simulating dwarf-dwarf galaxy flyby interactions …

SIMULATING DWARF-DWARF GALAXY FLYBY INTERACTIONS

by

ASHOK TIMSINA

JEREMY BAILIN, COMMITTEE CHAIRWILLIAM C. KEELDEAN TOWNSLEYPREETHI NAIR

PATRICK A. FRANTOM

A DISSERTATION

Submi�ed in partial ful�llment of the requirementsfor the degree of Master of Science

in the Department of Physics and Astronomyin the Graduate School of�e University of Alabama

TUSCALOOSA, ALABAMA

2018

Copyright Ashok Timsina 2018ALL RIGHTS RESERVED

ABSTRACT

�is thesis presents the N-body simulation results for the interaction between two equal-mass

dwarf galaxies. We studied how �yby interactions can cause a di�erent level of disturbance on

the dwarf galaxies with the help of four di�erent simulations and measured the departure from

their equilibrium state during the interactions. We performed the simulations using N-body code

ChaNGa. Initially, wemake 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 se�ing the components of velocity and

�nally we studied the distortion on the galaxies by using Fourier analysis looking at modesm = 1

and m = 2. �is analysis allowed us to determine the minimum tidal force required for galaxy

distortion.

ii

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 fromprimary 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 li�le daughter Florisha

Timsina and my brother Kamal Timsina who are my nearest surrounders and have provided me

with a strong love.

iii

ACKNOWLEDGMENTS

At �rst, I would like to express my heartiest gratitude and sincerity to my revered thesis

supervisor Dr. Jeremy Bailin, Associate Professor at �e University of Alabama, for his constant

encouragement, inspiration and patient guidance at every step of my research work. �e 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 o�er my sincere gratitude to all thesis commi�ee 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 DiMa�eo for providing code for my research.

I would like to thanks all the members of Astronomer at the University of Alabama for pro-

viding support and advice throughout my graduate career.

It is impossible to list here the name of all my friends who have givenme 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.

iv

CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

CHAPTER 3 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Tidal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

CHAPTER 4 CONCLUSION AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . 19

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

v

LIST OF TABLES

2.1 Distribution of particles, mass, scale length and scale height. . . . . . . . . . . . . 6

2.2 Numerical simulations with di�erent velocity vector for Galaxy B. . . . . . . . . . 8

vi

LIST OF FIGURES

2.1 �e face-on and edge-on view of isolated galaxy over di�erent phase of time. . . . 7

2.2 Stellar density map of Sim100 at di�erent points in time. . . . . . . . . . . . . . . 9

2.3 Stellar density map of Sim50 at di�erent points in time. . . . . . . . . . . . . . . . 10



3.1 �e variation of amplitude with time. . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Analysis of amplitude as a function of radius for the Fourier modes. . . . . . . . . 16

3.3 �e distortion of galaxies for di�erent interactions. . . . . . . . . . . . . . . . . . 16

3.4 �e variation of distance between two interacting dwarf galaxies at di�erent time. 17

3.5 �e variation of relative amplitude with maximum tidal force for Fourier modes. . 18

vii

CHAPTER 1

INTRODUCTION

Dwarf galaxies are small galaxies made of few thousand to several billion stars. �ey are faint,

having luminosity less than MV ∼ −11.0 (Whiting et al. 1997) which makes them di�cult 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).

�e numbers of dwarf galaxies in the Local Group and the structure of dwarf dark ma�er

halos can probe the mystery of dark ma�er. �e Local Group dwarf galaxies help us understand

the formation and evolution of galaxies by opening a window to their structure, chemical com-

position, 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 a�ect 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. �e 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 signi�-

cant e�ects on galaxies involved in interactions (Toomre & Toomre 1972; White 1978). �is 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. �e tidal forces do not involve direct collision but the in�uence of their force

1

�eld. To study the tidal force between the galaxies, we need to see the snapshot gallery of systems

characterized by di�erent 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 �elds. �is 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. �e 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

�ere are lots of events in the universe in which one galaxy in�uences another galaxy. When

galaxies pass each other quickly, then they experience a less noticeable perturbation to the smooth

potential. �ese types of interactions are called �yby galaxy interactions. �ey are the least vio-

lent producing less perturbation than mergers and both galaxies remain separated without colli-

sion. In �yby 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 �yby interactions

have insigni�cant e�ects which are not important for galaxy evolution. However, numerical sim-

2

ulations have revealed that at low redshi�s, �ybys are more common than mergers for massive

halos (Sinha & Holley-Bockelmann 2012). �erefore, though the �yby interactions have minor

in�uences 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), �ip the spin in the inner halo (Be� & Frenk 2012) or trigger spiral

arms in the galactic disk (e.g., Tutukov & Fedorova 2006). From linear perturbation theory, low

mass halo �ybys can trigger long-lasting e�ects in their evolution phases and such a�ributes can

persist for a long time even a�er the perturbing halo has moved far away (Vesperini & Wein-

berg 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 �ybys could be triggering the distortion (K.

Mc�inn, private communication).

In this work, I use numerical N-body simulations to study �yby 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 mea-

surement of tidal forces during di�erent �yby interactions and Chapter 4 gives the conclusion

and discussion of the �yby interaction simulations.

3

CHAPTER 2

SIMULATIONS

Numerical simulations are tools for studying the dynamics of galaxies. �e simulations pre-

sented here have been carried out by performing simulations using the N-body code Charm N-

body 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. �is speeds up the computational work and enables us to per-

form large simulations by allowing each processor to calculate gravitational forces for a fraction

of particles. �e N-body simulation determines the evolution of interacting particles based on

Newtonian gravitational forces. �e gravitational force on a particle is found by calculating and

summing the forces provided by each other particle. �e force acting on the ith particle due to

all other particles in the simulation is just given by equation (2.1).

Fi =∑i 6=

Gmjrj − 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) ∼

N2 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(NlogN) instead of O(N2) 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 �eld 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. �e velocity and position at time

step n+ 1 based on the leapfrog integrator can be wri�en as

vn+1 = vn + an+ 12∆t, (2.2)

rn+1 = rn +1

2(vn + vn+1)∆t (2.3)

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). �e dynamical state of each particle

is calculated exactly up to n + 12time steps (Springel et al., 2001b; Hernquist & Katz, 1989). �e

time step that the algorithm takes is η√

εa . Where η is a dimensionless constant that controls the

size of the time-steps and ε is the so�ening length. Without so�ening, 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, so�ening is very important in N-body simulations.

ChaNGa simulations run with a spline so�ening length. �e optimal so�ening length from

Power et al.(2003) is given by,

εopt =4r200√N200

(2.4)

Here εopt 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.

5

Table 2.1: Distribution of particles, mass, scale length and scale height of thin disk, intermediatedisk, thick disk and dark ma�er.

Component No. of particles Total mass Mass/particle Scale length Scale height�in Disk 250000 6.87× 109M� 2.75× 104M� 1.6 Kpc 0.1 Kpc

Intermediate Disk 150000 4.13× 109M� 2.75× 104M� 0.67 Kpc 0.2 Kpc�ick Disk 100000 2.70× 109M� 2.75× 104M� 0.67 Kpc 0.1 KpcDark Halo 500000 1.37× 1010M� 2.75× 104M� - 4.67 Kpc

2.1 Initial conditions and Code

�e 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. �is 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 DiMa�eo for generating the initial conditions to put into

the simulations. �is code forms galaxies which are in equilibrium in phase space. �e dwarf

galaxy consists of one million particles, divided evenly between dark ma�er and stars (no gas).

�e 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 ma�er particles with their masses and

scale structure. �e disks are exponential in radius and sech2 in height, and the dark ma�er

is spherically-symmetric with the Navarro-Frenk-White (NFW) density pro�le. We obtained 22

snapshots over 1.086 Gyr of the simulation and we checked the di�erent snapshots to make sure

that the system remained in equilibrium. �e analysis of snapshots was carried out using pyn-

body. Figure 2.1 shows that the isolated galaxy remained in equilibrium throughout its evolution.

�erefore, the initial conditions we used are valid. �e 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.

6

(a) Time = 0.0587 Gyr (b) Time =0.2543 Gyr

(c) Time = 0.499 Gyr (d) Time = 0.744 Gyr

Figure 2.1: �is is the isolated galaxy over di�erent phase of times. �e logarithmically-scaleddensity maps for pixels with >2 particles at di�erent times. It is apparent that the dwarf galaxyis stable. �e two panels show face-on and edge-on projections.

7

Table 2.2: Numerical simulations with di�erent velocity vector for Galaxy B.

Simulations Velocityvx(km/s) vy(km/s) vz(km/s)

Sim100 100 100 0Sim50 100 50 0Sim25 100 25 0Sim10 100 10 0

2.2 Interacting Galaxies

�e 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). �e 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. �ey 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 di�erent velocity vectors as shown

in Table 2.2.

2.3 Sim100

�e velocity vector for the �rst 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. �e density map of the two interacting galaxies at di�erent 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. �ey are still in equilibrium phase a�er crossing each other. �e closest approach

between the two dwarf galaxies is 66.88 kpc.

8

(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 di�erent points in time.

9





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. �e closest approach in this simulation is 33.917 kpc. �e 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 equi-

librium 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

�yby interaction only brie�y perturbs it. A quantitative analysis of the strength of the distortion

is presented in Chapter 3.

10





2.5 Sim25

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

velocity magnitude 103.078 km/sec. �is velocity vector leads Galaxy B to approach closer toward

GalaxyA than in the Sim50. �e densitymap 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.

11

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 e�ect in the interacting galaxies is dramatic. �e density map of this interaction is

shown in Figure 2.5. From the le� panels density map plot, at time 1.086 Gyr, shows that there

appear lots of distortion with vertical swath of stellar particles. �is is no longer �yby interaction

and it is more like merger. �is is determined by how much energy is transformed. During

interactions, orbital energy gets transformed into internal energy of the galaxy. �ough this is

not �yby 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.

12





13

CHAPTER 3

ANALYSIS

In this section, we analyze the Fourier amplitudes for modesm = 1 andm = 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 iso-

lation. In a Fourier analysis, as described for example by Kalnajs (1975), the observed distribution

is decomposed into components with given angular periodicity m. �e 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 di�erent

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 �yby interactions between two dwarf

galaxies, we use pynbody to �nd the m = 1 Fourier amplitude. We calculate a radial pro�le of

the amplitude, and then took just the average of the pro�le. Figure 3.2 (a) shows a sample plot

14

Figure 3.1: �e variation of amplitude with time. �is relation shows that the system is in equi-librium over time.

of amplitude with radius to exhibit how we measured the average amplitude from each snapshot

with modesm = 1. �is 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 �nd 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 �yby interaction is calculated and we �nd the ratio

between their amplitudes and �nally we studied the distortion in the interacting galaxies with

time. Figure 3.3 (a) shows the variation of relative amplitude with time for di�erent simulations

for Fourier mode m = 1. We can conclude that the interaction comes into play only a�er 0.6

Gyr. Before 0.6 Gyr, we cannot observed any lopsided e�ect in the interacting galaxies and a�er

this time, we observed lopsidedness. �e closer the interaction, the more distortion is observed

which leads to galaxies more extended on one side than the other. �us, Sim10 exhibits the most

lopsidedness compared to the other simulations.

15

(a) (b)

Figure 3.2: Analysis of amplitude as a function of radius for the Fourier modesm = 1 andm = 2of 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 theinteractions, and how the distortion evolves with time. �e distortion is measured with the helpof relative amplitude for the Fourier analysis for modem = 1 andm = 2 of interacting galaxiesof Sim100, Sim50, Sim25 and Sim10.

16

Figure 3.4: �e variation of distance between two interacting dwarf galaxies at di�erent times forthe di�erent simulations.

To study the ellipticity due to the interaction between the galaxies, we examine the relative

amplitude for Fourier mode m = 2 at di�erent times. Figure 3.3 (b) shows that the e�ect of

ellipticity appears only a�er 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. �is force appear because gravitational at-

traction between two objects increases with a decrease in distance and experiences a stronger

in�uence.

�e tidal force developed between two galaxies depends upon the masses, the distance be-

tween 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 =dF

dRL =

2GM2

R3L (3.1)

17

(a) (b)

Figure 3.5: �e variation of relative amplitude with maximum tidal force for Fourier modes m=1and m=2. Each data point represents an entire simulation. �e do�ed line is the equilibrium line.

�e distance between two interacting galaxies at di�erent times is shown in Figure 3.4. From

this �gure, 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 1.82×1028N . �e maximum tidal forces

for Sim50 and Sim25 are 1.39× 1029N and 3.44× 1030N corresponding to the distance 33.92 kpc

and 11.61 kpc respectively at same time 0.841 Gyr.

�e 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× 1029N tidal force.

18

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 di�erent simulations. We analyzed the perturbation with the relative amplitude for di�erent

simulations and we concluded that �yby interactions of galaxies disturb their equilibrium for

at least hundreds of Myr. �e 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. �e in�uence

of one galaxy over other galaxy during interactions shows higher �uctuation in the amplitude

which shows that the disturbance is powerful. �is can explain why nearly all galaxies in a group

are strongly lopsided and elliptical (Bournaud et al. 2005). �us, our study con�rmed the detailed

dynamical studies and simulations of �yby interactions between galaxies over time.

We also measured the tidal force for di�erent simulations and compared it to the e�ects of

the interaction on the mass distribution. �is con�rmed that the interaction is e�ective if one

galaxy passes very close to another galaxy because during interaction, the gravitational �eld of

one galaxy directly impacts the other. So, from these four simulations, we can say that at least

19

1× 1029N tidal force is required to notice the impact of one galaxy over another galaxy.

Here we analyzed Fourier mode withm = 1 andm = 2 only because during our simulations

we did not perform head on interactions which exert great e�ect on the system. But, when one

galaxy passes very close to another galaxy causing huge distortion with random sca�ering of

stellar particles, we probably would stop using Fourier analysis completely, since it would stop

being a useful description of the system.

We have �gured out how �yby interactions can cause a di�erent level of disturbance on the

dwarf galaxies with the help of four di�erent simulations. �is 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. �erefore, future

work will be to continue the simulations for longer.

20

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simulating dwarf-dwarf galaxy flyby interactions …

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