SIMULATING DWARF-DWARF GALAXY FLYBY INTERACTIONS by

SIMULATING DWARF-DWARF GALAXY FLYBY INTERACTIONS

by ASHOK TIMSINA JEREMY BAILIN, COMMITTEE CHAIR WILLIAM C. KEEL DEAN TOWNSLEY PREETHI NAIR PATRICK A. FRANTOM

A DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Physics and Astronomy in the Graduate School of The University of Alabama

TUSCALOOSA, ALABAMA

2018

Copyright Ashok Timsina 2018 ALL RIGHTS RESERVED

ABSTRACT This thesis presents the N-body simulation results for the interaction between two equal-mass dwarf galaxies. We studied how flyby interactions can cause a different level of disturbance on the dwarf galaxies with the help of four different simulations and measured the departure from their equilibrium state during the interactions. We performed the simulations using N-body code ChaNGa. Initially, we make sure that the interacting galaxies are in an equilibrium state separated by 100 kpc. We established the motion of one galaxy towards another galaxy, which is at rest. We designed the interactions to be increasingly strong by setting the components of velocity and finally we studied the distortion on the galaxies by using Fourier analysis looking at modes m = 1 and m = 2. This analysis allowed us to determine the minimum tidal force required for galaxy distortion.

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DEDICATION I dedicate this thesis to my parents (Damodar Timsina & Tara Devi Timsina) who have always been my nearest and closest with me whenever I needed. I owe a debt to my parents and for their love, blessings, inspiration, encouragement and the support from primary to university education. Also, I want to dedicated to Dr. Binil Aryal ; Professor of Tribhuvan University, who encouraged me to build my motivation towards the world of Astronomy. I also dedicate this thesis to my wife Sakuntala Gautam Timsina, my little daughter Florisha Timsina and my brother Kamal Timsina who are my nearest surrounders and have provided me with a strong love.

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ACKNOWLEDGMENTS At first, I would like to express my heartiest gratitude and sincerity to my revered thesis supervisor Dr. Jeremy Bailin, Associate Professor at The University of Alabama, for his constant encouragement, inspiration and patient guidance at every step of my research work. The work would not have been materialized in the present form without his constructive feedback and incisive observation from the very beginning. I would like to offer my sincere gratitude to all thesis committee members William C. Keel, Dean Townsley, Preethi Nair and Patrick A. Frantom for providing support and advice in my thesis work. I also would like to thank Paola DiMatteo for providing code for my research. I would like to thanks all the members of Astronomer at the University of Alabama for providing support and advice throughout my graduate career. It is impossible to list here the name of all my friends who have given me help, encouragement and advice during the time of work. However, I would be delighted to extend my thankfulness to my colleagues Mr. Prabanda Nakarmi, Mr. Nirmal Baral, Mr. Sujan Budhathoki, Mr. Sumedh Sharma and all my friends who helped me directly and indirectly for this work.

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CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

Galaxy interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Fly-by interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

CHAPTER 2

SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

Initial conditions and Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Interacting Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Sim100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4

Sim50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.5

Sim25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.6

Sim10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 3

ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.1

Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.2

Tidal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4

CONCLUSION AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . .

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF TABLES

2.1

Distribution of particles, mass, scale length and scale height. . . . . . . . . . . . .

6

2.2

Numerical simulations with different velocity vector for Galaxy B. . . . . . . . . .

8

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LIST OF FIGURES

2.1

The face-on and edge-on view of isolated galaxy over different phase of time. . . .

7

2.2

Stellar density map of Sim100 at different points in time. . . . . . . . . . . . . . .

9

2.3

Stellar density map of Sim50 at different points in time. . . . . . . . . . . . . . . .

10

2.4


11

2.5


13

3.1

The variation of amplitude with time. . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.2

Analysis of amplitude as a function of radius for the Fourier modes. . . . . . . . .

16

3.3

The distortion of galaxies for different interactions. . . . . . . . . . . . . . . . . .

16

3.4

The variation of distance between two interacting dwarf galaxies at different time. 17

3.5

The variation of relative amplitude with maximum tidal force for Fourier modes. .

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18

CHAPTER 1 INTRODUCTION

Dwarf galaxies are small galaxies made of few thousand to several billion stars. They are faint, having luminosity less than MV ∼ −11.0 (Whiting et al. 1997) which makes them difficult to observe. In the Local Group, there are a number of such dwarf galaxies either isolated or orbiting around a more massive associate (Miller 1996; Karachentseva et al 1985; Cote et al. 1997; Phillips et al. 1998; Ferguson & Sandage 1991). The numbers of dwarf galaxies in the Local Group and the structure of dwarf dark matter halos can probe the mystery of dark matter. The Local Group dwarf galaxies help us understand the formation and evolution of galaxies by opening a window to their structure, chemical composition, and kinematics. Marzke & Da Costa (1997) state that dwarf galaxies are very important for studying the evolution of recent galaxies because they are the most common type of galaxy in the universe, and are building blocks for larger galaxies. When two galaxies pass close to one another, they can still affect one another which is called an interaction. So, to study the evolution, the interaction between the dwarf galaxies is one of the primary issues which should be understood. The processes of galaxy formation and evolution involve many factors like how the stellar and halo components evolve and how the interactions between galaxies occur. In general, gravitational tidal forces are responsible for the most significant effects on galaxies involved in interactions (Toomre & Toomre 1972; White 1978). This force is responsible for generating the actual interaction between the visible parts of the two galaxies at closest approach. Distortion of the galaxy depends on the mass of galaxies and the distance of closest approach. The tidal forces do not involve direct collision but the influence of their force 1

field. To study the tidal force between the galaxies, we need to see the snapshot gallery of systems characterized by different structural and collisional parameters like velocity and time of impact.

1.1

Galaxy interaction Galaxy interaction is one of the dynamical processes which disturb the equilibrium. Gravity

is the dominant force in a galaxy interaction which causes the galaxies to become distorted or exchange mass. Interacting galaxies are deformed by their mutual gravitational fields. This may change the shape of galaxies which have been drawn out by tidal forces during the interaction. Larger perturbations during interaction lead to a merger of galaxies. Mergers are rare events in the universe which are violent. Mergers have been studied extensively, both theoretically and observationally (e.g., Lacey & Cole 1993; Guo & White 2008; Genel et al. 2008, 2009; Schweizer 1986; Casasola et al. 2004; Bridge et al. 2007; Ryan et al. 2008). By inferring a Mhalo − Mgal relation, the galaxy merger rates (Gau & White 2008; Wetzel et al. 2009, Behroozi et al. 2010) are studied. The merger between galaxies is a main leading factor for the evolution of galaxies. Observational studies show that the evolution is driven by several close encounters that would drive to change the morphology of the galaxies. Based on Moore et al. (1996), the harassment of low luminosity spirals create the dwarf ellipticals which has the potential to change the internal property of a galaxy within a cluster and the overall shape.

1.2

Fly-by interactions There are lots of events in the universe in which one galaxy influences another galaxy. When

galaxies pass each other quickly, then they experience a less noticeable perturbation to the smooth potential. These types of interactions are called flyby galaxy interactions. They are the least violent producing less perturbation than mergers and both galaxies remain separated without collision. In flyby interactions, the interaction encounter time is not enough to react for exchange of particles (Gonzalez-Garcia et al. 2005). During the 1950s, it was believed that flyby interactions have insignificant effects which are not important for galaxy evolution. However, numerical sim2

ulations have revealed that at low redshifts, flybys are more common than mergers for massive halos (Sinha & Holley-Bockelmann 2012). Therefore, though the flyby interactions have minor influences on galaxy structure, lots of them can add up for the galaxy evolution over a long period of time. Flyby interactions of galaxies can change the morphological structures e.g spiral to S0 galaxy (Bekki & Couch 2011), flip the spin in the inner halo (Bett & Frenk 2012) or trigger spiral arms in the galactic disk (e.g., Tutukov & Fedorova 2006). From linear perturbation theory, low mass halo flybys can trigger long-lasting effects in their evolution phases and such attributes can persist for a long time even after the perturbing halo has moved far away (Vesperini & Weinberg 2000). Johnson et al. (in prep) have found evidence for distorted outskirts in some dwarf galaxies that are not near any large galaxies, which is unexpected. But since there are sometimes other dwarf neighbors, one possibility is that dwarf flybys could be triggering the distortion (K. McQuinn, private communication). In this work, I use numerical N-body simulations to study flyby interactions between dwarf galaxies. Chapter 2 discusses the details of simulations with the structure of two interacting dwarf galaxies. Chapter 3 presents detailed Fourier analysis of their structure, and also the measurement of tidal forces during different flyby interactions and Chapter 4 gives the conclusion and discussion of the flyby interaction simulations.

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CHAPTER 2 SIMULATIONS

Numerical simulations are tools for studying the dynamics of galaxies. The simulations presented here have been carried out by performing simulations using the N-body code Charm Nbody GrAvity solver called ChaNGa (Jetley et al. 2008, 2010; Menon et al. 2015). It uses the Charm++ library to provide good runtime performance scaling on parallel systems. Charm++ provides a tree data structure to represent the N-body simulation space. During simulation, this tree is segmented and the pieces of the tree are distributed by the adaptive Charm++ runtime system to the processors for parallel computation of gravitational forces. On each processor, forces are calculated by ChaNGa. This speeds up the computational work and enables us to perform large simulations by allowing each processor to calculate gravitational forces for a fraction of particles. The N-body simulation determines the evolution of interacting particles based on Newtonian gravitational forces. The gravitational force on a particle is found by calculating and summing the forces provided by each other particle. The force acting on the ith particle due to all other particles in the simulation is just given by equation (2.1).

Fi =

X

Gmj

i6=

rj − ri (rj − ri )3

(2.1)

In N-body simulations, each particle interacts with (N −1) other particles which lie at (rj −ri ) distances. For N -particles the number of forces of interaction between them is N (N − 1) ∼ N 2 but due to opposite reaction force the number of unique interactions is reduced to

N (N −1) . 2

However, the Barnes-Hut algorithm (Barnes & Hut 1986) optimizes the summation, allowing it to scale as O(N logN ) instead of O(N 2 ) but retaining high accuracy. In ChaNGa, the gravitational 4

force calculation is based on the Barnes-Hut algorithm with PKDGRAV (Stadel 2001) and in this algorithm, the mass distribution of each tree node is expanded in multipoles up to hexadecapole for calculating the far field mass distribution within a tree node. In ChaNGa, an adaptive leapfrog time integrator (Springel et al., 2001b; Hernquist & Katz, 1989; Springel, 2005) is used to calculate particles’ time stepping. Each particle has its own time step. The velocity and position at time step n + 1 based on the leapfrog integrator can be written as

vn+1 = vn + an+ 1 ∆t,

(2.2)

1 rn+1 = rn + (vn + vn+1 )∆t 2

(2.3)

2

where v, r, ∆t, a, n are the velocity, position, time step, acceleration, and number of step respectively (Springel et al., 2001b; Hernquist & Katz, 1989). The dynamical state of each particle is calculated exactly up to n +

1 2

time steps (Springel et al., 2001b; Hernquist & Katz, 1989). The p time step that the algorithm takes is η a . Where η is a dimensionless constant that controls the size of the time-steps and is the softening length. Without softening, when two particles come very close to each other, the force between them becomes large which causes a problem for both collisional and collisionless calculations. So, softening is very important in N-body simulations. ChaNGa simulations run with a spline softening length. The optimal softening length from Power et al.(2003) is given by, 4r200 opt = √ N200

(2.4)

Here opt is measured within the virial radius r200 and N200 is a number of particles within the virial radius. ChaNGa can also perform collisional N-body simulations which include hydrodynamics and thermodynamics using the Smooth Particle Hydrodynamics technique (Lucy 1977; Gingold & Monaghan 1977) but we did not use hydrodynamics or thermodynamics in this work.

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Table 2.1: Distribution of particles, mass, scale length and scale height of thin disk, intermediate disk, thick disk and dark matter. Component Thin Disk Intermediate Disk Thick Disk Dark Halo

2.1

No. of particles 250000 150000 100000 500000

Total mass 6.87 × 109 M 4.13 × 109 M 2.70 × 109 M 1.37 × 1010 M

Mass/particle 2.75 × 104 M 2.75 × 104 M 2.75 × 104 M 2.75 × 104 M

Scale length 1.6 Kpc 0.67 Kpc 0.67 Kpc -

Scale height 0.1 Kpc 0.2 Kpc 0.1 Kpc 4.67 Kpc

Initial conditions and Code The initial conditions represent the position and velocity of particles at one point in time

which are used in numerical simulations. For our initial conditions, we want our galaxies in equilibrium. To create these, we use the iterative model of Rodionov et al. (2009), where both the kinetic constraints and the mass distribution can be arbitrary. This model creates the equilibrium phase models with the given mass distribution and with given kinematic parameters. Here, we use the code kindly provided by Paola DiMatteo for generating the initial conditions to put into the simulations. This code forms galaxies which are in equilibrium in phase space. The dwarf galaxy consists of one million particles, divided evenly between dark matter and stars (no gas). The star particles are distributed in the thin disk, intermediate disk and thick disk regions of the galaxy. Table 2.1 gives the distribution of stellar and dark matter particles with their masses and scale structure. The disks are exponential in radius and sech2 in height, and the dark matter is spherically-symmetric with the Navarro-Frenk-White (NFW) density profile. We obtained 22 snapshots over 1.086 Gyr of the simulation and we checked the different snapshots to make sure that the system remained in equilibrium. The analysis of snapshots was carried out using pynbody. Figure 2.1 shows that the isolated galaxy remained in equilibrium throughout its evolution. Therefore, the initial conditions we used are valid. The Fourier analysis we present in Chapter 3 also demonstrates that the system is in equilibrium (see Figure 3.1). We also checked various disk scale height over time and all the plot shows that the system is in equilibrium.

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(a) Time = 0.0587 Gyr

(b) Time =0.2543 Gyr

(c) Time = 0.499 Gyr

(d) Time = 0.744 Gyr

Figure 2.1: This is the isolated galaxy over different phase of times. The logarithmically-scaled density maps for pixels with >2 particles at different times. It is apparent that the dwarf galaxy is stable. The two panels show face-on and edge-on projections.

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Table 2.2: Numerical simulations with different velocity vector for Galaxy B. Velocity

Simulations

2.2

vx (km/s)

vy (km/s)

vz (km/s)

Sim100

100

100

0

Sim50

100

50

0

Sim25

100

25

0

Sim10

100

10

0

Interacting Galaxies The initial conditions for the simulation of two interacting galaxies are identical, each consist

of one million particles which is the sum of the number of stellar particles (500000) and the number of halo particles (500000). The locations of the two galaxies are given by the coordinate (0, 0, 0) and (-100, 0, 0) kpc respectively. Let us consider the galaxy with position (0, 0, 0) to be Galaxy A and the galaxy with position (-100, 0, 0) to be Galaxy B. Galaxy B is set in motion towards Galaxy A which is at rest. Initially, both galaxies are in equilibrium. They were designed to be increasingly strong interactions as we go from simulation Sim100 to Sim10 because they have closer approaches. We performed four simulations with different velocity vectors as shown in Table 2.2.

2.3

Sim100 The velocity vector for the first simulation is set to (100,100,0) km/sec for Galaxy B which

approaches toward Galaxy A with total velocity magnitude 141.42 km/sec making a parabolic path. The density map of the two interacting galaxies at different points in time is shown in Figure 2.2. From the density distribution over 1.086 Gyr, we can see that there is no distortion to either galaxy. They are still in equilibrium phase after crossing each other. The closest approach between the two dwarf galaxies is 66.88 kpc.

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(a) Time = 0.0587 Gyr

(b) Time = 0.2543 Gyr



(e) Time = 0.890 Gyr

(f) Time = 1.086 Gyr

Figure 2.2: Stellar density map of Sim100 at different points in time.

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(a) Time = 0.0587 Gyr

(b) Time = 0.2543 Gyr





Figure 2.3: Stellar density map of Sim50 at different points in time. 2.4

Sim50 During this simulation, we consider the velocity components equal to (100,50,0) km/sec, which

gives the total velocity magnitude 111.80 km/sec. In this case, Galaxy B approaches more closely toward Galaxy A due to the decrease in the y-components of the velocity in comparison with Sim100. The closest approach in this simulation is 33.917 kpc. The density map of the galaxy as shown in Figure 2.3. From time 0.744 Gyr shows that the involved galaxies are taken out of equilibrium due to the interaction, with very small elliptical and lopsided distortions. It doesn’t create S-shaped tidal tails that are typically thought of for tidally distorted dwarf galaxies, because the flyby interaction only briefly perturbs it. A quantitative analysis of the strength of the distortion is presented in Chapter 3.

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(a) Time = 0.0587 Gyr

(b) Time = 0.2543 Gyr





Figure 2.4: Stellar density map of Sim25 at different points in time. 2.5

Sim25 The simulation is conducted assuming the velocity vector (100,25,0) km/sec with the total

velocity magnitude 103.078 km/sec. This velocity vector leads Galaxy B to approach closer toward Galaxy A than in the Sim50. The density map of snapshots from this simulation as shown in Figure 2.4 shows noticeable distortion in both galaxies. From the density map we can see that there is large distortion from 0.6 Gyr and lasts for several Gyrs, and from the last panel of Figure 4, at time 1.086 Gyr, we can see vertical thickening.

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2.6

Sim10 For this simulation, we set the components of velocities for Galaxy B to (100,10,0) km/sec

which gives the total velocity magnitude 100.499 km/sec. From the simulation, the density map shows the effect in the interacting galaxies is dramatic. The density map of this interaction is shown in Figure 2.5. From the left panels density map plot, at time 1.086 Gyr, shows that there appear lots of distortion with vertical swath of stellar particles. This is no longer flyby interaction and it is more like merger. This is determined by how much energy is transformed. During interactions, orbital energy gets transformed into internal energy of the galaxy. Though this is not flyby interaction but we performed simulation because we don’t know how much energy will be transformed before we do the simulation. If they are point mass it doesn’t bound because energy get transfer into internal energy of the galaxy we don’t know before hand whether it is bound or not.

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(a) Time = 0.0587 Gyr

(b) Time = 0.2543 Gyr





Figure 2.5: Stellar density map of Sim10 at different points in time.

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CHAPTER 3 ANALYSIS

In this section, we analyze the Fourier amplitudes for modes m = 1 and m = 2 and also tidal force during each interaction. Fourier analysis is of the star particles, which correspond to the visible part of the galaxy. We have also calculated the distance of closest approach between the two interacting galaxies, which is very important for studying the distortion of galaxies from the equilibrium phase.

3.1

Fourier Analysis To study the structural properties and dynamics of the galaxy, cylindrical shells analysis in

the galactic disk is conducted based on relative Fourier amplitudes. We compare just the average amplitude obtained from each snapshot during each interaction to the average amplitude in isolation. In a Fourier analysis, as described for example by Kalnajs (1975), the observed distribution is decomposed into components with given angular periodicity m. The Fourier amplitude with m = 1 measures the lopsidedness, in which the galaxy is more extended one side than the other; the m = 2 amplitude measures ellipticity, in which the galaxy has deviated from the azimuthal symmetry. In this section, we compare the relative amplitude of each interaction at a different time for m = 1 and m = 2 respectively. From Figure 3.1 we can say that initially the system is in equilibrium. In order to check the lopsided amplitudes generated by flyby interactions between two dwarf galaxies, we use pynbody to find the m = 1 Fourier amplitude. We calculate a radial profile of the amplitude, and then took just the average of the profile. Figure 3.2 (a) shows a sample plot 14

Figure 3.1: The variation of amplitude with time. This relation shows that the system is in equilibrium over time. of amplitude with radius to exhibit how we measured the average amplitude from each snapshot with modes m = 1. This is the amplitude curve for Sim100 at time 0.744 Gyr. Similarly, we checked the ellipticity taking the Fourier amplitude with mode m = 2 using pynbody as shown in Figure 3.2 (b). m = 2 mode analysis has been performed intensively for N-body simulations, both for bar analysis of galaxies (Binney & Tremaine 1987, Combes 2008, Shlosman 2005) and two-armed spiral properties (Rohlfs 1977, Toomre 1981). In this case, we also use the same method to find the average value of amplitude as m = 1 and analyze how the distortion evolves with time. In order to understand the relative distortions in each simulation, the average amplitude of equilibrium and distorted galaxies due to flyby interaction is calculated and we find the ratio between their amplitudes and finally we studied the distortion in the interacting galaxies with time. Figure 3.3 (a) shows the variation of relative amplitude with time for different simulations for Fourier mode m = 1. We can conclude that the interaction comes into play only after 0.6 Gyr. Before 0.6 Gyr, we cannot observed any lopsided effect in the interacting galaxies and after this time, we observed lopsidedness. The closer the interaction, the more distortion is observed which leads to galaxies more extended on one side than the other. Thus, Sim10 exhibits the most lopsidedness compared to the other simulations.

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(a)

(b)

Figure 3.2: Analysis of amplitude as a function of radius for the Fourier modes m = 1 and m = 2 of interacting galaxies of Sim100 at time 0.744 Gys.

(a)

(b)

Figure 3.3: Here we show the main result of this work - how distorted galaxies get from the interactions, and how the distortion evolves with time. The distortion is measured with the help of relative amplitude for the Fourier analysis for mode m = 1 and m = 2 of interacting galaxies of Sim100, Sim50, Sim25 and Sim10.

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Figure 3.4: The variation of distance between two interacting dwarf galaxies at different times for the different simulations. To study the ellipticity due to the interaction between the galaxies, we examine the relative amplitude for Fourier mode m = 2 at different times. Figure 3.3 (b) shows that the effect of ellipticity appears only after 0.6 Gyr. Higher distortion is observed for Sim10.

3.2

Tidal force A tidal force developed between the galaxies is a cosmic process in which one galaxy gets

distorted gravitationally by another nearer galaxy. This force appear because gravitational attraction between two objects increases with a decrease in distance and experiences a stronger influence. The tidal force developed between two galaxies depends upon the masses, the distance between the galaxies and scale length. Here the mass and scale length of the two galaxies are the same so the tidal force acting between them is given by

∆F =

2GM 2 dF L= L dR R3

17

(3.1)

(a)

(b)

Figure 3.5: The variation of relative amplitude with maximum tidal force for Fourier modes m=1 and m=2. Each data point represents an entire simulation. The dotted line is the equilibrium line. The distance between two interacting galaxies at different times is shown in Figure 3.4. From this figure, we can say that for Sim100, the closest distance between two dwarf galaxies is 66.88 kpc at time 0.548 Gyr producing maximum tidal force of 1.82 × 1028 N . The maximum tidal forces for Sim50 and Sim25 are 1.39 × 1029 N and 3.44 × 1030 N corresponding to the distance 33.92 kpc and 11.61 kpc respectively at same time 0.841 Gyr. The relationship between the relative amplitude with maximum tidal force for Fourier mode m = 1 and m = 2 are shown in Figure 3.5. From this analysis, we can say that the noticeable distortion on galaxies will appear if there is at least 1 × 1029 N tidal force.

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CHAPTER 4 CONCLUSION AND DISCUSSION

Flyby interactions of dwarf galaxies are common in the universe and the studies of these simulations reveal that they can be important for the evolution and formation of galaxies. Here, we have analyzed the structure of the galaxies, and focused on using the Fourier amplitudes as a way of quantifying the trends for mode m = 1 (lopsidedness) and m = 2 (ellipticity) in the interacting galaxies and also measured the maximum tidal force created during their interaction for different simulations. We analyzed the perturbation with the relative amplitude for different simulations and we concluded that flyby interactions of galaxies disturb their equilibrium for at least hundreds of Myr. The amount of distortion in the interacting galaxy depends upon the distance of closest approach which is set with the direction of total velocity magnitude. In this work, we almost consider the same total velocity magnitude of Galaxy B changing their velocity components that is the direction of total velocity magnitude. When Galaxy B passes very close to Galaxy A i.e in the Sim10, more distortion appears with huge vertical thickening. The influence of one galaxy over other galaxy during interactions shows higher fluctuation in the amplitude which shows that the disturbance is powerful. This can explain why nearly all galaxies in a group are strongly lopsided and elliptical (Bournaud et al. 2005). Thus, our study confirmed the detailed dynamical studies and simulations of flyby interactions between galaxies over time. We also measured the tidal force for different simulations and compared it to the effects of the interaction on the mass distribution. This confirmed that the interaction is effective if one galaxy passes very close to another galaxy because during interaction, the gravitational field of one galaxy directly impacts the other. So, from these four simulations, we can say that at least 19

1 × 1029 N tidal force is required to notice the impact of one galaxy over another galaxy. Here we analyzed Fourier mode with m = 1 and m = 2 only because during our simulations we did not perform head on interactions which exert great effect on the system. But, when one galaxy passes very close to another galaxy causing huge distortion with random scattering of stellar particles, we probably would stop using Fourier analysis completely, since it would stop being a useful description of the system. We have figured out how flyby interactions can cause a different level of disturbance on the dwarf galaxies with the help of four different simulations. This is very useful for sorting out if any given system could have been distorted by a particular neighbor. Due to computational constraints, we could not run the simulations for long enough to know how long the distortions last. To compare to statistics of the population of dwarf galaxies, the lifetime of the disturbance is required until the galaxies return to equilibrium. Therefore, future work will be to continue the simulations for longer.

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