos desarrollado simulaciones de N-cuerpos para comprender la naturaleza de las estructuras en espi-ral y barra, y cómo ellas se ven afectadas por el paso de una perturbación. Estos se describen brevemente a continuación.

La primera parte de esta tesis también es parte de mi primer artículo ( Valencia-Enríquez et al.,2017), el cual se presenta en el tercer capítulo. Ahí usamos un conjunto de simulaciones en 3D de N-cuerpos sin y con barra; para luego analizar el crecimiento de estructuras usando métodos de Transformada de Fourier FT en 1D y 2D. Espectro-gramas y diaEspectro-gramas de la amplitud de los coeficientes de Fourier como función del tiempo, el radio y el ángulo de inclinación muestran que la morfología general de nue-stros modelos de galaxias se debe a la superposición de estructuras que tienen difer-entes valores de ángulo de paso (pitch angle), y número de brazos (m). Además, desde el marco de referencia de la barra, realicé una clasificación geométrica de órbitas. Esta

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capturan aproximadamente un tercio de la masa total del disco.

En el cuarto capítulo de esta tesis, que también es parte de mi segundo artículo, buscamos cuáles son los límites de estabilidad global en galaxias de disco para generar modelos susceptibles o estables a la formación de la barra. La configuración inicial de rotación (ej. parámetro de giroλ, y parámetro de giro crítico λcrit) de un modelo, que caracterizan los Parámetros de Estabilidad Global (GSP, por sus siglas en inglés), pueden determinar la evolución de un modelo. Sin embargo, la literatura no muestra medición y evolución de éstos parámetros en modelos de galaxias de disco. Por lo tanto, resolvemos este problema midiendo y analizando por primera vez la evolución de estos parámetros en modelos galácticos. Para generar los modelos galácticos, cambiamos el parámetro de giro para obtener modelos donde la curva de rotación en la parte interna es dominada por el disco o por el halo. Mostramos que mientras las Estabilidades Globales están más cerca de sus límites críticos, la formación de la barra se retrasa, las estructuras espirales son más fuertes, el patrón de velocidad de la barra es más lenta, y la barra atrapa más partículas en la fase de saturación.

Las simulaciones de N-cuerpos han demostrado que la barra puede ser activada por dos procesos: (1) por inestabilidades propias en el disco, o (2) por interacciones con otras galaxias. Ambos mecanismos han sido ampliamente estudiados. Sin embargo, la literatura no ha mostrado mediciones de los límites críticos de GSP a medida que las estructuras crecen; por lo tanto, para completar el estudio previo, se perturbó los modelos aislados con una galaxia elíptica para analizar la evolución de los GSP, ahora bajo perturbación. Esto es parte del quinto capítulo de esta tesis, y de ahí se deriva un tercer artículo. Este artículo está en fase final de revisión y lo pretendemos enviar en las próximas semanas. En ese estudio, se encontró que los límites críticos de GSP no se ven muy afectados en los modelos barrados, sin embargo, cuando la barra se des-encadena por la perturbación, el disco cae en el régimen inestable. Se muestra que la barra desencadenada por una perturbación liviana crece en dos fases: primero, la barra aparece como un rotador lento, luego evoluciona hacia un rotador rápido; segundo; luego, cuando la perturbación está lejos de la galaxia objetivo, la barra evoluciona de rotador rápido a lento. Sin embargo, cuando la barra se desencadena por la interacción con una perturbación pesada, la barra aparece como un rotador rápido y evoluciona hacia un rotador lento. Además, la captura de órbitas en el potencial de barra es difer-ente para modelos aislados y para modelos perturbados. El potencial de la barra en modelos aislados captura más cantidad de órbitas tipo compactas que órbitas tipo bucle (loop) o barra; mientras que la barra generada debido a una perturbación atrapa una gran cantidad de órbitas tipo bucle y barra, y muy pocas en tipo compacto.

La astronomía, más que ninguna otra cosa, enseña humildad a los hombres. Arthur C. Clarke

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Acknowledgments

I wish to thank my supervisor, Dr. Ivânio Puerari for his constant support. He is not only a supervisor but also he became a friend. I have learned from him a lot of things about galactic dynamics and how to be a better person and scientist. I am truly grateful to him for his guidance and patience throughout these years of work. I am also very grateful to Dr. Irapuan Rodrigues my co-advisor, for giving me the opportunity to work as a visitor student at IP&D, and for teaching me about N-body simulations and the use of the cluster.

I am also very grateful to Dr. Divakara Mayya, Dr. Abraham Luna, Dr. Octavio Valenzuela, Dr. Héctor Hernandez, Dr. Vahram Chavushyan and Dr. Sergiy Silich, they were part of my thesis evaluation committee and examiners, for undertaking the task to reviewing this manuscript and for their useful suggestions.

I wish to thank my generation group, David, Jorge, Ricardo, and Alejandro, for mak-ing this adventure fun and enjoyable. All my bicycle friends, especially, Jose Manuel and Juaquin, thanks for all the (crazy) trips and feats on the bike. My gratitude to Juan Salvador, my roommate, who taught me good things about life. Also, I would like to thank Leo, my kooky friend, whom I started this trip from the bachelor’s degree. I am also very grateful to Luis Fernando Quiroga, my partner in Brasil, he helped me to disentangle some issues about my thesis, thanks (parcero). And to all my friends and everybody that I met in my studies. I do not have to forget my first mentor Alberto Quijano Vodniza who introduce me to the astronomy world.

I also acknowledge financial support from CONACyT (Consejo Nacional de Ciencia y Tecnología) studentship (No. 262131), and all the support from INAOE.

I would like to thank all my family, who always have supported me. I want to especially thank my sister for the first economical support and my brother who has always believed in me. I also wish to thank Angela, my loved wife who gave me a beautiful daughter (Juliana) and taught me how to be a better person. I have no words to thank my parents, I owe all of this to both of you Mom and Dad!. You have been the best teachers in my life, and always will be. Thanks a lot.

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Chapter 1 Introduction

What’s right and good doesn’t come naturally. You have to stand up and fight for it as if the cause depends on you, because it does - Bill Moyers

1.1 Motivation and Aims of this work

I

N the observable universe, it is found that more than half of all spiral galaxies have

a bar-like structure. Classification of spiral galaxies by eye shows that about 30 per cent are strongly barred in the optical light. However, if we include weak bars that are visible only in careful Fourier decomposition of the light distribution, this fraction rises to 50 per cent or more (Marinova & Jogee, 2007). This fraction even increases up to

60₋70per cent when observing in the near-infrared (Eskridge et al.,2000).

Evaluate the stability of a disc galaxy as well as disentangling the formation and evolution of a barred galaxy from the observations is a difficult task. Analytical work and N-body simulations are alternatives to study the dynamics of the bar evolution. By using numerical models, we could constrain the space parameters from which the model will certainly generate a bar structure.

Analytical works have shown that global stabilities are responsible for the origin of a bar in a disc galaxy. Kalnajs (1971, 1977) made a full stability analyzes which led to an eigenvalue problem for normal modes of an axisymmetric stellar disc. However, it has been proved that it is very difficult to find those eigenvalues in observed discs and galaxy models (Sellwood & Wilkinson,1993). Lynden-Bell(1979) suggested that bars may grow slowly through the gradual alignment of eccentric orbits. Athanassoula

(2002b) showed that the main mechanism to the bar formation is the exchanging of en-ergy and angular momentum between the disc and the halo particles at the resonances. Generally, it has shown that the origin of a bar in an isolated disc galaxy depends on the ratio between the disc massMD and halo massMH (Athanassoula,2013) where the

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angular momentum redistribution plays a crucial role in the formation and evolution of the bar. However, it remains unclear what is the lower limit of these global stabilities to get a model stable or unstable to a bar formation, and how these global stabilities behave during the evolution of a disc.

On the other hand, we know that the galaxies are not completely isolated, in fact, they are interacting with each other. Numerical studies have shown that two interacting galaxies can eventually be merged. They can change their entire morphology if their masses are comparable (major merger), or one of the galaxies can be absorbed by the host galaxy (minor merger). If these two interacting galaxies have enough energy, they fly over each other generating a short, but intense, perturbation in both galaxies; so it causes an impact on their global stabilities. That perturbation can be similar in amplitude to that excited by a minor merger (Vesperini & Weinberg, 2000), or can trigger a bar in one or in both galaxies (Noguchi, 1987; Gerin et al., 1990; Sundin & Sundelius, 1991; Sundin et al., 1993; Lang et al., 2014). Gerin et al. (1990) and

Sundin & Sundelius(1991) showed that a direct tidal interaction transiently increases or decreases the strength of the bar depending on the masses and on the pericentre distance.

From the previous discussion we can see that the formation of bar galaxies depend on the angular momentum exchanged and on the properties of the model e.g. ratio be-tween disc mass and halo mass, halo concentration, velocity dispersion, etc, which can be characterized by the Global Stabilities. For our purpose, the most relevant studies are those ofEfstathiou et al.(1982) andMo et al.(1998) (hereafter EF82 and MO, respec-tively). EF82 used N-body techniques to investigate global stabilities of exponential discs embedded in a variety of halos and found that the bar instability for a stellar disc is characterized by the parameterm which depends on the maximum circular veloc-ity, the disc mass and the radial scale-length of the disc. MO studied the population of galactic discs expected in current hierarchical clustering models for structure formation. They found that the disc has a lower limitλcritofλdto the stability of a disc.

The manipulation of the spin parameter in a disc results in a variety of models such as maximum and sub-maximum disc. These models can be stable or unstable to the formation of a bar; it depends on the initial global stability parameters(λcrit, λd) (here-after GSP). Therefore,the main objective of this work is generate different models only changing the spin parameterλdof the disc in order to get different initial global

sta-bilities to be stable or unstable to bar formation, and then we follow these parameters through the time. Furthermore, we perturb the models by a flyby elliptical galaxy to analyze the bar growth under perturbation.

To comply this goal, we use N-body simulations because is the most feasible tool to analyze different remarkable frames that can be in the universe e.g isolated and inter-acting galaxies.

To get a completely understanding of bar formation let us to constrain some main pa-rameters of galaxies e.g their rotation curves or their masses (disc and halo) by knowl-edge of their critical limits of their stabilities.

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1.2. Structure of this work 3

1.2 Structure of this work

Through the second chapter we will give a general view of barred galaxies; their mor-phologies, dynamics, formation, and the possible parameters that allow us to character-ize what are the Global stabilities of a disc to be stable or instable to the bar formation. Finally, we will describe general aspects of collisions and encounters of stellar systems trying to connect as an encounter affects these stabilities.

Local stabilities play an important role in the growth of spiral structures; how-ever, the growth of a bar in a disc affects the local stabilities driven spiral structures (Valencia-Enríquez et al.,2017). Therefore, in the third chapter, we review the growth of spirals and bars by the local stability (Toomre parameterQ) and the Fourier Trans-form methods to get a completely understanding of the stabilities in a disc.

In the fourth chapter we will analyze the evolution of the Global Stabilities (Valencia-Enríquez et al., 2018a) with galactic models in which we only change the spin param-eter to get models from disc dominated to halo dominated ones. In addition, we follow the growth of the bar structure in models where a bar is formed .

The fifth chapter explores the bar formation in interacting models using a flyby in co-planar orbit of an elliptical galaxy on our isolated models described in chapter fourth (Valencia-Enríquez et al., 2018b).

Finally, in the sixth chapter we summarize the general conclusions to this work and the work to be done to improve and corroborate the results already obtained.

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Chapter 2 Introductory Concepts

"What we know is a drop, what we don’t know is an ocean." - Isaac Newton

U

NDERSTANDINGthe dynamics in disc galaxies has been a complicated task. From

earliest to the present day researchers had studied the kinematics and dynamics of disc galaxies to contribute with the understanding of these issues.

2.1 Dynamics of disc galaxies

The dynamics of disc galaxies are based on the motion of the material in the disc and the processes that form the structures in the disc like spirals and bars. In this section, we will give some introductory concepts that are usually used to study disc galaxies.

2.1.1 Epicycles

Stars and gas in disc galaxies move in nearly circular orbits. They wobble in their orbits because of self-gravitational effects; like molecular clouds which give random velocity components to the stars. These wobbles to the circular motion are periodic and, in a co-moving reference frame, trace out ellipses calledepicyclesas the stars move inward and outward in the disc. The frequency of oscillation is known as theepicycle frequency.

This motion may be viewed as a result of the conservation of angular momentum

L =mvθr for massm, radiusr, and tangential velocityvθ. The increasing tangential velocity causes that the centrifugal force exceeds the inward gravitational force lead-ing to an outward Coriolis force, so the star moves outward. Therefore, the balance between Coriolis, centrifugal, and gravitational forces result the star oscillate forward and backward in its orbit.

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In linear theory, the study of orbits in axisymmetric potentials using cylindrical co-ordinate system leads to get the equations of motions as

¨

R =₋∂Φef f

∂R ; z¨=− ∂Φef f

∂z (2.1)

Where

Φef f = Φ(R, z) +

L2

z

2R2 (2.2)

is the effective Potential. For nearly circular orbits an approximate solution of these equations lead to the quantities

κ2 ≡

∂2_Φ

ef f

∂R2

; ν2 ≡

∂2_Φ

ef f

∂z2

(2.3)

whereκis the epicycle or radial frequency andνis the vertical frequency of the motion. Since the circular frequency is given by

Ω2(R) = 1

R ∂Φ ∂R = L 2 z

R4, (2.4)

the epicycle of equation2.3may be written as

κ2₍_R_{) =}

RdΩ

2

dR + 4Ω

2

(2.5)

2.1.2 Resonances

The resonance condition occurs when a star has an angular velocity Ωand the spiral pattern has an angular velocity Ωp; then, the relative speed of the star to the pattern is given by the difference, Ω₋Ωp. Thus, the condition knew as resonance appears when the epicycle frequency is synchronous with the relative motion of the spiral pat-tern. Therefore, the difference between the angular velocity and the pattern speed is an integer multiple of the epicycle frequency. In other words

Ω₋Ωp =±

κ

m. (2.6)

For integer values ofm, which is the number of arms. In galaxies with two arms, the fundamental resonances are known as theInnerand Outer Lindblad Resonance(ILR, OLR). Hence, in the frame of reference rotating atΩ_p, two epicycles are completed in one revolution around the galaxy. That is, one epicycle is completed in the time it takes to go from one arm to the next. Here,

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2.1. Dynamics of disc galaxies 7

Ω₋Ω_p =₋κ/2 at OLR (2.7)

Ω₋Ωp = +κ/2 at ILR (2.8)

At the corotation resonance radius CR, a star is at the same point when it completes one epicycle. If the star is also placed in a spiral arm (wave), it receives the same gravitational pull at the same point in the epicycle. It is analogous when a swing is pumped at a just proper frequency to make it go higher and higher. In the same form, at the corotation radius, a star takes up energy from the "wave" making the epicycles every time larger.

In order to show the location of different resonances, we use the rotation curve of one of our models to plot the fundamental resonances in Figure 2.1. In this figure, the left panel depicts the rotation curvevc. The central curve in the right panel shows the angular velocity calculated from the rotation curveΩ =vc/r. We calculate the epicycle frequencyκfrom equations2.3and then we replace it in equations2.7and2.8to get the Lindblad resonances. Inner and Outer curves represent theΩp = Ω±κ/2, respectively. The straight line depicts the pattern speed of a structure. The intersection of this line with these curves represents the ILR, CR and OLR.

Figure 2.1: Left panel depicts the rotation curve vcof one of our models. Right panel shows the angular velocity calculated from the rotation curve Ω = vc/r (central curve). Inner and outer curves represent theΩ±κ/2. The straight horizontal line depicts the pattern speed of a structure. The intersection of this line with the central curve represents the ILR, OLR and CR. corotation.

The ILR and the OLR, therefore, define the limits where a spiral arm structure cannot propagate. Besides, bars appear to end before the corotation.

2.1.3 Spiral Density Waves

Density wave theory appears to solve the winding dilemma where the astronomers though that the spiral arms were material; therefore, the differential rotation of the disc causes the arms to become more and more tightly wound.

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The mathematical formulation for the density wave theory was developed by Lin & Shu (1964). A spiral density wave is a gravitational perturbation that propagates through the disc. The effect of the wave is to pile up material temporarily at the wave crest, and this piling up makes the spiral arm. This density concentration generates the spiral arm as a wave that is not material, and does not wrap up as quickly as material arms do.

2.2 Disc stabilities

This analyzes is taken from Binney & Tremaine (2008) andMo et al. (2010), and is based on standard first-order perturbation theory. Critical limits of stabilities play a significant role in transforming and regulating the properties of disc galaxies. Local stabilities are affected by disturbances with lengths much smaller than the size of the disc. They can be transient and can regulate the evolution of the disc by driven features as transient spiral structures and star formation due to the fragmentation and the col-lapse of gas clouds. In contrast, global stabilities can cause a significant transformation of the overall disc. Whenever a disc galaxy is globally unstable, it will evolve towards a new stable configuration, erasing information about the initial conditions under which the system was formed (Mo et al.,2010).

2.2.1 Basic Equations

The analyzes of the stabilities is based on the determination of the dispersion relations and then investigate the unstable modes. We assume the disc to be gaseous so that the fluid dynamics apply and we consider a thin, self-gravitating disc without halo. We also consider the unperturbed disc to be axisymmetric to work with cylindrical coordinates

(R, φ, z), where z = 0corresponds to the disc plane. We will start writing the basic dynamical equations as a sum of its unperturbed values and a small perturbation. And next, we expand the perturbations in terms of the eigenmodes (e.g. Fourier expansion) to determine the dispersion relation.

Now, we write the dynamical quantities as a sum of Σ = Σ0 + Σ1 (the surface

density),vR = vR0+vR1, vφ =vφ0 +vφ1,Φ = Φ0 + Φ1 (the gravitational potential),

andh=h0+h1(the enthalpy1), where the subscript ’0’ and ’1’ refer to unperturbed and

perturbed quantities, respectively. Neglecting the thickness of the disc, the continuity equation can be written as

∂Σ1

∂t +

1

R ∂

∂t(Σ0RvR1) + Ω ∂Σ1

∂φ +

Σ0

R ∂vφ1

∂φ = 0, (2.9)

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2.2. Disc stabilities 9

The Euler equations in cylindrical coordinates are

∂vR1

∂t + Ω ∂vR1

∂φ −2Ωvφ1 =− ∂

∂R(Φ1+h1) (2.10)

∂vφ1

∂t + Ω ∂vφ1

∂φ + κ2

2ΩvR1 =−

1

R ∂

∂φ(Φ1+h1), (2.11)

and the Poisson equation relatesΦandΣ

∇2_Φ

1 = 4πGΣ1δ(z) (2.12)

with δ(z) the Dirac delta function. In the above equations, h1 = c2sΣ1/Σ0, where cs is the sound speed, and Ω(R) and κ(R)are, respectively, the circular frequency and epicycle frequency of the unperturbed disc at radiusR.

In general, we can expand a perturbation in the form

Q1 =

X

Qa(R)ei(mφ−ωt), (2.13) whereQ1 is a physical quantity and denotes a perturbation quantity (e.g. Σ1), and the

summation is over all modesa, each of which is characterized by an angular frequency,

ω, and its azimuthal wavenumberm >0. Taking the real part of Eq.(2.13) and substi-tuting such solution into Eqs.(2.9)-(2.11), we have

i(mΩ₋ω)Σ_a+ 1

R d

dR(Σ0RvRa) + imΣ0

R vΦa= 0, (2.14)

vRa =

i $

(mΩ₋ω) d

dR(Φa+ha) +

2mΩ

R (Φa+ha)

, (2.15)

vφa =

1 $ κ2 2Ω d

dR(Φa+ha) +

m(mΩ₋ω)

R (Φa+ha)

, (2.16)

whereha=c2sΣa/Σ0, and

$≡κ2₋₍_m_Ω₋_ω₎2_. _(2.17)

Equation2.17is known as the relation dispersion. This set of equations is complete for

Σa,vRa,vφa onceΦais related toΣavia the Poisson equation.

Φa =−

2πG

|k_| Σa (2.18)

2.2.2 Local stability

If the size of the perturbation is much smaller than the size of the disc, the perturbation mode can be written as

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Σ1(R, φ, t) =A(R, t)ei(mφ+f(R,t)) (2.19)

wheref(R, t)is a shape function2, andA(R, t)is a moderate varying function ofRthat sets the amplitude of the density wave. The curves defined bymφ+f(R, t) = 2nπ(n= 0,±1, ...)represents the peaks of the density waves and delineate the ridges of the spiral density arms. Thus,[f(R+ ∆R, t)₋f(R, t)] = 2πwhere∆Ris the radial separation between adjacent arms at a given azimuth. Under the assumption of tight-winding

∆R R, then f(R+ ∆R, t) _≈ f(R, t) + (∂f /∂R)∆R so that∂f /∂R = 2π/∆R. Therefore, the perturbation in the neighborhood(R0, φ0)neglecting the variation with

angleφis

Σ1(R, φ, t)≈Σaeik(R0,t)(R−R0), (2.20) where

Σa=A(R0, t)ei(mφ0+f(R0,t)) (2.21)

k(R0, t)≡

∂f(R, t)

∂R

R0

= 2π

∆R. (2.22)

Hence, the spiral density wave resembles a plane wave in the direction R, with wave-vector keR and wavelength ∆R; the potential perturbation Φa is related to Σa byΦ_a = ₋2πGΣ_a/_|k_|, see Eq(2.18). Using the fact thatkR 1and inserting such solutions into Eqs.(2.14-2.16), we get

(mΩ₋ω)Σa+kΣ0vRa = 0, (2.23)

vRa =

(mΩ₋ω)k(Φa+ha)

$ (2.24)

vφa =

iκ2_k_(Φ

a+ha)

2Ω$ . (2.25)

For axisymmetric perturbation m = 0, usingha =cs2Σa/Σ0 and insertingvRa into Eq.(2.23), we getthe relation dispersionfor a gaseous disc in the tight-winding approx-imation:

ω2 =κ2−2πGΣ0|k|+kc2s. (2.26) Consider the case of a cold disc. Then, a cold fluid disc has cs = 0, so for ax-isymmetric disturbances, equation (2.26) becomes ω2 ₌ _κ2 ₋₂_πG_Σ

0|k|. Therefore,

if ω2 _> ₀_{, then} _ω _{is real and the disc is stable; on the other hand, if} _ω2 _< ₀_{, say}

ω2 ₌₋_p2_{, then}_ω_{' ±}_ip_{, and}_exp(₋_iwt_{) = exp(}_±_pt₎_{. Hence, the perturbation grows}

exponentially, and the disc is unstable.

2_{Shape function is the function which interpolates the solution between the discrete values obtained} by finite element methods.

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2.2. Disc stabilities 11

Next consider a fluid disc with non-zero sound speed. For axisymmetric disturbance in the equation2.26, the disc is unstable if and only ifw2 _< ₀_{, and the line of neutral}

stability is

κ2

−ω2_{+ 2}_πG_Σ

0|k|+kc2s = 0. (2.27)

Solving this equation fork, we get

Q≡ csκ

πGΣ0

>1(for a stable disc) (2.28)

The analyzes of the stability for the stellar disc is similar, in which the effective pressure is due to the random motions of the stars. In this case, the stability parameter that we get is

Q≡ σRκ

3.36GΣ0

>1(for a stable disc), (2.29)

where the parameter Q is also know asthe Toomre parameter (Binney & Tremaine,

2008, for a complete mathematical development).

2.2.3 Global stability

Now, we consider perturbations with wavelength that are comparable to the disc size. Therefore in the study of global stabilities, whether or not the disc is globally stable depends on the properties of the system (disc, halo, etc.), such as its density distribution

Σ0(R)for the disc, and the concentrationcfor the halo, and the spin parameterλ(see

chapter4or Valencia-Enríquez et al. 2018a).

In this case, it is not possible to get a universal dispersion relation because the pertur-bation is not linear like in the tight-winding approximation. However, in a few simple cases, the dispersion relation can be worked out analytically; one of such case is the Maclaurin disc, which is a razor thin fluid disc with an unperturbed surface density of the form

Σ0(R) =

Σc(1−R2/a2)1/2 (R≤a)

0 (R > a), (2.30)

withathe radius of the disc and rotates at a uniform angular speed, the circular velocity is that of a solid body. The fluid pressure P acts only in the disc plane and has a polytropic equation of state P = KΣ3

0, with K constant. The resulting equilibrium

angular frequency,Ω, follows from the Euler equation by settingvR= 0is:

Ω2 = Ω2₀₋ 3KΣ

2

c

a2 . (2.31)

The perturbation on such disc can be expanded asPm

l (ξ)eimφ, withξ ≡(1−R2/a2)1/2, andPm

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physical quantities correspond to ξ _≥ 0, only modes with l ₋m =even are required in the expansion. Inserting such solutions into basic equations (see section2.2.1), one can obtain the dispersion relation for perturbations on Maclaurin disc (Takahara,1976;

Binney & Tremaine,1987) forl = 2andm= 2is:

ω= Ω_±

q

Ω2

0/2−Ω2 (2.32)

and the mode is dynamically unstable (i.e.ωhas an imaginary par) if

Ω2 >Ω2₀/2. (2.33)

The density perturbation represented by this mode has the form

Σ1 =

3R2

a√a2₋_R2 cos(2φ−ωt), (2.34)

which corresponds to a rotating elliptical deformation of the disc. This mode is similar to the bars observed in disc galaxies; therefore it is called "the bar mode," or "bar instability."

Increasing the random motion of the particles can stabilize the disc against the bar instability, which suggests a criterion for bar instability based on energies of the disc. LetT be the rotational kinetic energy, Πthe kinetic energy in random motion and W

the self-gravitational energy. Since, these energies are related by the virial theorem

T + Π =₋W/2; we find that the disc can be stabilized with the following conditions

Π

T >0, and 0≤ T

|W_| ≤

1

2. (2.35)

In similar way, the conditions for stability for a Maclaurin stellar disc; Kalnajs

(1972) are

Π

T >5.776, and T

|W_| ≤

125

972 ≈0.13. (2.36)

Using N-body simulations of flattened galaxies, Ostriker & Peebles (1973) found that their model disc are stable against bar mode ifT /_|W_|< 0.14orΠ/T &5. How-ever, in the solar neighborhood, the random velocity of stars is about60km s−1_{, while}

the rotation velocity is about 220km s−1_{, then} _Π_/T _∼ ₀_.₂₇_{, much smaller than the}

value for the stability of the stellar disc, see equation2.36. This implies that there must be some unseen component of matter withΠ/T 1to stabilize the disc.

On the other hand,Efstathiou et al.(1982) find an alternative criterion for bar stabil-ity, characterized by the parameter

m ≡

Vmax p

GMd/Rd

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2.3. Bar Galaxies 13

whereVmaxis the maximum rotation velocity of a disc galaxy. This parameter measures the importance of the self-gravity in a disc galaxy. Springel & White(1999) showed that a disc can be stable against to bar formation if

0.7_≤m ≤1.2 (2.38)

2.2.4 Conclusion

From the previous discussion, we note that the local instabilities are responsible for the growth of spiral structures, while global stabilities cause the emergence of the bar structure in disc galaxies. The Toomre parameter Q > 1, determines when a disc is stable against local stabilities, e.g., wounded spirals. We will measure theQparameter to show the growth of spirals together with a bar instability. This parameter will be discussed in chapter3.

Concerning global stabilities, we observe that to find normal modes using significant perturbations for an original stellar disc is very difficult as Ostriker & Peebles(1973) showed. Instead, Efstathiou et al.(1982) compares the rotation curve of a disc galaxy with the circular velocity of the disc at the radiusRdas a point mass to get a stability criterion which is easy to measure in N-body simulations and real observed galaxies. Therefore, we will use the m parameter together with the spin parameterλ, shown in chapter4, to comply with our goal of the study of bar instability.

2.3 Bar Galaxies

2.3.1 Observations and Kinematics of bars

Bars can be found in all types of disc galaxies, from the earliest to the latest stages of the Hubble sequence. In the Hubble tunning-fork diagram, bar galaxies are character-ized by elongated bar-shaped stellar configuration. Hubble distinguished these from the spiral galaxies by calling themSB, with similar subdivisions as for the normal galaxies. de Vaucouleurs’ contribution to the Hubble scheme was to recognize intermediate bar types, where the central region is elongated in an oval-type distortion. In this classifi-cation, the name SA is given to normal spirals, SB to barred spirals, and SAB to oval distorted barred spirals. Examples of SB and SAB galaxies with different Hubble types are shown in Figure2.2.

More than half of all spiral galaxies contain a bar-like structure. Figure2.3shows the fraction of each spiral type judged to contain bars, compiled from A Revised Shapley-Ames Catalogue (Sandage & Tammann,1981). The Second Reference Catalogue (de Vaucouleurs et al., 1976) and the Uppsala General Catalogue (Nilson, 1973). Apart from the very latest types, there is rough agreement over the fractions containing strong bars; when combined with all stages, the SB family constitutes between 25% and 35%

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Figure 2.2: NGC 1300 (left) Barred galaxy SBb; NGC 1371 (middle) oval distorted galaxy SABa; NGC 4321 (right) oval distorted galaxy SABbc. Images from the STScI Digital Sky Survey (POSS2/UKSTU Blue). Size5×5arc-minutes.

of all disc galaxies. A small number of galaxies which appear unbarred at visual wave-lengths have been found to be barred when observed in the near-infrared (Eskridge et al., 2000) (the near-infrared light traces the disc mass). Bulges with boxy or “peanut-shaped” isophotes that are seen in_∼40%of disc galaxies that are viewed nearly edge-on are really bars whose ledge-ong axis is perpendicular to the line of sight (Kuijken & Merrifield,1995;Bureau & Freeman,1999).

Figure 2.3: The fractions of barred (SB, shaded) and intermediate (SAB, unshaded) types at different stages along the Hubble sequence of spiral galaxies identified in three independent morphological classifications.

There are two distinct types of bars: large bars which have a nearly constant surface brightness ("flat" bar), while smaller bars tend to have a decreasing surface brightness similar to the disc ("exponential" bar). The surface brightness along the major axis in early Hubble types galaxies (SBa-SBbc) is nearly flat with a sharp cutoff at the end of the bar (not centrally condensed), while the surface brightness profile in later

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Hub-2.3. Bar Galaxies 15

ble types (SBbc-SBm) is exponential similar to normal galaxies (Elmegreen, 1996). They are concentrated towards the center to the galaxy, it makes a larger relative con-tribution to the luminosity and mass of the inner region containing up to the third of the galaxy’s total luminosity in early-type disc galaxies, but this fraction is smaller in late-type galaxies.

Bars in a disc galaxy can be characterized by three main observational parameters: length, strength and pattern speed. Bars are elongated and larger relative to the size of the disc in early-type galaxies. For SB galaxies, the median axis ratio in the equatorial plane is about 2:1. The length of the bars has been determined by optical visual inspec-tion (Kormendy,1979;Martin,1995), locating the maximum of the isophotal ellipticity (Wozniak et al.,1995;Márquez et al.,1999;Laine et al.,2002), or by structural decom-positions of the galaxy surface brightness distribution (Prieto et al., 1997; Aguerri et al.,2001;Laurikainen et al.,2005).

The bar strength measures the non-axisymmetric forces produced by the bar po-tential. It is determined by several methods: measuring the torques of the bar from photometry (Combes & Sanders, 1981;Buta & Block, 2001; Salo et al., 2010; Díaz-García et al., 2016), or from kinematics (Seidel et al., 2015), measuring the bar el-lipticity (Martinet & Friedli, 1997; Aguerri, 1999; Abraham & Merrifield, 2000), or with Fourier decomposition of the galaxy light (Ohta et al.,1990;Marquez et al.,1996;

Aguerri et al.,2000;Laurikainen et al.,2005).

Bars are density waves in the disc; i.e., the bar rotates more-or-less rigidly at a single pattern angular velocity Ωp, whereas the angular velocityΩ(r)of material in the disc varies with radius. Observationally, this dynamical parameter is measured from the flow pattern of a tracer population, such as old disc stars (Tremaine & Weinberg,1984), by identifying Lindblad resonances obtained from the stellar rotation curve (Muñoz-Tuñón et al., 2004), by detecting changes in the morphology (Puerari & Dottori, 1997), by matching hydrodynamical simulations to observed surface gas distribution and by the gas velocity field (Sanders & Tubbs,1980;Treuthardt et al.,2008). Bar angular velocity or pattern speed is one of the most important parameters determining its dynamical structure. The bar pattern speedΩp is characterized by the ratio

R=RCR/ab (2.39)

of the corotation radiusRCR to the bar semi-major axisab. More precisely, the corota-tion is the distance from the center to the Lagrange point on the major bar axis where the gravitational attraction balances the centripetal acceleration in the frame rotating with the bar. Theoretical works based on stellar orbits in barred potential predict that weak bars cannot extend beyond the corotation that is _R > 1 (Contopoulos, 1980). Thus bars are often said to be “fast bars” if1.0<R <1.4and “slow bars” if_R>1.4

(Athanassoula, 1992;Debattista & Sellwood, 2000). Most observed bars have turned out to be fast bars; nevertheless, there are few bars in the literature compatible with being slow bars (Rautiainen et al.,2008;Aguerri et al.,2015).

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nearly circular orbits. Barred galaxies, on the other hand, are more complicated because the velocities, especially in the barred region, manifest strong non-circular streaming motions. Kormendy (1983) presented the first clear evidence for non-circular stellar streaming motions in a barred galaxy; e.g. the non-circular motions in the NGC 936, which is an SB0 galaxy, are about 20% of the circular streaming velocity and are con-sistent with orbits being elongated along the bar, with the circulation in the forward sense along the bar major axis. Sellwood & Sánchez(2010) estimated the magnitudes of forced non-circular motions over a broad range of bar strengths from strongly barred galaxies to mild bar-like distortions to place bounds on the shapes of halos in disc galaxies. These measurements establish that the bar distorts the axisymmetric potential of the disc sufficiently strongly to force the stars to stream on highly elliptical orbits inside the bar.

2.3.2 Dynamics in bars

Regarding dynamics, bars can be thought as a density wave. However, most bars are so strong that linear perturbation theory which is used to study spiral structure must be supplemented by other tools for describing bar dynamics.

From the linear perturbation theory, it is explained that weak bars can exist only between the Inner Lindblad resonance (ILR) and corotation (CR). Thus (i) weak bars must rotate sufficiently rapidly to avoid the inner Lindblad resonance, and (ii) they must end before the corotation (Binney & Tremaine,2008, or see also section2.3.1).

On the other hand, theoretical works have shown that strong bars are built by orbits elongated along the bar (namedx1). In some cases, they are oriented perpendicularly to

the bar (namedx2), and around the Lagrangian points (Contopoulos &

Papayannopou-los,1980;Skokos et al.,2002). Bar models are easy to construct in N-body simulations; they are created through inherent instabilities in self-consistent simulations of a realistic disc+halo galaxy model with a disc-dominated, flat rotation curve (O’Neill & Dubin-ski, 2003). We showed in section 2.2 that these instabilities are characterized by the Global Stability Parameters (GSP).

Athanassoula(1992) showed that the dust lanes are straight, but centered on the bar’s major axis in "fast bars", while they are curved in "slow bars" (larger values of_R). The high-velocity shear of the gas in such dust lanes along the bar does not permit the rapid star formation; thus the gas remains gravitationally stable. But, low-velocity shear at the end of the bar permits the gas pile up giving rise to the emergence of star formation. Additionally, the vertical structure of bars, viewed from the side (i.e., from a line of sight in the disc plane that is perpendicular to the long axis of the bar), appears boxy or peanut-shaped (Combes & Sanders,1981;Combes et al.,1990). Theoretical work and three-dimensional simulations have shown that the bar often bend out of the disc plane the orbits of the particles, leaving the bar substantially thicker than the surrounding disc (Raha et al.,1991) and show such morphologies.

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2.3. Bar Galaxies 17

2.3.3 N-body simulations

In the sixties, and partly through the seventies as well, theoretical works on galaxy dynamics were mainly analytical. There, they followed the motions of individual par-ticles, or would study collective effects aiming for self-consistent solutions of steady-state of quasi-steady-steady-state potentials, by following, e.g., the Boltzmann equation ( Bin-ney & Tremaine, 2008). In this way, the basis of orbital structure theory was set and a considerable understanding of dynamical effects were obtained. However, the ad-vent of numerical simulations made it clear that galaxies evolve with time, so that a quasi-steady-state approach can not give the complete picture (Athanassoula,2013).

N-body simulations have shown that not only a bar can be triggered by an inter-action in discs stable against their development in isolation (Noguchi, 1987; Gerin et al., 1990; Sundin & Sundelius, 1991; Sundin et al., 1993, and reference therein), but also an isolated model can develop a bar structure in the disc due to its global stabil-ities. Rotationally supported self-gravitating discs of stars are nearly always found to be violently unstable to global bi-symmetric distortions. Simulations reveal that this instability leads to a strong and rapidly rotating bar. In fact, controlling this instability is one of the biggest problems of galaxy dynamics. Athanassoula & Sellwood(1986) presented the role of random motion in controlling bi-symmetric instabilities in one simple mass distribution. Their experiments showed that it is possible to reduce the growth rate of the bar instability merely by increasing the random motion of disc par-ticles suggesting tentatively that Q ' 2₋2.5 at all radii is a sufficient criterion for non-axisymmetric stability in all disc. Where Qis lower than this, bar or spiral insta-bilities should be present; however, the Q parameter increases with the growth of the bar, reaching higher values ofQthan the limit set byAthanassoula & Sellwood(1986); therefore, this criterion only works for initial conditions in simulations and not assess the stability of a galaxy through the time.

On the other hand,Athanassoula(2002b) showed that bars could form in both max-imum and sub-maxmax-imum discs: the primary mechanism is the exchanging of energy and angular momentum between the disc particles and the halo particles at resonances. Generally, it has shown that the origin of a bar in an isolated disc galaxy depends on the ratio between the disc mass MD and halo massMH. The disc being maximum or sub-maximum disc (Athanassoula, 2013), the angular momentum redistribution plays a crucial role in the formation and evolution of the bar. Thus, bars will take longer to evolve in galaxies that have a large ratio of halo-to-disc mass. On the other hand, at later stages, after the secular evolution has started, the halo can increase the bar strength by absorbing a significant fraction of the angular momentum emitted from the bar re-gion. Thus, stronger bars will be found in galaxies with a larger halo-to-disc mass ratio. Besides, as we said before, the increasing of the velocity dispersion in the disc, and/or the halo leads to less angular momentum redistribution and therefore weaker bars form. Likewise, interactions trigger the bar formation in discs galaxies which are stable against their development in isolation (Noguchi, 1987; Gerin et al., 1990; Sundin &

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Sundelius, 1991; Sundin et al., 1993). Recently, Martinez-Valpuesta et al. (2017) showed that the evolution of the bar parameters (strength, length, and pattern speed) in disc galaxies, which forms a bar-like structure in isolation, are not much affected, while such parameters triggered by a perturbation show some difference with its coun-terpart. The angular velocity of the bar which was triggered by a flyby is slower than such structure formed by a self-instability of the disc. Besides, they showed that a slow flyby has a greater effect on the target galaxy.

The strength and the angular velocities of a bar changes due to the transfer of angular momentum at the resonances or due to the mass-loss produced by interactions (Gerin et al.,1990;Sundin et al.,1993;Miwa & Noguchi,1998). The variation of the bar pa-rameters generated by tidal effects depends on the mass of the perturber, the pericentre distance, and slightly of the relative phase of the bar and the companion at pericentre (Gerin et al., 1990; Sundin et al.,1993). Miwa & Noguchi(1998) found that bars in-duced by interactions are confined to the Inner Lindblad Resonant which produces slow bars. Simulations including gravity and hydrodynamics have shown that the gas com-ponent plays an important role in bars formed by tidal events. Moreover, cosmological simulations show that bars can be formed and destroyed several times during a galaxy’s lifetime depending on the accretion history (Romano-Díaz et al.,2008). The mass ratio between the primary galaxy and the perturber creating the interaction determines bar formation in the main galaxy, in the perturber, or even in both (Kazantzidis et al.,2011;

Lang et al.,2014;Łokas et al.,2014).

2.3.4 Growth of the bar

The formation and evolution of a bar in an isolated disc galaxy in N-body simulations have been widely followed. It has shown that the growth of a barred galaxy has three main phases (Martinez-Valpuesta et al., 2006). The first phase corresponds to the bar formation and extends _∼ 2Gyr; the bar strength and bar length grow quickly. The second phase is when the bar buckles; the bar becomes shorter and weaker ( Martinez-Valpuesta & Shlosman,2004). The final stage of the bar is the secular evolution. Sell-wood(1981) showed that the bar grows slowly by increasing its strength and length.

Combes & Sanders(1981), on the other hand, reported that bars tend to weaken in the long term. The rate at which bar parameters change depends on the properties of the model. Debattista & Sellwood (1998) showed that the bar slows by dynamical friction in a dense dark matter halo, whileAthanassoula et al.(2013) found that higher is the gas content in the disc, slower is the growth of the bar. They also concluded that the halo triaxiality triggers the bar formation earlier, and leads to considerably less increase of the bar strength. On the other hand, the pattern speed of the bar slows down during all these phases (Weinberg,1985;Little & Carlberg,1991;Athanassoula,2003b).

The growth of the bar in a disc galaxy can be characterized by three main observa-tional parameters: length, strength and pattern speed. Observaobserva-tionally, these parameters can be measured from real galaxies (see section2.3.1), as well as in N-body

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simula-2.4. Collisions and Encounters of Stellar System 19

tions. Get these parameters from N-body simulations is relatively easier because we get them from the phase space of the simulated galaxy (see appendixesB.2.2).

2.3.5 Conclusion

We have discussed some properties of barred disc galaxies in observations and simu-lations. Observation has shown that the profile of the bar can have two types: flat or exponential profiles. In this thesis, we set down an exponential disc in our models to follow the growth the bar instability.

On the other hand, N-body simulations have shown that bar instability can be con-trolled changing some properties of the disc, like the ratio mass between the disc and halo which is related by the central density of the halo or the disc, and dispersion ve-locity. However, all these works are not clear in what are the precise parameters to control or generate the bar instability. Therefore, one of our goals is to characterize the properties of the disc to be stable or unstable to bar formation. This issue will be addressed in chapters4and5.

Besides, the growth of the bar is described by the length, pattern speed, and the rotation parameter_R. These parameters will be followed in our models which form a bar.

2.4 Collisions and Encounters of Stellar System

In the hierarchical scenario of structure formation, galaxies with their dark halos un-dergo frequent interactions with each other. These interactions may have a dramatic impact on the morphologies, which is an essential part of the galaxy formation. In gen-eral, we can find two types of interactions in the universe: collisions and encounters.

A collision between two galaxies is characterized because their energy is transferred from order motion to random motion evolving as an inelastic collision; however, ac-cording to Newton’s laws, the total energy of the galactic system is conserved. In con-trast, an encounter which is characterized by the high velocity between the two galaxies may leave the two galaxies to survive with some considerable changes. The effect on the internal structure of a stellar system decrease as the encounter speed increase, or also the distance between the two galaxies. (Sundin et al., 1993;Binney & Tremaine,

2008).

2.4.1 Collisions

Galaxy collisions often lead to mergers, in which the final product of the collision is a single merged stellar system. There are two types of mergers: a minor merge, in which one of the galaxies is much smaller than the other, leaves the larger galaxy

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relatively unchanged. In contrast, a major merger, where the two merging galaxies have similar masses, transforms the two galaxies completely due to the violently change of the gravitational field. However, if the two galaxies have enough relative velocity, then they interact with sufficient orbital energy to escape with each other (see section2.4.2).

Dynamical friction

When stellar systems collide there is a permanent transfer of the system orbital energy to random motion of the constituent particles .This loss of energy is like a friction force acting on the moving galaxies; then, this phenomenon is called dynamical friction. Dynamical friction appears in all galactic interactions, but it is easier to understand for minor mergers, where the mass of one of the galaxies is larger. Let us assume an object of mass M (a satellite galaxy or globular cluster) with constant density profile ρ in motion through an infinite number of particles of gas, stars and dark matter. The mass of each particle mp in the gravitational sea is assumed to be very small compared to

M (mp M) such thatM moves forward without deviation. Under these conditions, particles closest to M are attracted by its stronger gravitational force that the rest of particles, which drives to an overdensity of particles on the object trajectory. This overdensity of particles lead to a gravitational force contrary to the way ofM, therefore the object loses kinetic energy and undergoes a velocity decrease.

2.4.2 High-speed encounters

Schematically, we can illustrate a high-speed encounter in Figure2.4. By "high-speed" we mean that the relative velocity between the two galaxies is much larger than the internal velocity dispersion of the target galaxy. Thus, the change in the internal energy can be approximated analytically. A typical example is the interaction of two galax-ies in a rich cluster of galaxgalax-ies, where the velocity dispersion is around of σcluster ∼

1000km s−1_{, and it is significantly larger than the internal dispersion of the individual}

members. In this section, we summarize the main analytical results for this case of encounters.

Consider two interacting galaxies of massesM sandM pwith comparable masses, impact parameterb and relative velocityvi (see Figure2.4). The starss of the galaxy

M sgain energy by the gravitational pull given by the galaxyM p, therefore the rate at which the starssgain energy isvq.g[rq(t)], wheregis the gravitational field at position

rq for a starq with velocityvq with respect to the center of M s. Thus, the change in the total energy is

∆E =

ti

Z

0

vq.g[rq(t)]dt (2.40)

wheretiis the tidal time required for the galaxies to reach the point of closest approach. When the relative velocityviincreases,ti decreases; thus the change of energy of stars

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2.4. Collisions and Encounters of Stellar System 21

Ms

q

Mp

b rq

rp

rs

R′

R−rq

R

Figure 2.4: Schematic illustration of an encounter with impact parameter bbetween a system

M sand its perturberM p. Figure fromMo et al.(2010).

is small.

In the point of closest approach, the duration of the encounter may be roughly esti-mated as

tenc '

max(rs, rp, b)

vi

, (2.41)

and the crossing time of the majority of stars of the interacting galaxies is

tc ∼

rs,p

σs,p

, (2.42)

wherer is the distance of a star to the center of its galaxy andσ indicates the internal velocity dispersion, and the subscripts s or p for galaxies M s or M p. If tc tenc, then the stars receives enough energy to distort their orbits. It is know as impulse approximation. In this mechanism the stars in each system barely move from their positions respect to the center of the galaxy during the encounter. Then, the change of

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the internal energy assuming an axisymmetric system is

∆E = 1

2

X

q

mq|∆vq|2, (2.43)

whereqdenotes theq-th particle of the galaxyM s. Before the encounter, we have by the virial theorem

Ki =−Ei, (2.44)

where the subscriptiindicates now the initial state. During the interaction, the kinetic energy increases in∆K; therefore, the final total energy is

Ef =Ei+ ∆K (2.45)

then the virial theorem for the final state is

Kf =−Ef =−(Ei+ ∆K) = Ki−∆K. (2.46)

So in the new equilibrium state the kinetic energy decreases from Ki + ∆K to

Ki −∆K, which means an amount of2∆K. When the impact parameter b is small (tidal approximation) the velocity increment tends to deform a sphere of stars into an ellipsoid whose long axis lies in the direction of perturbation point of closest approach. This distortion is similar to the way in which the Moon raise tides on the surface of the oceans. The total energy change is

∆E = 4

3G

2

M s

M p vi

2

r

b4 (2.47)

2.4.3 Conclusion

We know that galaxies are interacting with each other all the time. These interactions can change the entire galaxy modifying all the history of the galaxy. The analytic study of this issue is an arduous task. Therefore, the researchers have appealed to N-body simulations to understand the evolution of such encounters. They have shown that a bar galaxy can be generated when a disc galaxy is perturbed by any other galaxy; however, they ignore how the Global Stabilities evolve. Therefore, this issue will be treated in chapter5.

In chapter5, we model encounters of about600km s−1_{; in this case using equation}

2.41, the duration of the encounter is about_∼107_{years. However, we do not model an}

impulse approximation because the gravitational effect of the perturbation start before it approaches the pericentre; thus, we decide to use a model of an elliptical galaxy as a perturbation.

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Chapter 3 Detecting the growth of structures in

pure stellar disc models

Science is a way of thinking much more than it is a body of knowledge. - Carl Sagan

H

OW is the growth of structures in disc galaxies? What determines their

morpholo-gies? A discussion about initial stabilities in a disc galaxy would not be complete without an analyzes of their growth and evolution of local stabilities.

In this chapter, we have generated a series of high-resolution N-body simulations (_∼ 106 particles) in which we included halo, bulge, and disc components following the distribution functions described by Kuijken & Dubinski (1995). The simulations were analyzed using 1D and 2D Fourier Transform methods. These analyzes show the growth and evolution of spiral or bar structures. First, we describe the models and the FT1D and FT2D methods which are used to illustrate the growth of non-axisymmetric structures. Then, we present the results of our analyzes, the comparison with previous studies and a discussion. Finally, we summarize our findings at the end of this chapter. This work was published inValencia-Enríquez et al.(2017).

3.1 Methodology

3.1.1 Setting up of the initial conditions

We used the methodology delineated by Kuijken & Dubinski (1995) to generate the initial conditions of our models. In that work, they described methods for setting up self-consistent disc-bulge-halo galaxy models. Our models have been evolved from 0 to 5 gigayears, with four free parameters: the disc radial velocity dispersionσR, the disc scale height zd, the disc mass md, and the number of particles. Most of the structural

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Figure 3.1: This Figure shows a grid of all our 26 simulations. The names of the models are at the bottom left corner. The models have different disc central radial velocity dispersions

σR,0 and disc scale height zd. These values are given in the upper right corner of each panel. The other parameters for the models are given in Table3.1. The 16 black boxes correspond to models which were run with 1.2 million particles. The blue box represents the MW-A model (Kuijken & Dubinski,1995). The red box shows the models with 8 million particles. The green box corresponds to the four bar models which were run with different number of particles and disc mass (see table 1).

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3.1. Methodology 25

Figure 3.2: Upper panel: the rotation curve generated by model s27_z10D. All our models have similar rotation curves, except the more massive disc one (barred models). Bottom panel: initialQvalue for the 16 unbarred models with 1.2 million particles. These measurements are described in appendixB.

Dynamical stabilities in isolated and perturbed barred models.