simulating dwarf-dwarf galaxy flyby interactions …
TRANSCRIPT
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.
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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
2.4 Stellar density map of Sim25 at di�erent points in time. . . . . . . . . . . . . . . . 11
2.5 Stellar density map of Sim10 at di�erent points in time. . . . . . . . . . . . . . . . 13
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
(c) Time = 0.499 Gyr (d) Time = 0.744 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
(a) Time = 0.0587 Gyr (b) Time = 0.2543 Gyr
(c) Time = 0.499 Gyr (d) Time = 0.744 Gyr
(e) Time = 0.890 Gyr (f) Time = 1.086 Gyr
Figure 2.3: Stellar density map of Sim50 at di�erent 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. �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
(a) Time = 0.0587 Gyr (b) Time = 0.2543 Gyr
(c) Time = 0.499 Gyr (d) Time = 0.744 Gyr
(e) Time = 0.890 Gyr (f) Time = 1.086 Gyr
Figure 2.4: Stellar density map of Sim25 at di�erent points in time.
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
(a) Time = 0.0587 Gyr (b) Time = 0.2543 Gyr
(c) Time = 0.499 Gyr (d) Time = 0.744 Gyr
(e) Time = 0.890 Gyr (f) Time = 1.086 Gyr
Figure 2.5: Stellar density map of Sim10 at di�erent points in time.
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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