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이학박사 학위논문

Bar Formation, Gas Evolution,and Star Formation inBarred-Spiral Galaxies

막대나선은하의 막대 형성과 기체 진화 및 별 형성에

대한 연구

2018년 8월

서울대학교 대학원

물리·천문학부 천문학전공

서 우 영

ABSTRACT

Bar structures are common in spiral galaxies. The understanding of the evolution

of the bar galaxy is essential to the overall understanding of the disk galaxies. Many

previous studies have done research on the evolution of bar galaxies, but there are still

many parts that are not well understood, such as the temporal and spatial distributions

of star formation in a nuclear ring, the formation and evolution of gaseous structures

especially the ring size, and the effects of the gaseous component on a stellar bar. In this

thesis, using both hydrodynamic simulations and self-consistent simulations including

star formation and feedback, we try to understand unsolved problems for barred spiral

galaxies.

We first investigate the star formation in a nuclear ring without spiral arms. We

use hydrodynamic simulations to study temporal and spatial behavior of star formation

occurring in nuclear rings of barred galaxies where radial gas inflows are triggered solely

by a bar potential. The star formation recipes include a density threshold, an efficiency,

conversion of gas to star particles, and delayed momentum feedback via supernova

explosions. We find that star formation rate (SFR) in a nuclear ring is roughly equal

to the mass inflow rate to the ring, while it has a weak dependence on the total gas

mass in the ring. The SFR typically exhibits a strong primary burst followed by weak

secondary bursts before declining to very small values. The primary burst is associated

with the rapid gas infall to the ring due to the bar growth, while the secondary bursts

are caused by re-infall of the ejected gas from the primary burst. While star formation in

observed rings persists episodically over a few Gyr, the duration of active star formation

in our models lasts for only about a half of the bar growth time, suggesting that the bar

potential alone is unlikely responsible for gas supply to the rings. When the SFR is low,

most star formation occurs at the contact points between the ring and the dust lanes,

leading to an azimuthal age gradient of young star clusters. When the SFR is large, on

the other hand, star formation is randomly distributed over the whole circumference of

the ring, resulting in no apparent azimuthal age gradient. Since the ring shrinks in size

i

with time, star clusters also exhibit a radial age gradient, with younger clusters found

closer to the ring. The cluster mass function is well described by a power law, with a

slope depending on the SFR.

We then investigate the effects of spiral arms on the ring star formation. We use

hydrodynamic simulations to study the effect of spiral arms on the star formation rate

(SFR) occurring in nuclear rings of barred-spiral galaxies. We find that spiral arms can

be an efficient means of gas transport from the outskirts to the central parts, provided

that the arms are rotating slower than the bar. While the ring star formation in models

with no-arm or corotating arms is active only during about the bar growth phase,

arm-driven gas accretion makes the ring star formation both enhanced and prolonged

significantly in models with slow-rotating arms. The arm-enhanced SFR is larger by

a factor of ∼ 3 − 20 than that in the no-arm model, with larger values corresponding

to stronger and slower arms. Arm-induced mass inflows also make dust lanes stronger.

Nuclear rings in slow-arm models are ∼ 45% larger than in the no-arm counterparts.

Star clusters that form in a nuclear ring exhibit an age gradient in the azimuthal

direction only when the SFR is small, whereas no noticeable age gradient is found in

the radial direction for models with arm-induced star formation.

As a final project, we run fully self-consistent simulations of galaxies similar to the

Milky Way to study formation and evolution of a stellar bar in the presence of gas

as well as a nuclear ring and star formation therein. The gas suffers radiative heating

and cooling and is subject to star formation feedback. We consider two sets of models

with cold or warm disks that differ in the Toomre stability parameter, and vary the

gas fraction fgas by fixing the total disk mass. We find that a bar forms earlier and

more strongly as fgas increases in the cold disk, while the bar formation is progressively

delayed in the warm disks. The bar formation enhances the central mass concentration

(CMC) which in turns makes the bars decay temporarily, after which the bars grow

in size and strength again. Only the gas-free, warm-disk model undergoes buckling

instability since rapid bar and CMC growth in models with gas excites the vertical

stellar motions. The gas driven inward by the bar potential readily forms a star-forming

ii

nuclear ring. The ring is very small when it first forms and grows in size over time due to

addition of gas with higher angular momentum. The ring star formation rate is episodic

and bursty due to feedback, and well correlated with the bar strength. The bars and

nuclear rings formed in our models have properties similar to those in the Milky Way.

Keywords: galaxies: ISM − galaxies: kinematics and dynamics− galaxies: bar −

galaxies: spiral − galaxies: evolution − galaxies: star formation and feedback −

method: numerical simulation

Student Number: 2011-30128

iii

Contents

Abstract i

List of Figures ix

List of Tables xiii

1 Introduction 1

1.1 Barred Spiral Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Gaseous Structures: Nuclear Ring and Filamentary Spurs . . . . . . . . 3

1.3 Star Formation in the Bar Region . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Effects of Gas on Stellar Bars . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Purpose of this Thesis and Outline . . . . . . . . . . . . . . . . . . . . . 9

2 Star Formation in Nuclear Rings of Barred Galaxies 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Galaxy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Star Formation and Feedback . . . . . . . . . . . . . . . . . . . . 20

2.3 Star Formation in Nuclear Rings . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Overall Gas Evolution . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Star Formation Rate . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3 Parametric Dependence of SFR . . . . . . . . . . . . . . . . . . . 30

2.3.4 Star Formation Law . . . . . . . . . . . . . . . . . . . . . . . . . 35

v

2.4 Properties of Star Clusters and Gas Clouds . . . . . . . . . . . . . . . . 36

2.4.1 Azimuthal Age Gradient . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.2 Radial Age Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.3 Cluster Mass Functions . . . . . . . . . . . . . . . . . . . . . . . 45

2.4.4 Giant Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Effects of Spiral Arms on Star Formation in Nuclear Rings of Barred-

Spiral Galaxies 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 Overall Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.2 Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


4 Effects of Gas on Formation and Evolution of Stellar Bars and Nuclear

Rings in Disk Galaxies 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Models and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.1 Galaxy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3 Stellar Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.1 Bar Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3.3 Buckling Instability . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4 Gaseous Structures and Star Formation . . . . . . . . . . . . . . . . . . 104

4.4.1 Nuclear Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

vi

4.4.2 Filamentary Spurs . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.3 Ring Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . 109


4.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5 Conclusion 121

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Future Research Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.1 Various Galaxy Models . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.2 Interaction with other galaxies . . . . . . . . . . . . . . . . . . . 125

5.2.3 Effects of Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . 125

Bibliography 127

요 약 141

vii

List of Figures

1.1 NGC 1097 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Snapshots of logarithm of gas density as well as the locations of star

clusters in Model U20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Temporal and radial variations of the azimuthally averaged surface den-

sity in logarithmic scale for Model U20 and Model noSG . . . . . . . . . 25

2.3 Temporal variations of the SFR, the mass inflow rate MNR to the nuclear

ring, and the total gas mass MNR in the nuclear ring of Model U20 . . . 29

2.4 Temporal variations of the SFR for the uniform-disk models and the

exponential-disk models . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Temporal variations of the SFR for models with differing the bar growth

time and models with different momentum injection . . . . . . . . . . . 32

2.6 Dependence of the SFR on the mass inflow rate to the ring and the total

mass in the ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 (a) Dependence of the SFR surface density on the mean surface density

of dense clouds for Model U20, and (b) ΣSFR − Σcl relationship . . . . . 37

2.8 Histograms of the star clusters . . . . . . . . . . . . . . . . . . . . . . . 39

2.9 Positions and ages of young star clusters . . . . . . . . . . . . . . . . . . 40

2.10 Spatial distributions of the formation locations and the present positions

of star clusters in Model U20 . . . . . . . . . . . . . . . . . . . . . . . . 42

2.11 Ages of star clusters as functions of R . . . . . . . . . . . . . . . . . . . 44

2.12 Clusters mass functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

ix

2.13 Distribution of giant clouds in the nuclear ring of Model U20 . . . . . . 48

3.1 Snapshots of gas surface density for Model F00 and Model F10P20 . . . 64

3.2 Temporal changes of the gas surface density of the dust lanes for Models

F10P20 and F00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Temporal variations of the SFR in the ring, total stellar mass formed

until t, and total gas mass in the ring . . . . . . . . . . . . . . . . . . . 69

3.4 Total stellar mass formed until t = 1 Gyr and the averaged SFR . . . . 70

3.5 Spatial distributions of star clusters and the radial and temporal varia-

tions of the azimuthally-averaged surface density . . . . . . . . . . . . . 74

3.6 omparison of the ring SFR and the total stellar mass between Model

F10P20 and Model F10P20d . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 Radial profiles of Toomre Q parameter and rotation curves . . . . . . . 86

4.2 Snapshots of logarithm of the stellar surface density for the cold-disk

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3 Radial distributions of the normalized Fourier amplitudes for cold models 92

4.4 Snapshots of logarithmic stellar surface density for the warm models . . 95

4.5 Radial distributions of the normalized Fourier amplitudes for warm models 96

4.6 Temporal variations of the bar strength (upper panels) and the central

mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.7 Snapshots of logarithm of the stellar surface density at t = 5 Gyr . . . . 99

4.8 Temporal changes of the bar semi-major axis and the pattern speed . . 101

4.9 Contour of logarithmic slice density in x-z plane along the bar semi-major

axis for Model W00. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.10 Temporal variations of the radial and vertical velocity dispersion, and

the velocity dispersion ratio . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.11 Snapshots of logarithmic gaseous surface density for the model W05 . . 105

4.12 Temporal variations of the ring size . . . . . . . . . . . . . . . . . . . . . 108

x

4.13 (a) Gas surface density for the model W07 at t = 4.75 Gyr. (b) The gas

surface density along the green dashed line that denote filamentary spur

region, and (c) blue dotted line that denote dust lane in (a). . . . . . . 108

4.14 Spur structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.15 Temporal variation of the gas fraction . . . . . . . . . . . . . . . . . . . 112

xi

List of Tables

2.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Simulation Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Model Parameters and Simulation Outcomes . . . . . . . . . . . . . . . 60

4.1 Model Parameters and Simulation Outcomes . . . . . . . . . . . . . . . 87

xiii

Chapter 1

Introduction

1.1 Barred Spiral Galaxies

A barred spiral galaxy is a disk galaxy with a central bar-shaped structure and spiral

arms in the outer region. These bar structures are common in disk galaxies. If weak bars

are included, the bar fraction is up to 60% of spiral galaxies in local universe, and even

if limited to strong bars, the bar fraction is about 30% (e.g., de Vaucouleurs et al. 1991;

Reese et al. 2007; Sheth et al. 2008; Masters et al. 2010; Lee et al. 2012a; Ann & Lee

2013). These stellar bars are formed due to dynamical instabilities of self-gravitating

stellar disks (Miller et al. 1970; Hohl 1971). Observations showed that the shapes of

bars and spiral arms vary from galaxy to galaxy, but their correlation with physical

quantities is very weak (Comeron et al. 2010).

Stellar bars play a very important role in the formation of the gaseous structure in

central region, such as dust lanes, a nuclear ring, nuclear spirals, and filamentary spur

structures. Due to the non-axisymmetric bar torque, the gas readily forms dust lane

shocks in the leading side of the bar. After the formation of dust lanes, the gas moves

inward along the dust lanes. The inflowing gas speeds up gradually in the azimuthal

direction as it moves inward, and forms a ring very close to the galaxy center (e.g.,

Athanassoula 1992; Maciejewski et al. 2002; Maciejewski 2004; Regan & Teuben 2003;

Kim et al. 2012a). Figure 1.1 shows the image of NGC 1097 which is one of the famous

1

2 Introduction

Figure 1.1. Almost-true colour composite of NGC 1097 based on three images made

with the multi-mode VIMOS instrument on the 8.2-m Melipal (Unit Telescope 3) of

ESO’s Very Large Telescope.

Introduction 3

barred spiral galaxies. In this galaxy, a well developed bar structure and a spiral arm of

m = 2 mode starting at the bar end region are clearly visible, as well as a pair of dust

lanes from the bar end to the central region and a bright nuclear ring in the center.

Based on theoretical studies, numerical simulations, and recent data from advanced

observational techniques, we now understand a lot about the formation and evolution

of a bar and the detailed shape of the gas structure, but there are still many questions

that need to be answered.

1.2 Gaseous Structures: Nuclear Ring and Filamentary

Spurs

The nuclear ring is a gaseous ring structure with a small radius centered on the galactic

nucleus, commonly found in the bared spiral galaxies. The size of a nuclear rings range

from tens of pc to ∼ 3 kpc (e.g., Knapen 2005; Comeron et al. 2010; Comeron 2013).

Comeron et al. (2010) found that the ring size is related to the bar strength. In their

samples, galaxies with a strong bar allow only small rings, while galaxies with a weak

bar have wide range of a relative ring size (see also Knapen 2005). They also found

that the ring size is limited to the quarter of the bar length.

According to the theory, a nuclear ring is formed near the inner Lindblad resonance

(ILR) (e.g., Shlosman et al. 1990; Combes 1996; Buta & Combes 1996), and observations

also indicate that nuclear rings are typically located near the ILR (e.g., Combes & Gerin

1985; Knapen et al. 1995; Comeron et al. 2010) On the other hand, Regan & Teuben

(2003) argued that the location of the nuclear ring is more related to the existence of

x2-orbits rather than the ILR. Recently, Kim et al. (2012a) found that nuclear ring

forms not by resonant interactions of the gas with the bar potential, but instead by

the centrifugal barrier that the inflowing gas with non-vanishing angular momentum

cannot overcome. In this scenario, gas loses angular momentum when it passes through

the off-axis shocks, moves inward to the central region, and forms a nuclear ring where

the centrifugal force balances the external gravity. They also found that the ring forms

4 Introduction

closer to the center as sound speed cs increases because shock becomes stronger and the

gas crossing the shock loses more angular momentum, although the physical parameters

of the bar are exactly the same. It is not yet clear what exactly determines the ring

size.

Starting with Athanassoula (1992), a number of hydrodynamic simulations using

either smoothed particle hydrodynamics code (e.g., Englmaier & Gerhard 1997; Patsis

& Athanassoula 2000; Ann & Thakur 2005; Thakur et al. 2009; Kim et al. 2011; Shin

et al. 2017) or grid-based code (e.g., Athanassoula 1992; Piner et al. 1995; Maciejewski

et al. 2002; Maciejewski 2004; Regan & Teuben 2003, 2004; Li et al. 2015, 2017) help

to understand the formation and evolution of central gaseous structures. In general,

these studies use external static stellar potential from both analytic equation (most

of hydrodynamic simulations) or snapshot of specific epoch from N-Body simulations

(e.g., Shin et al. 2017; Li et al. 2017). Although these numerical methods have the

disadvantage of not being able to apply the temporal changes of the bar parameters

in realistic galaxies implemented by the N-body calculation(Martinez-Valpuesta et al.

2006; Minchev et al. 2012; Manos & Machado 2014), the potential is easily controlled

in detail and very high resolution is possible. Therefore, the method is very useful to

study specifically how the individual parameters of the bar affect the evolution of the

gas structure. Recently, using hydrodynamic simulations, several studies have found

parameters that directly affect the ring size.

Kim et al. (2012b) found that the radius of x2-type ring decreases systematically

with increasing bar strength, since the amount of angular momentum loss at off-axis

shocks also increases with increasing bar strength (see also Kim et al. 2012a). This is

an evidence that the ring size is not determined by the ILR, but instead by the amount

of angular momentum loss before the gas moves inward. Nonetheless, the x2-type ring

is formed only inside the ILR because the position of the ILR is almost similar to the

outer x2-orbit (Li et al. 2015).

Li et al. (2015) found that the ring size is affected by the bar pattern speed and cen-

tral bulge density. In their result, the ring size increases as the pattern speed decreases.

Introduction 5

Since the relative velocity becomes larger as the pattern speed decreases, off-axis shocks

become stronger in models with a relatively lower pattern speed. Although the shocks

become stronger, the angular momentum loss of the gas that moves along the dust

lanes is reduced because the dust lanes are shifted to a position where the bar torque is

weaker. As a result, the changes in the amount of angular momentum loss determines

the ring size, as well as the changes in the ring size according to the bar strength. The

central bulge density also affects the ring size. Since the centrifual barrier enhances as

the bulge central density increases, models with a massive bulge have a large ring. Li et

al. (2017) found that, if there are no central object, the inflowing gas does not form a

ring-like structure but instead moves directly to the galactic center, because there are

no x2-orbits in this case.

In general, common x2-type nuclear ring is formed in the region where the x2-

orbits exist, and the exact position within the range is determined by the angular

momentum loss during the process of entering the central region. However, in order to

understand whether these dependencies apply equally to time-varying environments, a

high-resolution self-consistent numerical simulations are required.

Some observed galaxies possess filamentary dust spurs that are connected perpen-

dicularly to the dust lanes (see figure 1.1) (e.g., Sheth et al. 2000, 2002; Zurita &

Perez 2008; Elmegreen et al. 2009), but their origin has been unidentified so far. And

some of these galaxies show star formation in the region connecting the dust lanes and

the spur structures. However, previous hydrodynamic simulations with a fixed external

stellar potential were not able to produce these structures (e.g., Kim et al. 2012a,b;

Seo & Kim 2013, 2014). In these simulations, especially with analytic potential, the

bar regions quickly reache a steady state and the gas moves along well-aligned closed

x1-orbits which are almost parallel to the dust lanes. Therefore, spur-like structures

are not formed because the gas cannot move across the bar major axis. However, if the

length, strength, and pattern speed of the bar change continuously with time, spur-like

structures may be formed because there are no closed steady x1-orbits.

6 Introduction

1.3 Star Formation in the Bar Region

Star formation in the bar region occurs in three different sites: bar ends (often called

ansae), gaseous dust lanes, and the nuclear ring. The bar end region has a high gas

density because it is not only the point at which the bar meets the spiral arm but

also the area where the x1-orbit is crowded (Kim et al. 2012a). Some SPH simulations

also show similar result (Englmaier & Gerhard 1997; Patsis & Athanassoula 2000)

Observations also showed star formation at the end of the bar. For instance, the bar

end of NGC 3627 has the star formation rate (SFR) of ∼ 0.23 M yr−1 (Kennicutt &

Evans 2012). In general, since the shear is large in the dust lane region, it is not suitable

for active star formation. However, if spur structures exist, star formation may occur at

the point where the spurs and dust lanes are connected. Zurita & Perez (2008) showed

that some of the star forming sites in NGC 1530 are located in dust lanes. The most

important and strong star forming region in the barred spiral galaxy is the nuclear ring.

Nuclear rings in barred spiral galaxies are one of the most intense star-forming sites

(e.g., Burbidge & Burbidge 1960; Sandage 1961; Phillips 1996; Buta & Combes 1996;

Knapen et al. 2006; Mazzuca et al. 2008; Comeron et al. 2010; Sandstrom et al. 2010;

Mazzuca et al. 2011; Hsieh et al. 2011). As can be seen from the results of previous

hydrodynamic simulations, the gas inside the bar region rapidly moves to the central

region of the galaxy due to the non-axisymmetric bar torque and forms a dense ring-like

structure, creating an environment sufficient for active star formation (see section 1.2).

There are several interesting and important observational results about star for-

mation in the rings. The SFR in a nuclear ring varies widely from galaxy to galaxy.

Mazzuca et al. (2008) found that the SFRs are widely distributed in the range of

0.1− 10 M yr−1. Observed SFRs show a weak correlation with the bar strength. The

SFR tends to be widely distributed when the bars are weak, while it is small in strong

barred galaxies (Mazzuca et al. 2008; Comeron et al. 2010). However, it does not seem

to be significantly influenced by the total gas mass of a nuclear ring. For example, in

the case of a weak barred galaxy NGC 1326 and a strong barred galaxy NGC 4314

with similar-sized rings, the mass of a ring in NGC 1326 is twice as large, but the SFR

Introduction 7

is ten times larger.

There is an interesting difference in the spatial distribution of star forming regions

in a nuclear ring. In some galaxies, the young star clusters in the ring exhibit an age

gradient. Boker et al. (2008) found that three of their five sample galaxies show an

evidence for an azimuthal age gradient of individual star forming region along the

ring, while the other two galaxies have uncertain information to determine such an

age gradient. They proposed two competing scenarios for the star formation in a ring:

namely, the “popcorn” model and “pearls on a string” model. In the “popcorn” scenario,

the gas accumulates in the ring until the entire ring becomes gravitationally unstable,

and star formation occurs in the entire ring region. In this case, star forming regions

exhibit no systematic sequence. On the other hand, in the “pearls on a string” scenario,

star formation occurs only in a specific overdensity region, usually located near the

contact points. In this case, since newly forming star clusters rotate along the ring, an

age gradient appears naturally. Mazzuca et al. (2008) also found that approximately

half of their 22 samples exhibit an azimuthal age gradient that encompass at least a

quarter of the ring, but that the presence of an age gradient is not significantly related

to the shape of the rings or the host galaxies. In particular, three of their samples (NGC

1343, NGC 1530, and NGC 4321) show a clear bipolar age gradient with the youngest

clusters located near the two contact points. (see also Benedict et al. 2002; Allard et

al. 2006; van der Laan et al. 2013)

The temporal changes of the SFR also varies from galaxy to galaxy. Allard et al.

(2006) estimated the star formation history (SFH) of the circumnuclear region of NGC

4321. Comparing the absorption line strength to a variety of simple stellar population

synthesis models, they found that the best-fitted model is short bursts of star formation

which have occurred for the last 500 Myr with a time separation of 100 Myr. Sarzi et

al. (2007) estimated the SFH in a ring of several galaxies including NGC 4321 using

similar method. They found that other galaxies had similar SFH to Allard et al. (2006),

but NGC 4314 stood apart from the other samples. In NGC 4314, there are no evidence

of clusters older than 40 Myr (see also Benedict et al. 2002), which means that there

8 Introduction

was no star formation before the recent active star formation.

A high-resolution numerical simulation is required for the analysis of specific tem-

poral and spatial distributions of star formation. Especially, in the case of azimuthal

age gradient, hydrodynamic simulation using fixed potential is very useful because it

needs to resolve the ring region in detail and the time scale showing the age gradient

is very short, ∆t < 10 Myr, whereas long-term evolution is less important.

1.4 Effects of Gas on Stellar Bars

The presence of gas not only influences the stability of the disk, but also interacts

directly with the stellar disk. In general, since the gaseous disk is colder than the stellar

disk, even if the total mass is the same, the disc becomes relatively unstable when

the gas fraction becomes larger (e.g., Rafikov 2001; Kim & Ostriker 2007; Romeo

& Wiegert 2011). In addition, as the gas moves inward to the central region by the

bar, the central mass concentration (CMC) changes as well as the amount of angular

momentum that the gas loses is transferred to the bar. As is well known, cold disk is

more bar-unstable (Binney & Tremaine 2007), the loss of an angular momentum has

a significant effect on the bar growth (Athanassoula 2002), and the difference in the

mass profile of the central region greatly affects the bar weakening (Shen & Sellwood

2004; Saha & Elmegreen 2018). Therefore, considering the effect of the presence of a

gas is very important in understanding the evolution of galaxies in a more realistic

environment. Self-consistent numerical simulations involving gases have been going on

for over twenty years and have given insight (e.g., Friedli & Benz 1995; Berentzen et al.

1998; Patsis & Athanassoula 2000; Bournaud & Combes 2002; Bournaud et al. 2005;

Debattista et al. 2000; Heller et al. 2007a,b; Villa-Vargas et al. 2010; Athanassoula et al.

2013; Robichaud et al. 2017; Pettitt & Wadsley 2018), but the effects of gas component

on the formation and evolution of the bar vary from study to study.

The role of the gas contribution to the bar formation differs from model to model.

Berentzen et al. (2007) varied the gas fraction from 0 to 8%, but they found that the

gas fraction does not significantly affect the bar formation. In their results, the bar

Introduction 9

growth time scale and peak strength are similar in all models regardless of the gas frac-

tion. However, Athanassoula et al. (2013) found that the disk stays near-axisymmetric

much longer and bar forms much later in gas-rich models. They argued that the dif-

ference in bar formation is the combined effects of the angular momentum redistribu-

tion.Robichaud et al. (2017) showed that bar forms earlier as the gas fraction increases

when AGN feedback exists, while there is no significant difference without AGN feed-

back, but they did not discuss the difference in detail.

The presence of the gas also has an important influence on the weakening and the

secular evolution of the bar. Bournaud et al. (2005) reported that the bars are transient

features with lifetime of 1 − 2 Gyr in model with the gas parameters of normal spiral

galaxies since the bar is fully dissolved because of the combined effects of the CMC

and gravity torques between gas and star particles. Berentzen et al. (2007) found that

although all models exhibit a similar drop in the bar strength after the bar growth,

the mechanisms of weakening are completely different in the gas-poor and the gas-rich

models. They showed that the bar weakening is caused by a stellar heating by the growth

of the CMC in the gas-rich models, while it is caused by the vertical bucking instability

in the gas-poor models. Athanassoula et al. (2013) found that a bar becomes weaker and

smaller as the gas fraction increases at late time, but even in the extreme case where

the initial gas fraction is 100%, a bar does not dissolve completely. Consequently, the

effects of the gas on the bar formation and evolution are not fully understood yet.

1.5 Purpose of this Thesis and Outline

In this thesis, by using both two-dimensional hydrodynamic simulations and three-

dimensional self-consistent simulations, we try to address the unsolved issues of the

formation and evolution of gaseous structures such as the ring size, the spatial and

temporal changes of star formation in the nuclear ring, and the effects of gas on stellar

bars. At first, we extend Kim et al. (2012a) by including self-gravity and simple star

formation and star formation feedback prescriptions. Feedback scheme considered only

type II supernovae and applied delayed explosion considering the average life time of

10 Introduction

OB stars. We focus on spatial distributions of star formation site and temporal changes

of star formation rate in a nuclear ring of strongly-barred galaxies with and without

spiral arm potentials.

Next, we run three dimensional high-resolution self-consistent simulations of isolated

galaxies consisting of a stellar disk, a gaseous disk, and a live halo. We also include

radiative heating and cooling for the gas component and star formation and related

feedback. In this work, feedback prescription considered both type I and II supernovae

and mass return via continuous stellar wind, and calculated the supernova explosion

of individual stars. We try to understand how the gas component affects the formation

and evolution of a stellar bar, and how a nuclear ring forms and evolves under the time-

varying bar potential. And we also try to find an evidence of the formation of filamentary

spur structures that were not seen in hydrodynamic simulations with external static

potentials.

The remainder of the thesis is organized as follows. In Chapter 2, we present the

temporal evolutions of the SFR and the spatial distribution of star-forming site in a

nuclear ring, and their dependence on the model parameters in an environment without

spiral potentials. In chapter 3, we present the temporal changes of the SFR in an

environment with spiral arm potentials, and their dependence on the strength and

pattern speed of the spiral arms. In chapter 4, we present how stellar bars are influenced

by the presence of the gaseous component, and how it changes with the gas fraction.

And, we describe how a nuclear ring evolves and filamentary spur structures form. In

Chapter 5, we provide the overall summary.

Chapter 2

Star Formation in Nuclear Rings

of Barred Galaxies1

2.1 Introduction

Nuclear rings in barred galaxies are sites of intense star formation (e.g., Burbidge &

Burbidge 1960; Sandage 1961; Phillips 1996; Buta & Combes 1996; Knapen et al. 2006;

Mazzuca et al. 2008; Comeron et al. 2010; Sandstrom et al. 2010; Mazzuca et al. 2011;

Hsieh et al. 2011). These rings are thought to form as a result of nonlinear interactions

of gas with a non-axisymmetric bar potential (e.g., Combes & Gerin 1985; Shlosman

et al. 1990; Athanassoula 1992; Heller & Shlosman 1994; Knapen et al. 1995; Buta &

Combes 1996; Combes 2001; Piner et al. 1995; Regan & Teuben 2003). Due to the

bar torque, the gas readily forms dust-lane shocks in the bar region and flows inward

along the dust lanes. The inflowing gas speeds up gradually in the azimuthal direction

as it moves inward, and shapes into a ring very close to the galaxy center (e.g., Kim

et al. 2012a). Consequently, nuclear rings have very large surface densities and short

dynamical time scales, capable of triggering starburst activity.

There are some important observational results that may provide clues as to how

star formation occurs in the nuclear rings. First of all, observations indicate that the

1Seo & Kim 2013, ApJ, 769, 100

11

12 Star Formation in Barred Galaxies

star formation rate (SFR) in the nuclear rings appears to vary with time, and differs

considerably from galaxy to galaxy. Analyses of various population synthesis models

for a sample of galaxies reveal that the strength of observed emission lines from the

nuclear rings is best described by multiple starburst activities over the last 0.5 Gyr

or so, rather than by a constant star formation rate (e.g., Allard et al. 2006; Sarzi et

al. 2007). For 22 nuclear rings, Mazzuca et al. (2008) found that the present SFRs

are widely distributed in the range 0.1–10 M yr−1. It appears that the SFR is largely

insensitive to the total molecular mass in a ring, but can be strongly affected by the bar

strength. For instance, the ring in a strongly-barred galaxy NGC 4314 has a relatively

low SFR at ∼ 0.1 M yr−1 (Benedict et al. 2002), which is about an order of magnitude

smaller than that in a weakly-barred galaxy NGC 1326 (Buta et al. 2000), although the

total molecular mass contained in the ring is within a factor of two. In fact, the SFRs

given in Mazzuca et al. (2008) combined with the bar strength presented by Comeron

et al. (2010) show that strongly-barred galaxies tend to have a very small SFR in the

rings, while weakly-barred galaxies have a wide range of the SFRs.

Second, based on the spatial distributions of star-forming regions in nuclear rings,

Boker et al. (2008) proposed two models of star formation: “popcorn” and “pearls on

a string” models (see also Sandstrom et al. 2010). In the first popcorn model, star

formation occurs in dense clumps that are randomly distributed along a nuclear ring.

This type of star formation, presumably caused by gravitational instability of the ring

itself (Elmegreen 1994), does not produce a systematic gradient in the ages of young

star clusters along the azimuthal direction (see also, e.g., Benedict et al. 2002; Brandle

et al. 2012). In the second pearls-on-a-string model, on the other hand, star formation

takes place preferentially at the contact points between a ring and dust lanes. This may

happen because gas clouds with the largest densities are usually placed at the contact

points due to orbit crowding (e.g., Kenney et al. 1992; Reynaud & Downes 1997; Kohno

et al. 1999; Hsieh et al. 2011). Since star clusters age as they orbit along the ring, this

model naturally predicts a bipolar azimuthal age gradient of star clusters starting from

the contact points (see also, e.g., Ryder et al. 2001; Allard et al. 2006; Mazzuca et al.

Star Formation in Barred Galaxies 13

2008; Boker et al. 2008; Ryder et al. 2010; van der Laan et al. 2013). Mazzuca et al.

(2008) found that ∼ 50% of the nuclear rings in their sample galaxies show azimuthal

age gradients and that such galaxies have, on average, a larger value of the mean SFR

than those without noticeable age gradients.

Another interesting observational result concerns radial locations of star clusters

relative to the nuclear rings. In some galaxies such as NGC 1512 (Maoz et al. 2001)

and NGC 4314 (Benedict et al. 2002), young star clusters are located at larger radii

than the dense gas of nuclear rings. Martini et al. (2003) also reported that out of 123

barred galaxies in their sample, eight galaxies have strong nuclear spirals, all of which

have star-forming regions outside the rings. By analyzing multi-waveband HST archive

data of NGC 1672, Jang & Lee (2013) recently identified hundreds of young and old

star clusters with ages in the range ∼ 1 − 103 Myr. They found that the clusters in

the nuclear regions exhibit a systematic positive radial age gradient, such that older

clusters tend to be located at larger galactocentric radii, farther away from the ring.

Proposed mechanisms for the radial age gradient include the decrease in the ring size

due to angular momentum loss (Regan & Teuben 2003) and migration of clusters due

to tidal interactions with the ring (van de Ven & Chang 2009).

Numerical simulations have been a powerful tool to study formation and evolution of

bar substructures such as dust lanes, nuclear rings, and nuclear spirals (e.g., Sanders &

Huntley 1976; Athanassoula 1992; Piner et al. 1995; Englmaier & Gerhard 1997; Patsis

& Athanassoula 2000; Maciejewski et al. 2002; Regan & Teuben 2003, 2004; Ann &

Thakur 2005; Thakur et al. 2009; Kim et al. 2012a,b). In particular, Athanassoula

(1992) showed that dust lanes are shocks formed at the downstream side from the bar

major axis. Dust lanes tend to be shorter and located closer to the bar major axis

as the gas sound speed increases (Englmaier & Gerhard 1997; Patsis & Athanassoula

2000; Kim et al. 2012a).

Very recently, Kim et al. (2012a) ran various models with differing bar strength and

demonstrated that nuclear rings form not by resonant interactions of the gas with the

bar potential, as was previously thought, but instead by the centrifugal barrier that


the inflowing gas with non-vanishing angular momentum cannot overcome. According

to this idea, a more massive bar forms stronger dust-lane shocks which remove angular

momentum more efficiently from the gas, so that the inflowing gas is able to move

inward closer to the galaxy center, forming a smaller nuclear ring. This turns out entirely

consistent with the observational result of Comeron et al. (2010) that “stronger bars

host smaller rings”. Magnetic stress at the dust lanes takes away angular momentum

additionally, leading to an even smaller ring compared to the unmagnetized counterpart

(Kim & Stone 2012). Kim et al. (2012a) also showed that nuclear spirals that form inside

nuclear rings unwind with time due to the nonlinear effect (Lee & Goodman 1999), with

an unwinding rate higher for a stronger bar. Thus, the probability of having more tightly

wound spirals is larger for galaxies with a weaker bar, consistent with the observational

result of Peeples & Martini (2006).

While the numerical studies mentioned above are useful to understand gas dy-

namics in the central regions of barred galaxies, they are without self-gravity and/or

prescriptions for star formation. There have been only a few numerical studies that

considered star formation in nuclear rings in a self-consistent way. Heller & Shlosman

(1994) studied star formation in galactic disks that are unstable to bar formation. Us-

ing a smoothed particle hydrodynamics (SPH) combined with N -body method, they

found that star formation in barred galaxies occurs episodically, with a time scale of

∼ 10 Myr, and that the associated SFR is well correlated with the mass accretion rate

to the central black hole (BH). These were confirmed by Knapen et al. (1995) who also

found that turbulence driven by star formation tends to widen nuclear rings. Friedli &

Benz (1995) used another SPH+N -body method to run various models with differing

parameters, finding that star formation in the nuclear regions first experiences a burst

phase before entering a quiescent phase (see also Martin & Friedli 1997). Since these

authors employed a small number (∼ 104) of gas particles in their models, however,

they were unable to resolve the nuclear regions well. Kim et al. (2011) ran SPH simula-

tions for star formation specific to the central molecular zone in the Milky Way. While

Dobbs & Pringle (2010) studied cluster age distributions in spiral and barred galaxies,


their results were based on SPH simulations that did not consider star formation and

feedback.

In this chapter, we extend Kim et al. (2012a) by including self-gravity and a pre-

scription for star formation feedback. We focus on temporal and spatial distributions

of star formation occurring in nuclear rings of strongly-barred galaxies. Unlike the pre-

vious SPH simulations with star formation, our models use a grid-based, cylindrical

code with high spatial resolution in the central regions. We also allow for time delays

between star formation and feedback, which is crucial to study age gradients of star

clusters that form in nuclear rings. Our main objectives are to address important ques-

tions such as what controls the SFR in the nuclear rings and what are responsible for

the presence (or absence) of the age gradients of star clusters in the rings, mentioned

above.

We take a simple galaxy model in which a self-gravitating gaseous disk with either

uniform or exponential density distribution is placed under the influence of a non-

axisymmetric bar potential. We implement a stochastic prescription for star formation

that takes allowance for a threshold density as well as a star formation efficiency. Star

formation feedback is treated only through direct momentum injections from super-

nova (SN) explosions occurring 10 Myr after star formation events. By considering an

isothermal equation of state, we do not consider gas cooling and heating, and radiative

feedback, which may be important in regulating star formation in disk galaxies (e.g.,

Ostriker et al. 2010; Ostriker & Shetty 2011; Kim et al. 2011; Shetty & Ostriker 2012).

In our models, the bar potential is turned on slowly over time, which not only represents

a situation where the bar forms and grows but also helps avoid abrupt gas responses.

In each model, we measure the SFR in the ring and study its dependence on various

quantities such as the gas mass in the ring, mass inflow rates to the central regions,

bar growth time, etc. We also explore temporal and spatial variations of star-forming

regions and their connection to the SFR. In addition, we study physical properties of

star clusters and gas clouds in the rings and compare them with observational results

available.


We remark on a few important limitations of our models from the outset. First of all,

our gaseous disks are two-dimensional and razor-thin. This ignores potential dynamical

consequences of vertical gas motions and related mixing, which was shown important

in inducing non-steady gas motions across spiral shocks (e.g., Kim & Ostriker 2006;

Kim et al. 2006, 2010). Second, we adopt an isothermal equation of state for the gas,

corresponding to the warm phase, and do not consider radiative cooling and heating

required for production and transitions of multiphase gas (e.g., Field 1965; Wolfire et

al. 2003; McKee & Ostriker 2007). We also ignore the effects of outflows, winds, and

radiative feedback from young stars, which may be of crucial importance in setting up

the equilibrium pressure in galactic planes, thereby regulating star formation in disk

galaxies (e.g., Ostriker et al. 2010; Ostriker & Shetty 2011; Shetty & Ostriker 2012).

Finally, we in the present work do not consider the effects of spiral arms that may

supply gas to the bar regions. With these caveats, the numerical models presented in

this chapter should be considered as a first step toward more realistic modeling of star

formation in nuclear rings.

This chapter is organized as follows. In Section 2, we describe the numerical methods

and parameters used for our time-dependent simulations. In Section 3, we present the

temporal evolution of SFRs, and their dependence on the model parameters. The age

gradients of star clusters in the azimuthal and radial directions as well as properties of

star clusters and dense clouds are discussed in Section 4. In Section 5, we summarize

our main results and discuss their astronomical implications.

2.2 Model and Method

To study star formation in nuclear rings of barred galaxies, we extend the numerical

models studied in Kim et al. (2012a) by including self-gravity, conversion of gas to

stars, and feedback from star formation. In this section, we briefly summarize the

current models and describe our handling of star formation and feedback. The reader

is referred to Kim et al. (2012a) for more detailed description of the numerical models.


2.2.1 Galaxy Model

We initially consider an infinitesimally-thin, rotating disk. The disk is assumed to be

unmagnetized and isothermal with sound speed of cs = 10 km s−1. The external

gravitational potential responsible for the disk rotation consists of four components:

a stellar disk, a stellar bulge, a non-axisymmetric stellar bar, and a central BH with

mass MBH = 4 × 107 M. This gives rise to a rotation curve that is almost flat at

vc ∼ 200 km s−1 in the bar region and its outside. The presence of the BH makes

the rotation velocity rise as vc ∝ (MBH/r)1/2 toward the galaxy center. The bar po-

tential is modeled by a Ferrers (1887) prolate spheroid with semi-major and minor

axes of 5 kpc and 2 kpc, respectively. The bar is rigidly rotating with a pattern speed

Ωb = 33 km s−1 kpc−1, which places the corotation resonance radius at r = 6 kpc and

the inner Lindblad resonance (ILR) radius at r = 2.2 kpc. The corresponding orbital

time is torb = 2π/Ω = 186 Myr. In our models, the bar potential is turned on over the

bar growth time scale τbar, while the central density of the spheroidal component (bar

plus bulge) is kept fixed. We vary τbar to study situations where the bar grows at a

different rate. The mass of the bar, when it is fully turned on, is set to 30% of the total

mass of the spheroidal component within 10 kpc. All the models are run until 1 Gyr.

As in Kim et al. (2012a), we integrate the basic equations of ideal hydrodynamics

in a frame corotating with the bar. We use the CMHOG code in cylindrical polar

coordinates (r, φ). CMHOG is third-order accurate in space and has very little numerical

diffusion (Piner et al. 1995). To resolve the central region with high accuracy, we set

up a logarithmically-spaced cylindrical grid over r = 0.05 kpc to 8 kpc. The number of

zones in our models is 1024 in the radial direction and 632 in the azimuthal direction

covering the half-plane from φ = −π/2 to π/2. The corresponding spatial resolution is

0.25 pc, 5 pc, and 40 pc at the inner boundary, at r = 1 kpc where most star formation

takes places, and at the outer radial boundary, respectively. We adopt the outflow and

continuous boundary conditions at the inner and outer radial boundaries, respectively,

while taking the periodic boundary conditions at φ = ±π/2.

Our models consider conversion of gas to stars, as will be explained in Section 3.3.2


Table 2.1. Model Parameters

Model Σ0( M pc−2) τbar/torb fmom

(1) (2) (3) (4)

noSG 20 1 0.

U05 5 1 0.75

U10 10 1 0.75

U20 20 1 0.75

U30 30 1 0.75

M25 20 1 0.25

M50 20 1 0.50

FB05 20 0.5 0.75

FB20 20 2 0.75

FB40 20 4 0.75

E30 30 1 0.75

E50 50 1 0.75

E100 100 1 0.75

Note. — Gas surface density in Models with the

prefix “E” initially have an exponential distribu-

tion Σ = Σ0 exp (−r/3.5 kpc). All the other mod-

els have a uniform density distribution Σ = Σ0.


in detail. Since the total gas mass transformed to stars is significant, it is important to

evolve them under the combined gravitational potential of the gas and stars. At each

time step, we calculate the stellar surface density on the grid points via the triangular-

shaped-cloud assignment scheme (Hockney & Eastwood 1988) from the distribution

of stellar particles. We then solve the Poisson equation to obtain the gravitational

potential of the total (gas plus star) surface density, using the Kalnajs (1971) method

presented in Shetty & Ostriker (2008).2 To allow for dilution of gravity due to finite

thickness H of the combined disk, we take H/r = 0.1 as a softening parameter in the

potential calculation.

To explore the dependence of SFR upon the total gas content and the way the

gas is spatially distributed, we initially consider gaseous disks with either uniform

surface density Σ0 or an exponential distribution Σ0 exp(−r/Rd) with the scale length

of Rd = 3.5 kpc. We also vary the fraction fmom of the radial momentum from SNe

imparted to the disk in the in-plane direction relative to what would be the total radial

momentum in a three-dimensional uniform medium (see Section 3.3.2). We run a total

of 13 models that differ in Σ0, τbar, and fmom. Table 4.1 lists the model parameters.

Column (1) lists each model. Models with the prefix “E” have an exponential disk,

while all the others have a uniform disk. Model noSG is a control model that does not

include self-gravity and star formation. Column (2) lists Σ0 of the disk. Column (3)

gives the bar growth time τbar in units of torb, while Column (4) gives fmom. We take

Model U20 with Σ0 = 20 M pc−2, τbar/torb = 1, and fmom = 0.75 as our fiducial

model. All the models initially have a Toomre Q parameter greater than unity, so that

they are gravitationally stable in the absence of a bar potential. However, nuclear rings

that form near the center achieve large density, enough to undergo runaway collapse to

form stars.

2We ignore the gravity from the initial gas distribution in order to make the initial rotation curve

the same with that in the non-self-gravitating counterpart.


2.2.2 Star Formation and Feedback

To model star formation and ensuing feedback, we first identify high-density regions

whose average surface density 〈Σ〉 within a radius RSF exceeds a critical density. The

natural choice for the threshold density would be

Σth =c2s

2GRSF= 1160 M pc−2

( cs10 km s−1

)2(RSF

10 pc

)−1

, (2.1)

from the Jeans condition. While it is desirable to choose a small value for the sizes

of star-forming regions, we take RSF = 10 pc because of numerical resolution: a star-

forming cloud at r ∼ 1 kpc encloses typically ∼ 13 grid points.

Not all clouds with 〈Σ〉 ≥ Σth immediately undergo gravitational collapse and

star formation since we need to consider the star formation efficiency as well as the

computational time step (Kim et al. 2011). The SFR expected from a cloud with mass

Mcloud = πR2SF〈Σ〉, from the Schmidt (1959) law, is

SFR = εffMcloud

tfffor 〈Σ〉 ≥ Σth, (2.2)

where εff is the star formation efficiency per free-fall time, tff , defined by

tff =

(3π

32G〈ρ〉

)1/2

= 3.4 Myr

(〈Σ〉

1160 M pc−2

)−1/2

, (2.3)

assuming a disk scale height of 100 pc. We take εff = 0.01, consistent with theoretical

and observational estimates (e.g., Krumholz & McKee 2005; Krumholz & Tan 2007).

The star formation probability of an eligible cloud with 〈Σ〉 ≥ Σth in a time interval

∆t is then given by p = 1−exp(−εff∆t/tff) ≈ εff∆t/tff (e.g., Hopkins et al. 2011). For a

given computational time step ∆t, the probability p calculated in our models is typically

∼ 10−6−10−5, much smaller than unity. In each time step, we thus generate a uniform

random number N ∈ [0, 1), and turn on star formation only if N < p. When a cloud

undergoes star formation, we create a particle with mass M∗, and convert 90% of the

cloud mass to the particle mass. The initial position and velocity of the particle are set

equal to the density-weighted mean values of the parent cloud within RSF. Each particle

has a mass in the range M∗ ∼ 105 − 107 M, which is about ∼ 1 − 102 times larger


than the masses of observed clusters in nuclear rings (e.g., Maoz et al. 2001; Benedict

et al. 2002; see also Portegies Zwart et al. 2010). Therefore, a massive single particle in

our models can be regarded as representing an unresolved group of star clusters rather

than an individual cluster.

We treat SN feedback using simple momentum input to the surrounding gaseous

medium. We consider only Type II SN events since our models run only until 1 Gyr.

Since we do not resolve individual stars in a cluster or their group, we assume that all

SN explosions occur simultaneously. Stars with mass between 8 M and 40 M explode

as Type II SNe (Heger et al. 2003), which comprise about 7% of the cluster mass under

the Kroupa (2001) initial mass function. The mean mass of SN progenitors is then

∼ 14 M, indicating that the number of SNe exploding from a cluster (or their group)

with mass M∗ is NSN = M∗/(200 M), with the total ejected mass Mejecta = 0.07M∗

returning back to the ISM. Between star formation and SN explosions, we allow a time

delay of 10 Myr, corresponding to the mean life time of Type II SN progenitors (e.g.,

Lejeune & Schaerer 2001). Note that the orbital time of gas in nuclear rings is typically

∼ 25 Myr in our models, so that star clusters move by about 150 in the azimuthal

angle from the formation sites before experiencing SN explosions.

Each feedback, corresponding to NSN simultaneous SNe, injects mass and radial

momentum in the form of an expanding shell. In the momentum-conserving stage, a

single SN with energy 1051 erg would drive radial momentum

Prad,3D = 3× 105ε7/80 (Σ/1 M pc−2)−1/4 M km s−1 (2.4)

to the surrounding gas if the background medium is uniform and in three dimensions,

where ε0 is the SN energy in units of 1051 erg (e.g., Chevalier 1974; Shull 1980; Cioffi

et al. 1988). The dependence of Prad,3D on Σ is due to the fact that the shell expansion

in the Sedov phase is slow when the background surface density is large. Since our

model disks are razor-thin by ignoring the vertical direction, the momentum imparted

to the gas in the simulation domain would be smaller than Prad,3D. Let fmom denote

the fraction of the total radial momentum that goes into the in-plane direction. If the

expansion is isotropic, fmom ∼ 75% (Kim et al. 2011), but fmom can be smaller in


a vertically stratified disk since it is easier for a shell to expand along the vertical

direction. The total momentum of a shell from each feedback is thus set to

Psh = 3× 105fmom N 7/8SN

(Σ

1 M pc−2

)−1/4

M km s−1. (2.5)

In this chapter, we take fmom = 0.75 as a standard value, but run some models with

lower fmom to study the effect of fmom on the SFR (see Table 4.1).

As the initial radius of a shell, we take Rsh = 40 pc, corresponding to the shell

size at the end of the Sedov phase when NSN = 103 and the background density is

Σ = 103 M pc−2, the typical mean density of nuclear rings when star formation is

active. When feedback occurs from a particle, we redistribute the mass and momentum

within a circular region with radius Rsh centered at the particle by taking their spatial

averages. We then add the shell momentum density

Σvsh =

Pmax

(RR2

sh

)R, r ≤ Rsh,

0, r > Rsh,(2.6)

to the gas momentum density in the in-plane direction, and Mejecta/(πR2sh) to the

gas surface density, while reducing the particle mass by Mejecta. In equation (2.6), R

denotes the position vector relative to the particle location and Pmax = 2Psh/(πR2sh) is

the momentum per unit area at r = Rsh. Note that vsh(R) ∝ R2 ensures an initially

divergence-free condition at the feedback center (e.g., Kim et al. 2011).

2.3 Star Formation in Nuclear Rings

In this section, we first describe overall evolution of our numerical simulations. We then

present the temporal variations of SFRs and their dependence on the gas mass, the bar

growth time, and fmom, as well as the relationship between the SFR surface density

and the gas surface density. The properties of star clusters and gas clouds in the rings

are analyzed in the next section.


2.3.1 Overall Gas Evolution

We begin by presenting evolution of our standard model U20 that has a uniform den-

sity Σ0 = 20 M pc−2 initially. Evolution of the other models is qualitatively simi-

lar, although more massive disks show more active star formation. In all models, star

formation occurs mostly in the nuclear rings.3 Figure 2.1 plots snapshots of the gas

distributions as well as the positions of star clusters in Model U20 at a few selected

epochs. The left panels show the gas density in logarithmic scales in the 5 kpc regions,

with the solid ovals indicating the outermost x1-orbit that cuts the x- and y-axes at

xc = 3.6 kpc and yc = 4.7 kpc, respectively. The middle and right panels zoom in the

central 2 kpc regions. In the right panels, small dots indicate star clusters older than

10 Myr, while asterisks correspond to young clusters with age < 10 Myr: the upper

right colorbar represents their ages. In all panels, the bar is oriented vertically along

the y-axis and remains stationary. The gas inside the corotation resonance is rotating

in the counterclockwise direction.

Early evolution of Model U20 before t = 0.1 Gyr is not much different from non-self-

gravitating models presented in Kim et al. (2012a). The imposed non-axisymmetric bar

potential perturbs gas orbits to create overdense ridges at the far downstream side from

the bar major axis. Only the region inside the outermost x1-orbit responds strongly to

the bar potential. Before the bar is fully turned on (i.e., t < τbar), no closed x1- and

x2-orbits exist since the external gravitational potential varies with time. During this

time, the gas streamlines are highly transient as they try to adjust to the time-varying

potential. The overdense ridges grow with time and move slowly toward the bar major

axis, developing into dust-lane shocks. The gas passing through the shocks loses angular

momentum and flows radially inward along the dust lanes. The radial velocity of the

inflowing gas is so large that it is not halted at the ILR (Kim et al. 2012a). It gradually

rotates faster due to the Coriolis force, and eventually forms a nuclear ring after hitting

the dust lane at the opposite side. Produced by supersonic collisions of two gas streams,

3In Model U30, the bar-end regions, often called ansae, form stars as well, although the associated

SFR is less than 3% compared to the ring star formation.


Figure 2.1. Snapshots of logarithm of gas density (color scale) as well as the locations

of star clusters in Model U20 at t = 0.15, 0.2, 0.5, 0.8 Gyr. The left panels show the 5 kpc

regions, while the middle and right panels zoom in the central 2 kpc regions. The upper

left colorbar labels log(Σ/Σ0). The solid ovals in the left panels draw the outermost

x1-orbit that cuts the x- and y−axes at xc = 3.6 kpc and yc = 4.7 kpc. The dotted

curve in (d) is an x1-orbit with xc = 1.7 kpc and yc = 4.4 kpc that traces the inner

ring. Small dots in the right panels denote clusters older than 10 Myr, while asterisks

represent clusters younger than 10 Myr, with the upper right colorbar displaying their

ages.


Figure 2.2. Temporal and radial variations of the azimuthally averaged surface density

in logarithmic scale for (a) Model U20 and (b) Model noSG. The horizontal dashed line

in each panel marks the location of the inner Lindblad resonance. The colorbars label

log(Σ/Σ0). The decreasing rate of the ring size is smaller when self-gravity and star

formation are included.


the contact points between the ring and the dust lanes have the largest density in the

ring, and can thus be preferred sites of star formation.

Over time, the nuclear ring shrinks in size and the contact points rotate in the

azimuthal direction.4 To illustrate this more clearly, Figure 2.2 plots the temporal and

radial variations of the azimuthally averaged surface density in the central regions

of Model U20 and its non-self-gravitating counterpart, Model noSG. The horizontal

dashed line marks the location of the ILR. The ring is beginning to form at t ∼ 0.1 Gyr

at the galactocentric radius of r ∼ 1 kpc, well inside the ILR. At early time when

the bar is growing, the ring is not in an equilibrium position and its shape is quite

different from x2-orbits: the major axis of the ring is inclined to the x-axis by −30

at t = 0.15 Gyr (Fig. 2.1a). As the ring material continuously interacts with the bar

potential, the contact points rotate in the counterclockwise direction. At t ∼ 0.2 Gyr,

the ring settles on one of the x2-orbits, and the contact points are located close the bar

minor axis.

At the same time, the ring becomes smaller in size due to the addition of low-angular

momentum gas from outside as well as by collisions of the ring material whose orbits

are perturbed by thermal pressure (Kim et al. 2012a). When self-gravity is absent,

the decreasing rate of the ring radius is d ln rNR/dt ∼ −1.5 Gyr−1 until t <∼ 0.8 Gyr.

After this time, the ring is so small that strong centrifugal force inhibits further decay

of the ring. In Model U20 with self-gravity included, on the other hand, strong self-

gravitational potential of the ring makes the gas orbits relatively intact, reducing the

ring decay rate to d ln rNR/dt ∼ −0.4 Gyr−1. The decrease of the ring size in turn causes

the contact points to move radially inward and rotate further in the counterclockwise

direction (but not more than ∼ 30).

As Figure 2.1 shows, the bar region (i.e., inside the outermost-x1 orbit) becomes

evacuated quite rapidly. This is because the bar potential is efficient in removing angular

4As pointed out by the referee, Figure 2.1 shows that the ring starts out fairly elongated and titled

relative to the bar and becomes rounder with time. Note that the shape of the ring in Model U20 at

t = 0.15 Gyr is remarkably similar to that of a highly-elongated nuclear ring in ESO 565-11 observed

by Buta et al. (1999), although the latter is ∼ 2–3 times bigger than the former.


momentum from the gas only in the bar region, while its influence on the gas orbits

outside the outermost-x1 orbit is not significant. By t ∼ 0.3 Gyr, most of the gas in the

bar region is transferred to the ring. The amount of the gas added to the bar region

from outside is much smaller than that lost to the ring, which causes the mass inflow

rate to the ring to decrease dramatically with time (see Section 4.4.3).

Figure 2.1d shows that at late time there is a substantial amount of gas trapped in

around an x1-orbit with xc = 1.7 kpc and yc = 4.4 kpc, shown as a dotted line. This

elongated gaseous structure circumscribing the dust lanes and nuclear ring is called

the inner ring (e.g., Buta 1986; Regan et al. 2002): we term the corresponding x1-orbit

the inner-ring x1-orbits. The formation of the inner ring in our model is as follows. As

mentioned before, the dust-lane shocks form first at far downstream and moves toward

the bar major axis as the bar potential grows. During this time, much of the gas in

the bar region infalls to the nuclear region. Near the time when the bar attains its

full strength, the dust lanes find their equilibrium positions on an x1-orbit, still at the

leading side from the bar major axis. The inner ring starts to form at this time, by

gathering the residual material located between the outermost and inner-ring x1-orbits.

Some gases located outside the outermost x1-orbit experience collisions near the bar

ends where x1-orbits crowd, and are then able to lower their orbits to the inner-ring

x1-orbit, increasing the inner-ring mass. In Model U20, the gas added to the inner ring

from outside of the outermost x1-orbit is about 70% of the total inner-ring mass at

t = 1 Gyr. Most of the gas inside the inner-ring x1-orbit had already transited to the

nuclear ring by experiencing the dust-lane shocks before the bar was fully turned on.

2.3.2 Star Formation Rate

Figure 2.3 plots the time evolution of the SFR, the mass inflow rate to the nuclear

ring MNR ≡∫ 2π

0 Σvrrdφ (measured at r = 1.5 kpc), and the total gas mass inside

the nuclear ring MNR in Model U20. Here vr denotes the radial velocity of the gas.

In plotting these profiles, we take a boxcar average, with a window of 20 Myr. In this

model, the bar potential grows over the time scale of τbar = 0.19 Gyr, and the first star


formation takes place at the contact points at t = 0.12 Gyr. As the bar grows further,

the dust-lane shocks become stronger, increasing the amount of the infalling gas to the

ring. MNR attains a peak value ∼ 8 M yr−1 at t = 0.15 Gyr, which coincides with the

time of highest density of the dust lanes (see Fig. 5 of Kim et al. (2012a)). The decay

of MNR after the peak is caused by the fact that only the gas inside the outermost x1-

orbit responds strongly to the bar potential, while the outer region is not much affected.

Similarly, the SFR exhibits a strong burst with a maximum value ∼ 8 M yr−1, which

occurs ∼ 30 Myr after the peak of MNR. The associated SN feedback produces many

holes in the gas distribution, driving a huge amount of kinetic energy to the surrounding

medium. Note that the nuclear ring, albeit somewhat patchy, is overall well maintained

despite energetic momentum injections (Fig. 2.1b).

When the mass inflow rate to the ring is very large, star formation occurring at

the contact points alone is unable to consume the whole inflowing gas. As we will show

in Section 2.4.1, the maximum gas consumption rate afforded to the contact points is

estimated to be about 1 M yr−1 for the parameters we adopt. Any surplus inflowing

gas passes by the contact points and is subsequently added to the ring that is clumpy.

Some overdense regions in the ring are soon able to achieve the mean density larger

than the critical value and undergo star formation, increasing the SFR rapidly. Since the

star-forming regions are randomly distributed in the ring, there is no obvious azimuthal

age gradient of star clusters in this high-SFR phase.

Particles spawned from star formation orbit about the galaxy center under the total

gravity, but they do not feel gas pressure that is quite strong in the nuclear ring. In

addition, the gaseous ring becomes smaller in size with time. Thus, the orbits of star

particles increasingly deviate from the gaseous orbits over time. When stars in clusters

explode as SNe, they are not always located in the ring. A majority of star clusters are

still in the ring, while there are some clusters (∼ 10%) located exterior to the ring. SNe

occurring in the ring have relatively small radial velocities due to a large background

density, and the shell expansion is limited by the surrounding dense gas. On the other

hand, SNe exploding outside the ring can have very large expansion velocities enough


Figure 2.3. Temporal variations of (a) the SFR, (b) the mass inflow rate MNR to

the nuclear ring, and (c) the total gas mass MNR in the nuclear ring of Model U20.

The ordinate is in linear scale in the left panels, while it is in logarithmic scale in the

right panels. In (a) and (b), the horizontal dotted lines indicate the reference rate of

1 M yr−1.


to send the neighboring gas out to the bar-end regions. The expelled gas sweeps up the

gas on its way to the bar-end regions, passes through the dust-lane shocks again, and

falls back radially inward along the dust lane. This increases MNR temporarily during

t = 0.2–0.35 Gyr, with the associated short bursts of star formation at t = 0.27 and

0.32 Gyr in Model U20.

At t = 0.3 Gyr, the bar region is almost emptied, except for the inner ring, as most

of the gas is already lost to the ring. Other than intermittent infalls of the expelled and

swept-up gas from SNe, the mass inflow rate from the inner ring to the nuclear ring and

the related SFR become fairly small. In addition, MNR is reduced to below 2× 108 M

in Model U20 since star formation consumes the gas in the ring, which also decreases

the SFR (Fig. 2.3). As most of the gas in the bar region inside the outermost x1-orbit

is almost lost to star formation, the galaxy evolves into a quasi-steady state where star

formation is limited to small regions near the contact points (Fig. 2.1d).

2.3.3 Parametric Dependence of SFR

Figure 2.4 compares the SFRs from (left) uniform-disk and (right) exponential-disk

models with different Σ0. In all models, the SFR displays a primary burst followed

by a few secondary bursts, with time intervals of ∼ 50 − 80 Myr, before becoming

reduced to below 1 M yr−1. The primary burst is associated with the rapid gas infall

due to angular momentum loss at the dust-lane shocks, while the secondary bursts are

caused by the re-infall of the ejected gas via SN feedback out to the bar region. Models

with larger Σ0 start to form stars earlier and have a larger value of the maximum star

formation rate, SFRmax. The duration of active star formation, ∆tSF, defined by the

time span when SFR ≥ SFRmax/2, is also larger for models with larger Σ0, since the

gas available for star formation is correspondingly larger. Models U20 and E50 initially

have a similar gas mass inside the outermost x1-orbit, but Model E50 has larger SFRmax

since the gas is more centrally concentrated and thus infalls more readily to the nuclear

ring. Columns (2) and (3) of Table 2.2 list SFRmax and ∆tSF for all models. The phase

of active star formation lasts only for ∼ 0.5τbar in all models.


Table 2.2. Simulation Outcomes

Model SFRmax ∆tSF M tot∗ Mx1 MNR β Γ

(M yr−1) (Myr) (108M) (108M) (108M)

(1) (2) (3) (4) (5) (6) (7) (8)

noSG - - - 10.66 8.03 - -

U05 0.53 - 1.4 2.62 0.87 13.7 3.1

U10 3.55 42 3.3 5.33 1.39 14.4 2.6

U20 7.46 83 9.0 10.66 1.42 12.5 2.2

U30 11.8 83 13.3 15.99 2.36 12.6 2.2

M25 7.77 87 8.2 10.66 1.08 9.6 2.3

M50 7.78 77 8.7 10.66 1.55 10.8 2.2

FB05 9.83 68 8.9 10.66 1.62 12.3 2.2

FB20 4.04 163 8.7 10.66 1.73 9.7 2.2

FB40 1.94 392 7.2 10.66 2.75 7.4 3.0

E30 5.10 74 6.1 7.55 1.55 12.4 2.2

E50 10.4 77 10.1 12.59 1.41 6.3 2.2

E100 21.9 84 23.0 25.18 1.78 6.3 2.0

Note. — SFRmax and ∆tSF denote the peak rate and the duration of active star

formation with SFR ≥ SFRmax/2, respectively; M tot∗ is the total mass in stars at

t = 1 Gyr; Mx1 is the total gas mass inside the outermost x1-orbit at t = 0; MNR is

the mass of the nuclear ring at t = 1 Gyr; β = d log(t/yr)/d(r/kpc) is the radial age

gradient of clusters; Γ = −d logN/d logM∗ is the power-law slope of the cluster mass

functions.


Figure 2.4. Temporal variations, over t = 0−0.5 Gyr, of the SFR for (a) the uniform-

disk models and (b) the exponential-disk models with differing Σ0. The horizontal dot-

ted lines indicate SFR = 1 M yr−1. Models with larger gas mass inside the outermost

x1-orbit form stars earlier and at a larger rate.

Figure 2.5. Temporal variations of the SFR for (a) models with differing the bar

growth time τbar and (b) models with different momentum injection fmom. Note that

the range of the abscissa is 1 Gyr in (a) and 0.5 Gyr in (b). The horizontal dotted lines

indicate SFR = 1 M yr−1.


Figure 2.6. Dependence of the SFR on (a) the mass inflow rate MNR to the ring and

(b) the total mass MNR in the ring. The dashed line draws SFR = MNR in (a), and

SFR ∝ M0.3NR in (b). The SFR is almost equal to MNR for the whole range of the SFR,

while it is not well correlated with MNR when SFR ≤ 1 M yr−1.


Figure 2.5 shows how (left) the bar growth time and (right) the amount of the

momentum injection affect the temporal behavior of the SFR. In models where the

bar grows more rapidly, dust-lane shocks form earlier and initiate stronger gas inflows.

This causes star formation in models with smaller τbar to occur at a higher rate and

for a shorter period of time (see Table 2.2). The peak SFR is attained approximately

at t ∼ (0.8 − 1)τbar. This suggests that galaxies in which the bar forms more slowly

are likely to have star formation less active instantaneously but extended for a longer

period of time. Figure 2.5b shows that the SFR computed in our models is largely

insensitive to fmom, although smaller fmom makes the secondary bursts less active.

To directly address what controls star formation in nuclear rings, we plot in Figure

2.6 the dependence of the SFR (left) on the mass inflow rate to the ring and (right)

on the gas mass in the ring for Models U10, U20, and U30. Color indicates the star

formation epoch of each symbol. Even though there are large scatters especially when

the SFR is low, the SFR is almost equal to MNR over two orders of magnitude variations

in MNR. The scatters in the SFR−MNR relation are due to the fact that star formation

is stochastic in our models and that it takes the gas some finite time (∼ 10− 30 Myr)

to travel from r = 1.5 kpc (where MNR is measured) to the nuclear ring. On the other

hand, the SFR does not show a good correlation with MNR. While SFR ∝ M0.3NR for

SFR >∼ 1 M yr−1, it is almost independent of MNR for SFR <∼ 1 M yr−1. Note that

the change in MNR is less than a factor of 5 in Figure 2.6b, while the SFR varies by more

than two orders of magnitude. This suggests that it is the mass inflow rate to the ring,

rather than the ring mass, that determines the SFR in the nuclear ring. Conversely,

the SFR can be a good measure of the mass inflow rate driven by the bar potential.

Column (4) of Table 2.2 gives the total mass in stars M tot∗ formed until the end

of the run for each model. Columns (5) and (6) list the total gas mass Mx1 inside the

outermost x1-orbit in the initial disk and the mass of the nuclear ring MNR at t =

1 Gyr, respectively. Note that Mx1 is approximately the maximum gas mass available

for star formation in the ring. We find that the relation M tot∗ = Mx1 − MNR, with

MNR = 2 × 108 M fixed, explains the numerical results fairly well, indicating that


most of the gas inside the outermost x1-orbit flows inward to form stars, with some

residual gas remaining in the nuclear ring. Compared to Model U20 with τbar/torb = 1,

Model FB40 with τbar/torb = 4 has M tot∗ about 20% smaller, since the gas in the bar

region is still flowing in to the nuclear ring at the end of the run.

As will be discussed in more detail in Section 4.5.2, the overall temporal trend of

the SFR (that is, rapid decline after a primary burst except for a few secondary bursts)

found in our numerical models is largely similar to the numerical results of previous

studies (e.g., Heller & Shlosman 1994; Knapen et al. 1995; Friedli & Benz 1995), but

appears inconsistent with observations of Allard et al. (2006) and Sarzi et al. (2007)

who found that star-forming nuclear rings live long, with multiple episodes of starburst

activities. Since the ring SFR is controlled by the mass inflows rate to the rings, this

implies that rings in real galaxies should be continually supplied with fresh gas from

outside for quite a long period of time. Candidate mechanisms for additional gas inflows,

over a time scale much longer than the bar growth time, include spiral arms and cosmic

gas infalls, which are not included in this chapter.

2.3.4 Star Formation Law

To explore the dependence of the local SFR on the local gas surface density, we define

clouds as regions in the simulation domain whose density is larger than 300 M pc−2.

This density roughly corresponds to the mean density of boundaries of gravitationally

bound clouds in our models (see Section 2.4.4). While this choice of the minimum

density for clouds is somewhat arbitrary, these clouds may represent giant molecular

clouds and their complexes including hydrogen envelopes (e.g., Williams et al. 2000;

McKee & Ostriker 2007).

At each time, we calculate the mean surface density Σcl of, and the total area Acl

occupied by, the clouds distributed along the ring. The mean SFR surface density is

then given by ΣSFR = SFR/Acl. Figure 2.7a plots the resulting ΣSFR as a function of

Σcl for Model U20. Color represents the time when each point is measured, while the

curved arrow indicates the mean evolutionary direction in the ΣSFR–Σcl plane. When


the first star formation takes place (t = 0.12 Gyr), the ring has Σcl ∼ 560 M pc−2 and

ΣSFR ∼ 0.5 M yr−1 kpc−2. The radial gas inflow along the dust lanes increases ΣSFR

rapidly until it achieves a peak value at t = 0.18 Gyr. The corresponding increase of

Σcl is smaller since star formation reduces the gas content in the ring. After the peak,

ΣSFR decreases with decreasing MNR, but it has a larger value, by a factor of ∼ 2−4 on

average, than that at the same Σcl before the peak. This is because active SN feedback

stirs the ring material vigorously, tending to increase density contrast between clumps

and the background material. After the ΣSFR peak, therefore, the total area covered by

the gas with Σ > 300 M pc−2 becomes smaller than before, resulting in larger ΣSFR.

Figure 2.7b plots the ΣSFR–Σcl relationship measured at every 0.1 Gyr for (dia-

monds) the uniform-disk models and (asterisks) the exponential-disk models. Models

with larger Mx1 tend to have larger ΣSFR and larger Σcl. Note that our numerical results

are overall consistent with the Kennicutt-Schmidt law for normal galaxies (squares) and

circumnuclear starburst galaxies (pluses) adopted from Kennicutt (1998), and not much

different from the observed ΣSFR–Σcl relation for spatially-resolved star-forming regions

(circles) in the nuclear ring of NGC 1097 taken from Hsieh et al. (2011).5 There are

large scatters in ΣSFR, amounting to ∼ 1–2 orders of magnitude, both in observational

and simulation results. For star formation in nuclear rings of barred galaxies, the gas

surface density probably sets the mean value of ΣSFR, as the Kennicutt-Schmidt law

implies, while the scatters in ΣSFR are likely due to the temporal variations of the mass

inflow rate to the ring.

2.4 Properties of Star Clusters and Gas Clouds

2.4.1 Azimuthal Age Gradient

As mentioned in Introduction, observations indicate that some galaxies have well-

defined azimuthal age gradients of star clusters in nuclear rings (e.g., Ryder et al.

5In plotting the ΣSFR–Σ relation, Hsieh et al. (2011) used the maximum density of a cloud, instead

of the mean density, for Σ.


Figure 2.7. (a) Dependence of the SFR surface density ΣSFR on the mean surface

density Σcl of dense clouds for Model U20. The colorbar indicates the epoch of star

formation for each symbol, and the curved arrow denotes the mean evolutionary track

in the ΣSFR −Σcl plane. (b) ΣSFR −Σcl relationship from our models compared to the

observed Kennicutt-Schmidt law. Squares and pluses represent normal and circumnu-

clear starburst galaxies adopted from Kennicutt (1998), respectively, while circles are

for spatially-resolved star-forming regions in NGC 1097 from Hsieh et al. (2011).


2001; Allard et al. 2006; Boker et al. 2008; Ryder et al. 2010; van der Laan et al. 2013),

while others do not (e.g., Benedict et al. 2002; Brandle et al. 2012). Our simulations

show that the presence or absence of the azimuthal age gradient is decided by the SFR

in the ring (or, more fundamentally, on MNR), independent of Σ0 and the initial gas

distribution.

Figure 2.8 plots for the uniform-density models the histograms of star clusters

formed in each selected time bin, with bin width of 0.15 Gyr, as a function of the

angular position where they form. The arrow at the bottom of each panel marks the

location of a contact point that is moving in the positive azimuthal direction with time,

as described in Section 3.3.1. Model U05 with Σ0 = 5 M yr−1 has SFR <∼ 0.1 M yr−1

(Fig. 2.4), and star-forming regions in this model are almost always localized to the

contact point. In Models U20 and U30, on the other hand, star-forming regions are

widely distributed along the azimuthal direction at early time (t = 0.15 − 0.45 Gyr)

when SFR > 1 M yr−1, while they are preferentially found near the contact points at

late time (t >∼ 0.45 Gyr) when SFR < 1 M yr−1.

Star clusters age as they orbit along the ring and emit copious UV radiations during

about ∼ 10 Myr after birth. If star-forming regions are localized to the contact points,

therefore, clusters would appear as “pearls on a string” (e.g., Boker et al. 2008), with

an age gradient along the rotational direction of the nuclear ring. This is illustrated in

the left panels of Figure 2.9 which plot the spatial locations of star clusters with color

indicating their ages (< 10 Myr), overlaid on the density distribution in linear scale, in

Model U10 at t = 0.35 Gyr and Model U20 at t = 0.51 Gyr. There is clearly a positive

bi-polar age gradient starting from the contact points that are located at φ ∼ 30

and 210. On the other hand, when the star-forming regions are randomly distributed

throughout the ring, as in the “popcorn” model of Boker et al. (2008), star clusters

with different ages would be mixed. In this case, there is no apparent age gradient along

the ring, as exemplified in the right panels of Figure 2.9 for Model U30 t = 0.33 Gyr

and Model U20 at t = 0.18 Gyr.

Why does then the SFR (or MNR) matter for the azimuthal distributions of star-


Figure 2.8. Histograms of the star clusters that formed in each selected time bin (with

bin width of 0.15 Gyr) for all uniform-density models as a function of the azimuthal

angle where they form. The arrow at the bottom of each panel indicates the mean

position of a contact point.


Figure 2.9. Positions and ages of young star clusters at selected epoches in Models

U05, U20, and U30, overlaid on the gas density distribution in linear scale. The left

panels show a clear azimuthal age gradient, while there is no age gradient in the right

panels. Color indicates the cluster ages.


forming regions? The answer lies at the fact there is a limit on the rate of gas con-

sumption at the contact points that occupy very small areas in the ring. When MNR

is sufficiently small, most of the inflowing gas to the ring can be converted into stars

at the contact points, and the resulting SFR is correspondingly small. When MNR is

very large, on the other hand, the contact points cannot transform all the inflowing

gas to stars instantaneously. The excess inflowing gas overflows the contact points and

is transferred to other regions in the ring. The ring becomes denser not only by the

addition of the overflowing gas but also by its own self-gravity. Some clumps in the ring

achieve surface density above the threshold value, and start to form stars. We find that

the rings have the Toomre stability parameter as low as ∼ 0.5 when the SFR is near

its peak, suggesting that gravitational instability promotes star formation.

Observations show that galaxies with no azimuthal age gradient have, on average,

slightly larger SFRs in the rings (Mazzuca et al. 2008), although the spread in SFR for

individual galaxies is too large to make this conclusive. Note, however, that the critical

SFR determining the presence or absence of the azimuthal age gradient depends on

many parameters. The maximum SFR at the contact points can be estimated as follows.

Let ∆r and ∆φ denote the radial thickness and the azimuthal extent of a contact point,

respectively. Then, the maximum SFR expected from two contact points is simply

M∗,CP = 2εffΣCPrNR∆r∆φ/tff , (2.7)

where ΣCP is the surface density of the contact points. The mean density of star-forming

clouds is ∼ 4000 M pc−2, which we take for ΣCP. For εff = 0.01, ∆r = 50 pc, ∆φ =

30, and rNR = 1 kpc typical in our models, equation (2.7) yields M∗,CP ∼ 1 M yr−1,

consistent with our numerical results. Note that the specific value of M∗,CP depends

on the parameters we adopt. In particular, M∗,CP ∝ c3sr

2NR/R

3/2SF if the ring width is

proportional to the ring radius, suggesting that galaxies with a weak bar (to have a

smaller ring) and strong turbulence would have large M∗,CP.


Figure 2.10. Spatial distributions of (a) the formation locations and (b) the present

positions of star clusters in Model U20 at t = 1 Gyr. The dashed lines draw the

ring at this time. Open circles denote the clusters that have passed the central region

with r < 0.3 kpc during their orbits, while filled circles represent those that have not.

Colorbar indicates the formation epoch of the clusters in unit of Gyr. At late time, stars

form preferentially near the contact points. Gravitational interactions lead to diffusion

of the clusters.


2.4.2 Radial Age Gradient

Although the presence of an azimuthal age gradient depends on the SFR, we find that

star clusters always display a radial age gradient. Figure 2.10 plots the spatial distribu-

tions of (left) the formation locations and (right) the present positions of star clusters

on the x-y plane at t = 1 Gyr in Model U20. Each cluster is colored according to its

formation time. The open circles denote the clusters that have passed the central region

with r < 0.3 kpc at least once during their orbital motions: such clusters would have

been destroyed at least partially by strong tidal fields near the galaxy center if their

internal evolution such as core collapse, evaporation, disruption, etc. had been consid-

ered. On the other hand, the filled circles are for clusters that have never approached

the central region, and thus are most likely to survive the galactic tide. The dashed

lines draw the ring at t = 1 Gyr. Star-forming regions at late time are concentrated

on the contact points, while they are well distributed at early time. Note that clusters

that form early before the nuclear ring settles on an x2-orbit have initial kick velocities

quite different from those on x1- or x2-orbits at their formation locations. Although

the ring soon takes on the x2-orbit, these clusters move on eccentric orbits and wander

around the nuclear region. Figure 2.10b shows that young clusters are preferentially

found near the ring, while old clusters are located away from it, indicative of a positive

radial gradient of their ages.

To show this more clearly, Figure 2.11 plots the age of clusters as functions of their

present radial positions (circles) at t = 1 Gyr as well as their formation locations (plus

symbols) for Models U10 and U20. Again, the open circles denote the clusters that

have passed by the galaxy center, while the filled circles are for those that have not.

Note that the age distributions of the present-day and formation-epoch locations of

the clusters are not much different from each other, although the former shows a large

spatial dispersion. The dispersion is larger for older clusters. To quantify the radial age

gradient, we bin the clusters according to their ages, with a bin size of ∆ log(t/yr) = 0.2,

and calculate the mean age and position in each bin. Our best fits of the ages to the

formation-epoch positions are β ≡ d log(t/yr)/d(r/kpc) ∼ 14.4 and 12.5 for Models


Figure 2.11. Ages of star clusters as functions of their current radial locations (circles)

at t = 1 Gyr and their formation positions (pluses) for Models U10 and U20. Open

circles are those that have passed by the galaxy center at a very close distance during

orbital motions, while the filled circles are for those that have not. The dashed lines are

the fits, with slopes of β = d log(t/yr)/d(r/kpc) = 14.4 and 12.5, for the initial cluster

positions, for Models U10 and U20, respectively.


U10 and U20, respectively. Column (7) of Table 2.2 gives β for all models. This radial

age gradient results primarily from the decrease in the ring size with time, such that

old clusters formed at larger galactocentric radii. Clusters diffuse out radially through

gravitational interactions themselves and also with dense clouds in the ring, without

much effect on the radial age gradient.

2.4.3 Cluster Mass Functions

Figure 2.12 plots the mass functions of all the clusters that have formed in each of

the (left) uniform-disk and (right) exponential-disk models until t = 1 Gyr. The upper

panels are for the clusters formed while star formation is very active with SFR ≥

1 M yr−1, whereas those produced when SFR < 1 M yr−1 are presented in the

lower panels. In general, the mass distribution of clusters is described roughly by a

power law, with its index depending on the SFR and Mx1 . When the SFR is larger

than 1 M yr−1, the slope of the mass function is Γ ≡ −d logN/d logM∗ ∼ 2–3, with

a smaller value corresponding to larger Mx1 . When SFR ≤ 1 M yr−1, clusters have a

much steeper mass distribution with Γ >∼ 3. Column (8) of Table 2.2 gives Γ.

The dependence of the power-law index on the SFR and Mx1 can be understood as

follows. When the SFR is large, there are numerous dense regions distributed through-

out the ring. Such regions grow by accreting the surrounding material. Since star for-

mation occurs in a stochastic manner in our models, some dense clouds have a chance

to grow as massive as, or even larger than ∼ 107 M, leading to a relatively shallow

mass function. On the other hand, when the SFR is small, there are not many dense

regions. Since the growth of density is quite slow in these models, they form stars at

densities slightly above the critical value. In this case, most clusters have mass around

∼ 105 − 106 M, with a steep mass distribution.

2.4.4 Giant Clouds

Finally, we present the properties of giant clouds located in nuclear rings. High-resolution

radio observations show that nuclear rings consist of giant molecular associations at


Figure 2.12. Clusters mass functions for (left) the uniform-density and (right) the

exponential-density models. The upper and lower panels plot the clusters formed when

the SFR is larger or smaller than 1 M yr−1, respectively. The mass function becomes

shallower with increasing SFR.


scale of ∼ 0.2–0.3 kpc in which most star formation takes place. They typically have

masses of ∼ 107 M and are gravitationally bound (e.g., Hsieh et al. 2011).

To identify giant clouds in our models, we utilize a core-finding technique developed

by Gong & Ostriker (2011). This method makes use of the gravitational potential of the

gas, and thus allows smoother cloud boundaries than the methods based on isodensity

surfaces (see, e.g., Smith et al. 2009). At a given time, we search for all the local minima

of the gravitational potential and find the largest closed potential contour encompassing

one and only one potential minimum. We then define the potential minimum and

outermost contour as the center and boundary of a cloud, respectively. If the distance

between two neighboring minima is less than 0.1 kpc, we combine them. Figure 2.13

plots, for example, giant clouds identified by this technique for Model U20 at t =

0.36 Gyr. The left panel shows the gas surface density in linear scale, while the right

panel displays boundaries of giant clouds as contours overlaid over the gravitational

potential of the gas. A total of 14 giant clouds are identified. The mean values of their

masses M , radii R, and one-dimensional velocity dispersions σ are 107 M, 100 pc,

and 20 km s−1, respectively, corresponding to supersonic internal motions. The average

density of the cloud boundaries is found to be ∼ 300 M pc−2. The average value of

the virial parameter is α = 5σ2R/(GM) ∼ 2, so that they are gravitationally bound,

consistent with the observed cloud properties in nuclear rings of barred galaxies (e.g.,

McKee & Ostriker 2007).

2.5 Summary and Discussion

2.5.1 Summary

We have presented the results of two-dimensional hydrodynamic simulations of star

formation occurring in nuclear rings of barred galaxies. We initially consider an in-

finitesimally thin, isothermal gas disk placed under the external gravitational potential.

The external potential consists of a stellar disk, a stellar bulge, a central BH, and a

non-axisymmetric stellar bar. We do not study the effect of spiral arms in the present


Figure 2.13. Distribution of giant clouds in the nuclear ring of Model U20 at t =

0.36 Gyr. Left: the gas surface density is shown in linear scale. Right: cloud boundaries

found by the method described in the text are overlaid on the gravitational potential

of the gas.


work. The bar potential is modeled by a Ferrers prolate spheroid with the semi-major

and minor axes of 5 kpc and 2 kpc, respectively, and rotates about the galaxy center

with a patten speed of 33 km s−1 kpc−1. The bar mass, when it is fully turned on,

is set to 30% of the total stellar mass in the spheroidal component, corresponding to

a strongly barred galaxy. We fix the gas sound speed to cs = 10 km s−1 and the BH

mass to 4 × 107 M. Our simulations incorporate star formation recipes that include

a density threshold corresponding to the Jeans condition, a star formation efficiency,

conversion of gas to particles representing star clusters or their groups, and delayed

momentum feedback via SN explosions. To explore various situations, we consider both

uniform and exponential density models, and vary the gas surface density, bar growth

time, and the total momentum injection in the in-plane direction. The main results of

this work can be summarized as follows.

The imposed bar potential readily induces a pair of dust-lane shocks in the bar

region inside the outermost x1-orbit. At early time when the bar potential is weak, the

dust-lane shocks are placed at the far downstream side from the bar major axis. As the

bar potential increases, the dust-lane shocks become stronger and slowly move toward

the bar major axis. The gas passing through the shocks loses a significant amount of

angular momentum, infalls radially along the dust lanes, and forms a nuclear ring.

The continuous gas inflows provide a fuel for star formation in the ring. After the bar

potential reaches its full strength, the dust lanes settle on an x1-orbit, while the nuclear

ring follows an x2-orbit. The remaining gas located inside the outermost x1-orbit and

outside the dust lanes is gathered to form an elongated inner ring in the bar region,

whose shape is well described by an x1-orbit, as well. Some of the gas located outside

the outermost x1-orbit transits to the inner ring near the bar ends where x1-orbits

crowd. Similarly, the gas in the inner ring loses angular momentum when it collides

with other gas near the bar ends, slowly infalling to the nuclear ring through the dust

lanes.

The contact points between the dust lanes and the nuclear ring is a tunnel through

which the inflowing gas on x1-orbits switches to the x2-orbit of the nuclear ring. About


the time when the bar potential is fully turned on, the contact points are located near

the bar minor axis. Over time, the nuclear ring shrinks in size due to the addition of

low angular momentum gas from outside and by collisions of the ring material, which

in turn makes the contact points rotate slowly in the counterclockwise direction. Since

the contact points have largest density in the nuclear ring, they are preferred sites of

star formation, although star-forming regions can be distributed throughout the ring

when the mass inflow rate is high.

The bar potential transports the gas in the bar region to the nuclear ring very

efficiently, but does not have strong influence on the gas orbits outside the bar region.

This not only makes the bar region evacuated rapidly but also reduces the mass inflow

rate MNR dramatically after ∼ 0.3 Gyr. The SFR in nuclear rings displays a single

primary burst followed by a few secondary bursts before becoming reduced to small

values. The primary burst is associated with the massive gas inflow along the dust

lanes caused by the growth of the bar potential. The duration and maximum rate of

the primary burst depend on the growth time of the bar potential, in such a way that

a slower bar growth results in a more prolonged and reduced SFR. The secondary

bursts are due to the re-entry of the ejected and swept-up gas by SN feedback from the

nuclear ring out to the bar-end regions. Time intervals between the secondary bursts

are roughly ∼ 50 − 80 Myr. The peak SFR is attained at t ∼ (0.8 − 1)τbar, and the

duration of active star formation is roughly a half of the bar growth time. The SFR

is almost equal to MNR. It has a weak dependence on the total gas mass MNR in the

ring when SFR >∼ 1 M yr−1, and is not correlated with MNR when SFR <∼ 1 M yr−1.

This suggests that star formation in the ring is controlled primarily by MNR rather

than MNR. The relationship between the SFR surface density and the surface density

of dense clumps in nuclear rings found from our numerical models are consistent with

the usual Kennicutt-Schmidt law for circumnuclear starburst galaxies.

The presence or absence of azimuthal age gradients of young star clusters in nuclear

rings depends on the SFR (or MNR) in our models. When MNR is small, most of the

inflowing gas to the nuclear ring is consumed at the contact points. In this case, young


star clusters that form would exhibit a well-defined azimuthal age gradient along the

ring. When MNR is large, on the other hand, the contact points are unable to transform

all of the inflowing gas to stars. The extra gas overflows the contact points and goes

into the nuclear ring. The ring becomes massive and forms stars in clumps that become

dense enough. In this case, no apparent age gradient of star clusters is expected since

star-forming regions are randomly distributed over the whole length of the ring. The

critical value of MNR that determines the presence or absence of the azimuthal age

gradient is estimated to be ∼ 1 M yr−1 in our models, although it depends on various

parameters such as the ring radius, critical density, etc. (eq. [2.7]).

Star clusters produced also exhibit a positive radial age gradient, such that young

clusters are located close to the nuclear ring, while old clusters are found away from the

ring. The primary reason for this is that the nuclear ring becomes smaller in size with

time, and thus star-forming regions gradually move radially inward. In our models, the

radial age gradient amounts to β = d log(t/yr)/d(r/kpc) ∼ 6 − 15. Radial diffusion of

clusters via mutual gravitational interactions and also with the gaseous ring does not

affect the radial age gradient much.

When the SFR is large (> 1 M yr−1), some dense clouds are able to grow by

accreting surrounding material and form massive clusters. The cluster mass function

is well described by a power law, with slope Γ = −d logN/d logM∗ ∼ 2 − 3. A larger

slope corresponds to a more massive disk in which more gas is available for the ring

star formation. When the SFR is small, on the other hand, most clusters form near the

threshold density, leading to a steeper slope with Γ >∼ 3. Giant clouds in nuclear rings

have typical masses 107 M and sizes 0.1 kpc. Driven by momentum injection from

SNe, their one-dimensional internal velocity dispersions are supersonic at ∼ 20 km s−1.

They are gravitationally bound with the virial parameter of α ∼ 2.

2.5.2 Discussion

We find that the SFR in nuclear rings shows a strong primary burst, with its duration

and peak value dependent on the bar growth time, and subsequently a few weak and


narrow bursts, after which the SFR becomes very small. The peak of the primary burst

is attained roughly when the bar potential is fully turned on. This burst behavior of

the SFR appears to be a generic feature of star formation in nuclear rings of strongly-

barred galaxies found in numerical simulations. For instance, N -body+ SPH models

presented by Heller & Shlosman (1994), Knapen et al. (1995), and Friedli & Benz (1995)

showed that the SFR reaches its peak value, with narrow bursts superimposed, about

the time when the stellar bar fully develops, after which it is reduced to small values.

In numerical modeling for star formation in the nuclear region of the Milky Way, Kim

et al. (2011) found that the SFR is maximized at t ∼ 0.15 Gyr, with a peak value

∼ 0.22 M yr−1, and then drops to a relatively constant value ∼ 0.05− 0.07 M yr−1.

The sustained star formation in Kim et al. (2011) is thought to arise because the Milky

Way has a very weak bar that takes a long time to clear out gas in the bar region. In

this case, the gas infall may proceed continuously over an extended period of time, a

situation similar to the case with a slowly-growing bar.

There is observational evidence that star formation in nuclear rings occurs contin-

ually over a long period of time (a few Gyrs) with successive ∼ 4− 10 bursts separated

by a few tenths of Gyrs each (e.g., Allard et al. 2006; Sarzi et al. 2007; see also van der

Laan et al. 2013). This is in sharp contrast to our numerical results that show that star

formation in nuclear rings is dominated by one primary burst before declining to small

values, with ∼ 0.1 Gyr duration of active star formation. This discrepancy is mostly

likely due to the fact that our models consider only a bar potential for angular mo-

mentum transport and thus are too simple to describe more complicated mass inflows

in real disk galaxies. If the mass inflow rate to a nuclear ring really controls the SFR

in the ring, as found by our numerical models, the observational results call for a need

to feed the rings with gas episodically for a long time interval. The bar potential alone

is unlikely responsible for gas supply needed for star formation in real nuclear rings.

Unless bars are dynamically young, present star formation in nuclear rings of nearby

barred galaxies requires additional gas feeding. One obvious such mechanism is spiral

arms that can remove angular momentum at spiral shocks to transport gas from outer


disks to the bar regions (e.g., Lubow et al. 1986; Kim & Stone 2012), which is not

considered in the present work. Accretion of halo gas to the disk may not only rejuve-

nate bars (e.g., Bournaud & Combes 2002) but also enhance the SFR in the rings (e.g.,

Jiang & Binney 1999; Fraternali & Binney 2006, 2008). Such gas flows might actually

exist, as evidenced by the presence of an enhanced number of carbon stars in the outer

spiral arms of M33 (Block et al. 2007). Temporal variations in the bar strength (e.g.,

Bournaud & Combes 2002) and in the bar pattern speed (e..g, Combes & Sanders 1981)

are also likely to affect the mass inflow rate to the ring and thus the SFR.

Some galaxies such as IC 4933 (Ryder et al. 2010) show age gradients of star clusters

along the azimuthal direction in nuclear rings, while there are other galaxies such as

NGC 7552 (Brandle et al. 2012) that do not show a clear age gradient. Mazzuca et al.

(2008) analyzed Hα data for 22 nuclear rings and found that about half of their sample

galaxies contain azimuthal age gradients, although most of them are not throughout the

entire ring. They also found that the mean SFR in galaxies with azimuthal age gradients

is 2.2± 0.7 M yr−1, which is slightly larger than the mean value of 3.6± 1.1 M yr−1

for galaxies with no apparent age gradient. While this appears consistent with our

numerical results, the large dispersions in the mean SFRs suggest that there is no

fixed SFR that can distinguish between galaxies with and without age gradients. In

addition, the critical SFR for the absence or presence of azimuthal age gradients is

about 1 M yr−1 in our models, while most galaxies in the sample of Mazzuca et al.

(2008) have SFR > 1 M yr−1. As equation (2.7) suggests for the maximum SFR,

M∗,CP, allowed at the contact points, however, the critical SFR depends on many

factors that may vary from galaxy to galaxy. For example, NGC 1343 with the most

clear bi-polar age gradient in the Mazzuca et al. (2008) sample has the current SFR of

∼ 6.8 M yr−1. Its ring radius is ∼ 2 kpc (Comeron et al. 2010), which increases the

critical SFR by a factor of 4, assuming that the ring width is proportional to the ring

size and the other parameters remain the same. In addition, M∗,CP ∝ Σ3/2CP ∝ c3

s/R3/2SF ,

so that the level of interstellar turbulence and the size of star-forming regions RSF may

change M∗,CP considerably.


We find that star clusters that form in nuclear rings naturally develop a positive

radial gradient of their ages owing primarily to the decrease in the ring size over ∼ Gyr

in our models. This is consistent with the results of Jang & Lee (2013) who found that

clusters with ages <∼ 1 Gyr in the nuclear region of NGC 1672 are older systematically

with increasing radius. Note that the radial age gradient holds over a timescale of

∼ Gyr and may not apply to clusters in a small age range since the decay of the

ring size is quite slow. Indeed, Mazzuca et al. (2008) found that two (NGC 5953 and

7570) of their sample galaxies show a negative radial age gradient of H II regions in

the rings.6 Our numerical results plotted in Figure 2.11 also show that when limited to

clusters with age ∼ 107–107.5 yr at t = 1 Gyr, younger clusters can be found at larger

radii, which is due to the stochastic nature of star formation and ensuing gravitational

interactions.

Our results show that the SFR in the nuclear rings is tightly correlated with the

mass inflow rate to the ring rather than the total gas mass in the ring (Fig. 2.6). This

result is seemingly consistent with the results of Benedict et al. (2002) who found that

the SFR in the nuclear ring of a strongly-barred galaxy NGC 4314 is smaller, by a

factor of 30, than that in a weakly-barred galaxy NGC 1326 (Buta et al. 2000), even

if the gas mass in the ring is smaller by only a factor of two. It is interesting to note

that the gas mass contained in most “gas-rich” nuclear rings of barred galaxies in the

BIMA SONG sample is in a remarkably narrow range of ∼ (1 − 6) × 108 M (Sheth

et al. 2005).7 The SFR data presented in Mazzuca et al. (2008) combined with the

bar strength given in Comeron et al. (2010) show that strongly-barred galaxies usually

have very small present-day SFRs and the SFRs in weakly-barred galaxies vary in a

wide range, although the number of galaxies in their sample is too limited to make a

conclusive statement. It will be interesting to see how the bar strength as well as gas

inflows by spiral shocks influence the SFR in nuclear rings.

While star formation is concentrated in nuclear rings in our models, observations

6Note that NGC 5953 is a non-barred galaxy.7While Sheth et al. (2005) reported that the ring in NGC 6946 has a mass of ∼ 109 M, a higher-

resolution observation of Schinnerer et al. (2006) gives the ring mass of ∼ 4 × 108 M.


indicate that star formation in some galaxies occurs not only in nuclear rings but also

in the bar region including dust lanes (e.g., Martin & Friedli 1997; Sheth et al. 2000;

Zurita & Perez 2008; Elmegreen et al. 2009; Martınez-Garcıa & Gonzalez-Lopezlira

2011). While dust lanes themselves are known hostile to star formation due to strong

velocity shear (e.g., Athanassoula 1992; Kim et al. 2012a), Sheth et al. (2000) proposed

that stars form in interbar dust spurs in filamentary shape that impact the dust lanes

from the trailing side of the bar (see also Sheth et al. 2002; Zurita & Perez 2008).

Indeed, Elmegreen et al. (2009) inferred that some clusters in the nuclear ring of NGC

1365 actually formed in one of the dust lanes by the impact of spurs and subsequently

migrated inward to the nuclear ring. The origin of these interbar spurs is yet unclear.

Apparently, there is no filamentary interbar feature in our models. They may originate

from gas inflows due to spiral shocks from the region outside the bar (Elmegreen et

al. 2009), from interactions of gas with magnetic fields that are pervasive in the bar

region (Beck et al. 1999, 2005), and/or from other dynamical processes that involve gas

cooling, self-gravity, etc., which are not considered in the present work. It will be an

important direction of future work to study how spiral arms and magnetic fields affect

the gas inflows and star formation in the bar and nuclear regions.

Chapter 3

Effects of Spiral Arms on Star

Formation in Nuclear Rings of

Barred-Spiral Galaxies1

3.1 Introduction

Barred-spiral galaxies often host star-forming nuclear rings at their centers (e.g., Phillips

1996; Buta & Combes 1996; Knapen et al. 2006; Mazzuca et al. 2008; Comeron et al.

2009; Sandstrom et al. 2010; Mazzuca et al. 2011; Hsieh et al. 2011; van der Laan et

al. 2011). These rings are most likely produced by the radial infall of gas at large radii

caused by angular momentum loss due to its nonlinear interactions with an underlying

stellar bar potential (e.g, Combes & Gerin 1985; Shlosman et al. 1990; Athanassoula

1992; Heller & Shlosman 1994; Knapen et al. 1995; Buta & Combes 1996; Kim et al.

2012a; Kim & Stone 2012). They appear smaller in more strongly barred galaxies (e.g.,

Comeron et al. 2010) and unrelated to resonances with the bars (e.g., Pinol-Ferrer et

al. 2014). This suggests that the ring location is determined primarily by the amount of

angular momentum loss rather than the resonances, as confirmed by numerical simula-

tions (e.g., Kim et al. 2012a,b). Large gas surface density and small dynamical timescale

1Seo & Kim 2014, ApJ, 792, 47

57

58 Star Formation in Barred-spiral Galaxies

of nuclear rings make them one of the most intense star forming regions in disk galaxies.

Observations indicate that the star formation rate (SFR) in a nuclear ring varies

widely from galaxy to galaxy (e.g., Mazzuca et al. 2008; Comeron et al. 2010), although

the total gas content in a ring is similar at ∼ (1− 6)× 108 M (e.g., Buta et al. 2000;

Benedict et al. 2002; Sheth et al. 2005; Schinnerer et al. 2006). Data presented in

Mazzuca et al. (2008) and Comeron et al. (2010) suggest that strongly barred galaxies

tend to have a small SFR, while weakly barred galaxies have a wide range of SFR at

∼ 0.1 − 10 M yr−1. Ring star formation appears to be long lived, occurring either

continuously (van der Laan et al. 2013) or episodically (Allard et al. 2006; Sarzi et al.

2007) for ∼ 1 − 3 Gyr. In some galaxies, young star clusters exhibit an azimuthal age

gradient such that they tend to be older farther away from the contact points between

a ring and dust lanes, while there is no noticeable age gradient in many other galaxies

(e.g, Boker et al. 2008; Mazzuca et al. 2008; Ryder et al. 2010; Brandle et al. 2012).

Yet, what determines the ring SFR as well as the presence or absence of the azimuthal

age gradient along a ring is not clearly understood.

In a recent attempt to understand these observational results, In Chapter 2, we

ran hydrodynamic simulations for star formation occurring in nuclear rings of barred

galaxies without spiral arms, and found that the ring SFR is controlled mainly by the

mass inflow rate to the ring rather than the total gas mass in the ring. In these bar-only

models, the massive gas inflows caused by the bar growth result in a strong burst phase

of SFR that lasts only for ∼ 0.2 Gyr, after which both the mass inflow rate and SFR

are decreased to very small values below ∼ 0.1 M yr−1. The main reason for this

short burst of star formation is that only the gas inside the bar region (more precisely,

inside the outermost x1 orbit) can respond to the bar potential to initiate the rapid gas

infall, while the gas outside the bar is not much affected by the bar potential (e.g, Kim

et al. 2012a; Kim & Stone 2012). In the chapter 2, we also found that an azimuthal age

gradient of young star clusters is expected when the SFR is less than a critical value

affordable at the contact points. These bar-only models may explain ring star formation

in galaxies with low SFR, but would require that the bars should be dynamically young

Star Formation in Barred-spiral Galaxies 59

in galaxies with high SFR, which is quite unlikely since the lifetime of bars is quite long

(several Gyrs) in observations (e.g., Gadotti & de Souza 2005; Perez et al. 2009) and

N -body simulations (e.g., Shen & Sellwood 2004; Bournaud et al. 2005; Berentzen et

al. 2007; Athanassoula et al. 2013).

If the mass inflow rate to the ring is really a critical factor in determining the ring

SFR, long-lived star formation requires that fresh gas should be supplied to the nuclear

rings continually or continuously. There are several additional gas-feeding mechanisms

at work in real galaxies that may change the temporal evolution of the SFR consid-

erably. These include galactic fountains (e.g, Fraternali & Binney 2006, 2008), cosmic

accretion of primordial gas (e.g, Dekel et al. 2009; Richter 2012), and angular momen-

tum dissipation by spiral arms (e.g., Roberts & Shu 1972; Lubow et al. 1986; Hopkins

et al. 2011; Kim & Kim 2014). For instance, Fraternali & Binney (2008) estimated

the gas infalling rates of ∼ 2.9 and ∼ 0.8 M yr−1 for NGC 891 and NGC 2403, re-

spectively, most of which was ejected to the halos via supernova (SN) feedback, while

Richter (2012) found high velocity clouds feed a normal galactic disk with gas at a rate

of ∼ 0.7 M yr−1. Since the gas supply in the form of fountains and cosmic accretion

occurs over a whole disk plane, the accreted gas should make its way to the galaxy

center anyway to help promote star formation in the rings. Very recently, Kim & Kim

(2014) showed that stellar spiral arms can play such a role, transporting the gas inward

at a rate ∼ 0.05 − 3.0 M yr−1 depending on the arm strength and pattern speed,

which can potentially enhance the ring SFR.

In this chapter, we investigate star formation in a nuclear ring of a disk galaxy that

possesses both spiral arms and a bar. This is a straightforward extension of the Chapter

2 that considered only the bar potential. By varying the arm strength and pattern speed

while fixing the bar parameters, we quantity the effect of the spiral arms on the ring

SFR and gaseous structures that form. In Section 3.2, we describe our galaxy models

and numerical method. In Section 3.3, we present the results on overall evolution of

our galaxy models, star formation occurring in nuclear rings, and age gradients of star

clusters. We summarize and discuss our results in Section 4.5.1.


Table 3.1. Model Parameters and Simulation Outcomes

Model F Ωarm ∆τSN 〈Σdl〉〈MNR〉 M∗ 〈SFR〉 rNR

(%) (km s−1 kpc−1) (Myr) (M pc−2) (109 M) (109 M) (M yr−1) (kpc)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

F00 0 - 10 21.8±9.34 0.25±0.01 0.92 0.19±0.21 0.67

F05P10 5 10 10 41.3±34.6 0.21±0.06 1.53 0.98±0.76 0.90

F05P20 5 20 10 34.8±31.2 0.23±0.03 1.27 0.62±0.35 0.89

F05P33 5 33 10 23.1±13.3 0.27±0.01 1.02 0.31±0.20 0.72

F10P10 10 10 10 66.7±55.5 0.26±0.08 2.07 1.32±1.09 1.00

F10P20 10 20 10 75.4±79.2 0.27±0.05 1.70 1.32±1.07 0.95

F10P33 10 33 10 24.0±20.5 0.28±0.01 1.01 0.31±0.17 0.73

F15P10 15 10 10 101.7±133.8 0.24±0.07 2.84 2.44±1.93 0.98

F15P20 15 20 10 94.2±90.1 0.25±0.07 2.31 2.11±1.41 1.02

F15P33 15 33 10 24.4±19.1 0.26±0.01 1.16 0.34±0.40 0.70

F20P10 20 10 10 123.9±314.1 0.23±0.08 3.55 3.54±1.69 1.08

F20P20 20 20 10 114.3±220.3 0.27±0.07 3.04 2.95±1.84 0.94

F20P33 20 33 10 30.1±27.7 0.23±0.03 1.42 0.75±0.42 0.81

F10P20d 10 20 5 68.3±80.4 0.25±0.06 1.76 1.45±1.21 0.95

Note. — F denotes the dimensionless arm strength; Ωarm is the arm pattern speed; ∆τSN is the time

interval between the cluster formation and SN explosion; 〈Σdl〉 is the mean surface density of the dust-lane

segments at R = 2.0−2.5 kpc averaged over t = 0.4−1.0 Gyr; 〈MNR〉 is the gas mass in the ring averaged

over t = 0.4 − 1.0 Gyr; M∗ is the total stellar mass formed until t = 1.0 Gyr; 〈SFR〉 is the ring SFR

averaged over t = 0.4− 1.0 Gyr; rNR is the ring radius at t = 1.0 Gyr.


3.2 Model and Method

We consider disk galaxies with both spiral arms and a bar. Our galaxy models are

identical to those in the Chapter 2 except that we initially consider an exponential

gaseous disk rather than a uniform disk and that we additionally include stellar spiral

perturbations. The reader is referred to the Chapter 2 for the detailed description of the

simulation setups and numerical methods. Here we briefly describe our current models

and methods.

The gaseous disk is infinitesimally-thin, self-gravitating, unmagnetized, and rotating

about the galaxy center. The initial profile of gas surface density is taken to

Σ0 = 29.4 exp(−R/9.7 kpc) M pc−2, (3.1)

which describes nearby disk galaxies reasonably well (Bigiel & Blitz 2012). We adopt

an isothermal equation of state with sound speed of cs = 10 km s−1.

The axisymmetric part of the external gravitational potential gives rise to a ro-

tational velocity profile that resembles normal disk galaxies with velocity of vc '

200 km s−1 at the flat part. The non-axisymmetric part consists of two components:

a bar and spiral arms. As in the Chapter 2, the bar potential is modeled by a Fer-

rers prolate spheroid whose parameters are fixed to the central density concentration

index n = 1, the semi-major and minor axes 5 kpc and 2 kpc, respectively, the mass

1.5×1010 M, and the pattern speed Ωbar = 33 km s−1 kpc−1. For the spiral potential,

we take a two-armed trailing logarithmic model of Shetty & Ostriker (2006):

Φs(R,φ; t) = Φs0 cos

(m

[φ+

lnR

tan p∗− Ωarmt+ φ0

]), (3.2)

for R ≥ 6 kpc and Φs = 0 at R < 5 kpc, with Φs between 5 kpc and 6 kpc tapered by a

Gaussian function. Here, m, p∗, Ωarm, and φ0 denote the number, the pitch angle, the

pattern speed, and the initial phase of the arms, respectively. The amplitude Φs0 of the

arm potential is controlled by the dimensionless arm-strength parameter F defined by

F ≡ mΦs0

v2c tan p∗

, (3.3)


which measures the radial force due to the spiral arms relative to the centrifugal force

from the background galaxy rotation (e.g., Kim & Kim 2014). In this work, we fix

m = 2, p∗ = 20, φ0 = 147, and vary F and Ωarm.

Our calculations incorporate a prescription for star formation and ensuing feedback

via SNe. We determine star-forming regions based on the critical density corresponding

to the Jeans condition, and allow for a star formation efficiency of 1% (e.g., Krumholz

& McKee 2005; Krumholz & Tan 2007). When a cloud is determined to undergo star

formation, we spawn a sink particle corresponding to a star cluster, and convert 90% of

the gas mass to the particle. The mass of each particle is typically in the range of∼ 105−

107 M. Each particle interacts gravitationally with each other while orbiting under

the influence of total gravity, and injects radial momentum to the surrounding gaseous

medium, mimicking multiple simultaneous SN explosions from a cluster. We consider a

time delay, ∆τSN, between star formation and SN feedback. Under the Kroupa (2001)

initial mass function, the mass-weighted mean main-sequence lifetime of stars with

M ≥ 8 M that explode as SNe is estimated to be ∆τSN = 10 Myr, but we also run

a case with ∆τSN = 5 Myr to study its effect on the ring SFR. In our models, the

amount of the radial momentum per single SN in the in-plane direction is taken to be

2.25 × 105 M km s−1. This corresponds to the snow-plow phase of a shell expansion

due to an injection of SN energy 1051 erg in the in-plane direction (e.g., Chevalier 1974;

Cioffi et al. 1988; Thornton et al. 1998; Kim et al. 2013; Kimm & Cen 2014).

Although it is challenging to measure the pattern speeds of bars and spiral arms,

observations indicate that they are either corotating or the arms are rotating more

slowly than the bar (e.g., Fathi et al. 2009; Martınez-Garcıa & Gonzalez-Lopezlira

2011). To explore various situations, we run a total of 14 models with differing F

between 0 and 20%, Ωarm between 10 and 33 km s−1 kpc−1, and ∆τSN between 5

and 10 Myr. Columns (1)–(4) of Table 4.1 list the name and the parameters of each

model. Columns (5)–(9) give some of the simulation outcomes, which will be explained

later. Model F00 is a bar-only model, while the other models possess spiral arms as

well. Model F10P20d is a control model with ∆τSN = 5 Myr. Note that the models


with Ωarm = 33 km s−1 kpc−1 have the arms and bar corotating, with the co-rotation

resonance (CR) radius at RCR,bar = RCR,arm = 6 kpc, while the models with Ωarm = 10

and 20 km s−1 kpc−1 have the RCR,arm = 20 and 10 kpc, respectively.

As in the Chapter 2, we integrate the basic ideal hydrodynamic equations using the

CMHOG code in the frame corotating with the bar (Piner et al. 1995). To resolve the

ring regions with high accuracy, we set up a logarithmically-spaced grid that extends

from R = 0.05 to 30 kpc. The number of zones in our models is 1290×632 in the radial

and azimuthal directions covering the half-plane with φ = −π/2 to π/2, leading to the

grid size of 5 pc at R = 1 kpc where a ring preferentially forms. We adopt the outflow

and periodic boundary conditions at the radial and azimuthal boundaries, respectively.

In order to avoid strong transients in the gas flow caused by a sudden introduction of

the bar, the bar is slowly introduced over one bar revolution time of 0.19 Gyr.

3.3 Simulation Results

We in this section first describe the overall evolution of our fiducial model F10P20 with

F = 10% and Ωarm = 20 km s−1 kpc−1 in comparison with its no-arm counterpart,

Model F00. Evolution of other models with arms is quantitatively similar to that of

Model F10P20. We then present the results on star formation histories and distributions

of star clusters that form in nuclear rings.

3.3.1 Overall Evolution

Figure 3.1 shows snapshots of the gaseous surface density in logarithmic scale at t =

0.15, 0.4, and 0.7 Gyr of Models F00 with no arms (left) and Model F10P20 with arms

(middle). The right panels zoom in the central 2 kpc regions of Model F10P20 to display

the positions of star-forming regions younger than 20 Myr (colored asterisks) as well as

all star clusters that have formed (small dots). The bar is pointing toward the y-axis

and remains stationary in the simulation domain. The solid oval in the lower-left panel

draws the outermost x1 orbit under the given potential, which cuts the x- and y-axes

at 3.6 kpc and 4.7 kpc, respectively. The total gas mass enclosed by the outermost x1


Figure 3.1. Snapshots of gas surface density at t = 0.15 Gyr (top), 0.40 Gyr (middle),

and 0.70 Gyr (bottom) for Model F00 (left column) and Model F10P20 (middle col-

umn). The right panels expand the central 2 kpc regions of Model F10P20 to show the

distributions of young clusters with age less than 20 Myr (asterisks) and older clusters

(dots) that formed. The CR of the bar is at RCR,bar = 6 kpc inside which the gas

is rotating in the counterclockwise direction. A pair of the red arrows in each of top

panels indicate high-density ridges, termed dust lanes, located at the downstream side

from the bar major axis. The oval in the lower-left panel draws the outermost x1 orbit,

outside of which the gas distribution is not much perturbed in Model F00. The left and

right colorbars label log Σ and the age of young clusters, respectively.


orbit is 1.2× 109 M in the initial disk. Note that the inner ends of the gaseous spiral

arms are connected to the bar ends for most of the time, even though the arms and the

bar have different pattern speeds.

An introduction of the bar and spiral potentials provides strong perturbations for

gas orbits that would otherwise remain circular. The gas in the bar region is readily

shocked to form “dust lanes” referring to high-density ridges, indicated as arrows in

the upper panels of Figure 3.1, located at the leading side of the bar major axis (e.g.,

Athanassoula 1992). Gas passes through the dust-lane shocks almost perpendicularly

and loses angular momentum, moving radially inward to form a nuclear ring at the

location where centrifugal force balances the gravity (Kim et al. 2012a,b). At the same

time, the gas in the arm region develops spiral shocks whose pitch angle is smaller

than that of the stellar arms. The offset between the pitch angles of gaseous and stellar

arms is larger for smaller F and larger Ωarm (Kim & Kim 2014). In general, spiral

shocks in the arm region are much weaker than the dust-lane shocks in the bar region

owing largely to a smaller angle between the gas streamlines and the shock fronts in the

former, so that gas infall due to the spiral shocks occurs much more slowly than that

associated with the bar. At about 0.11 Gyr, stars start to form in the ring where plenty

of gas is accumulated by the bar potential to meet the Jeans condition for gravitational

collapse.

At early time (t = 0.15 Gyr), the effect of spiral arms on the bar region is almost

negligible since they are still weak and growing. When the arms become strong enough

to induce spiral shocks, the gas originally located outside the bar region but inside

the CR of the arms (i.e., RCR,bar < R < RCR,arm) starts to move radially inward

by losing angular momentum due to the spiral potential and associated spiral shocks,

while the gas outside RCR,arm drifts outward. The inflowing gas due to the arms moves

on along x1 orbits after entering the bar region. The gas is piled up at the bar ends

where x1 orbits crowd. Mutual collisions of gas orbits there further take away angular

momentum from the gas, intermittently sending gas blobs along the dust lanes to the

nuclear ring. When this happens, the dust lanes become inhomogeneous, as illustrated


in the t = 0.4 Gyr snapshot of Model F10P20 in Figure 3.1. This not only enhances the

gas surface density of the dust lanes but also fuels episodic star formation in the ring

at late time (see below).

Dust lanes are located at the downstream side of galaxy rotation from the bar

major axis, and are more straight in more strongly barred galaxies (Knapen et al. 2002;

Comeron et al. 2009; Kim et al. 2012b). In our models, they typically have a width of

∼ 50 pc and extend from the bar ends to the nuclear ring located at R ∼ 1 kpc. Figure

3.2 plots temporal variations of the mean gas surface density, Σdl, of the dust-lane

segments at R = 2.0 − 2.5 kpc in Models F00 (dashed) and F10P20 (solid). In Model

F00, Σdl is large only when the gas in the bar region experiences the massive infall to

the nuclear ring (t ∼ 0.1 − 0.2 Gyr) and when the feedback from the initial starburst

activities sends the ring material out to the dust lanes (t ∼ 0.2− 0.3 Gyr), after which

Σdl decays to very small values. The early behavior of Σdl in Model F10P20 is similar

to that in Model F00, but the gas inflows induced by spiral arms make the dust lanes in

the former much more pronounced at t >∼ 0.4 Gyr than in the no-arm counterpart. This

trend can also be seen in Figure 3.1 where the dust lanes in Model F00 are strong at

t = 0.15 Gyr but can be barely identified at t >∼ 0.4 Gyr, while they are vividly apparent

at any time in Model F10P20. Column (5) of Table 4.1 gives the mean surface density

〈Σdl〉 as well as standard deviations of the dust lanes for all models, where the angle

brackets 〈〉 represent a time average over t = 0.4− 1.0 Gyr.

In the t = 0.7 Gyr snapshot of Model F00, there is a well-defined, elongated gaseous

feature, called a gaseous inner ring, that lies inside the outermost x1 orbit and encom-

passes the dust lanes (e.g., Buta 1986, 2013; Regan et al. 2002). The inner ring has

roughly the same size as the bar (e.g., Buta & Combes 1996). As explained in the Chap-

ter 2, it begins to form after dust lanes find their equilibrium positions by collecting the

residual gas that did not experience the dust-lane shocks. In Model F10P20, however,

the inner ring is strongly perturbed by the arm-induced mass inflows at late time. It

is also influenced by feedback from star formation in the nuclear ring. As Figure 3.1

shows, the spiral shocks at t = 0.4 Gyr are abundant with dense clumps produced by


Figure 3.2. Temporal changes of the gas surface density of the dust lanes Σdl averaged

over R = 2.0− 2.5 kpc for Models F10P20 and F00. In Model F00, the dust lanes are

strong only when the gas in the bar region infalls (t ∼ 0.1−0.2 Gyr) and when the ring

gas is expelled to the bar region by star formation feedback (t ∼ 0.2 − 0.3 Gyr). The

arm-induced mass inflows make the dust lane strong at late time in Model F10P20.


a wiggle instability of the shock fronts, which occurs as a consequence of the poten-

tial vorticity accumulation in the gas flows moving across curved shock fronts multiple

times (Kim et al. 2014a). These clumps collide and merge with each other as they move

along the arms, and become loose at the interface between the arm and bar regions.

When they enter the bar region, they thus have significant inward radial velocities,

providing strong perturbations to the gas already in the inner ring. They eventually

settle on x1 orbits, lose angular momentum by hitting the bar ends, and move further

in to the nuclear ring.

3.3.2 Star Formation

3.3.2.1 Enhanced SFR by Arms

Figure 3.3 plots the temporal evolution of the SFR in the ring, the total stellar mass

M∗(t) formed until time t, and the total gas mass MNR in the ring for models with

F = 10% but differing Ωarm.2 The results of Model F00 with F = 0 are compared as

dashed lines. The overall behavior of the SFR in the bar-only model is characterized by

a strong primary burst and a few subsequent secondary bursts before declining to small

values at t >∼ 0.3 Gyr, with the duration of the burst phase corresponding to the bar

growth time (see Chapter 2). The presence of spiral arms especially when the pattern

speed is small can make the SFR rejuvenated at t >∼ 0.4 Gyr. As we mentioned above,

the mass infalls from the bar ends to the ring occur intermittently, resulting in episodic

star formation in the ring at late time. Since the typical inflow velocity due to the arms

is ∼ 1 km s−1 (Kim & Kim 2014), the arm-induced SFR in the ring can persist longer

than the Hubble time, as long as RCR,arm is located sufficiently far away from the bar

ends. Note that ring SFR is not enhanced much in Model F10P33, since RCR,arm is

located just outside the bar ends.

Figure 3.3(c) shows that in all models MNR is maintained relatively constant at

2We calculate the ring mass as MNR ≡∫ 1.5 kpc

0.5 kpc

∫ΣRdφdR, since the ring density in our models is

about two orders of magnitude larger than the density of the surrounding medium near the galaxy

center.


Figure 3.3. Temporal variations of the (a) SFR in the ring, (b) total stellar mass

M∗(t) formed until t, and (c) total gas mass MNR in the ring for models with F = 10%

and varying Ωarm. The results of the bar-only model are compared as dashed lines.

Arm-induced gas inflows make the SFR rejuvenated at t >∼ 0.4 Gyr. The ring SFR is

not well correlated with MNR.


Figure 3.4. (a) Total stellar mass M∗ formed until t = 1 Gyr and (b) the mean values

(symbols) and standard deviations (errorbars) of the SFR averaged over t = 0.4−1 Gyr

as functions of the arm strength F and pattern speed Ωarm. The horizontal dashed line

in each panel marks the value in the bar-only model. In (b), the data are displaced

slightly in the horizontal direction for clarity. The dotted lines are our best fits (Eq.

(3.4)).


∼ (2 − 4) × 108 M throughout evolution, similar to observations (e.g., Sheth et al.

2005). Column (6) of Table 4.1 gives 〈MNR〉 for all models. Since the SFR depends

on Ωarm considerably, this suggests that it is not the gas mass in the ring but the

mass inflow rate to the ring that controls the ring SFR, even when the effect of spiral

arms is included. With the typical ring radius of 1 kpc and thickness of 50 pc, this

translates into the averaged gas surface density Σring ∼ 650− 1250 M pc−2, which is

approximately equal to the critical density Σc = 103 M pc−2 for Jeans collapse. When

Σring > Σc, star formation takes place in the ring, reducing the gas density. When

Σring < Σc, star formation is halted in the ring which has to wait until fresh gas is filled

in to resume star formation. For the bar-only models, in the Chapter 2, we found that

the ring shrinks in size steadily over time due to the addition of gas with low angular

momentum. Enhanced SFR by arms turns out to reduce the shrinking rate of the ring

size (see Section 3.3.2).

Figure 3.4 plots M∗ at t = 1 Gyr and the mean values 〈SFR〉 (symbols) along

with the standard deviations (errorbars) of the SFR averaged over t = 0.4 − 1.0 Gyr

for all models. These values are also tabulated in Columns (7) and (8) of Table 4.1.

The horizontal dashed line in each panel represents the case with no arm. Note that

M∗ = 0.9×109 M in Model F00, which is about 75% of the initial gas mass inside the

outermost x1 orbit. While M∗/(1 Gyr) corresponds to the mean SFR throughout entire

evolution, 〈SFR〉 measures the mean value of the arm-induced SFR after the initial gas

infall due to the bar potential is almost finished. Clearly, both M∗ and 〈SFR〉 are larger

for models with larger F and/or smaller Ωarm. For Ωarm = 10 and 20 km s−1 kpc−1,

for example, the time-averaged ring SFR over t = 0.4 − 1.0 Gyr is enhanced by a

factor of 5.2, 6.9, 12.8, 18.6, and 3.3, 6.9, 11.1, 15.5 for models with F = 5, 10, 15, 20%,

respectively, compared to the no-arm counterpart. The dotted lines are our best fits to

〈SFR〉:〈SFR〉

M yr−1= 0.19 + 25F1.2 log

(11.5− 10

Ωarm

Ωbar

). (3.4)

The increasing trend of the ring SFR with F is similar to that of the gas inflow rates

driven by the arms reported by Kim & Kim (2014). This makes sense since a smaller


pattern speed implies a larger RCR,arm and since stronger spiral shocks can remove

a larger amount of angular momentum from the gas. When the arms are corotating

with the bar, on the other hand, the presence of spiral arms does not affect the ring

SFR much. This shows that the enhancement of the ring SFR due to spiral arms is

significant only when they have different pattern speeds.

3.3.2.2 Age Gradients

The result of the Chapter 2 showed that young star clusters exhibit a noticeable age

gradient in the azimuthal direction along a nuclear ring only when the SFR is larger than

the critical value M∗,CP, which is set by the maximum SFR affordable at the contact

points. In the high-SFR phase (SFR > M∗,CP), star-forming regions are distributed

randomly throughout the ring, hence no age gradient of clusters is expected. In the low-

SFR phase (SFR < M∗,CP), on the other hand, they are localized to the contact points,

leading to a well-defined azimuthal age gradient. These two modes of star formation are

referred to respectively as “popcorn” and “pearls-on-a-string” models by Boker et al.

(2008). For our adopted parameters, in the Chapter 2, we found M∗,CP ∼ 1 M yr−1,

although it depends sensitively on the ring size and the gas sound speed. The result of

the Chapter 2 also showed that star clusters with age < 1 Gyr show a positive radial

age gradient such that older clusters are located at larger R owing to a secular decrease

of the ring size.

We find that the condition for the presence or absence of the azimuthal age gradient

of young clusters is not affected by arm-enhanced star formation. By analyzing the

dependence of the cluster ages younger than 10 Myr on the azimuthal positions in all

models with spirals, we find that star formation occurs in the pearls-on-a-string fashion

for ∼ 80% of the low-SFR phase and for ∼ 20% of the high-SFR phase. In the high-

SFR phase, the mass inflow rate along the dust lanes is too large for star formation

to consume all the inflowing gas at the contact points: overflowing gas produces star-

forming clumps distributed randomly along the nuclear ring. In the low SFR phase,

however, most of the inflowing gas undergoes star formation at the contact points.


Since clusters age as they move along the ring, this naturally leads to an azimuthal age

gradient.

However, the arm-enhanced star formation tends to remove the radial age gradient

of star clusters. The upper panels of Figure 3.5 display the spatial distributions of star

clusters at t = 1 Gyr in Model F00 (left) and Model F20P20 (right), with the color

representing their age. The lower panels plot the corresponding temporal changes of

the azimuthally-averaged gas surface density Σ(R) ≡∫

Σdφ/(2π) in linear scale. In

the no-arm model, younger clusters are located preferentially at smaller R due to a

secular decrease in the ring size. This results in a radial age gradient amounting to

d log(t/yr)/d(R/kpc) ∼ 15 in Model F00. In Model F20P20, on the other hand, gas

driven in by the spiral arms has larger angular momentum than in the ring gas, so that

the ring does not decrease in size after ∼ 0.4 Gyr. In addition, active star formation

feedback disperses the ring gas widely in the radial direction, making the ring larger

than in the no-arm model. Column (9) of Table 4.1 gives the average ring radius rNR

at t = 1 Gyr, where rNR ≡∫RΣdR/

∫ΣdR, with the radial integration taken over

R = 0.5 − 1.5 kpc, showing that the radii of the rings in spiral-arm models with

Ωarm<∼ 20 km s−1 kpc−1 are larger by about 45% than in Model F00. Consequently,

star clusters in models with arm-enhanced star formation exhibit no apparent radial

age gradient, as the upper-right panel of Figure 3.5 illustrates.

3.3.2.3 Effects of ∆τSN

To explore the effects of the time delay between star formation and SN explosion,

we run Model F10P20d with ∆τSN = 5 Myr, while the other parameters are taken

identical to those in Model F10P20. Figure 3.6 compares the temporal changes of the

SFR and M∗(t) from these two models. Due to a shorter delay, feedback occurs earlier in

Model F10P20d, which tends to reduce SFR and M∗ at early time compared to those

in Model F10P20. Although detailed star formation histories are different, the time

averaged SFRs over t = 0.4 − 1.0 Gyr are 1.32 and 1.45 M yr−1 for Models F10P20

and F10P20d, respectively, which agree within 10%. The total stellar mass formed at


Figure 3.5. (Upper) Spatial distributions of star clusters at t = 1 Gyr, with color

denoting their age and (lower) the radial and temporal variations of the azimuthally-

averaged surface density Σ. The left and right panels are for Model F00 and F20P20,

respectively. In the upper panels, the dashed oval fits the ring at t = 1 Gyr in each

model.


Figure 3.6. Comparison of the ring SFR and the total stellar mass M∗(t) between

Model F10P20 with ∆τSN = 10 Myr and Model F10P20d with ∆τSN = 5 Myr. While

the detailed histories of the SFR are different, the time-averaged SFR and M∗ are

within 10%.


the end of the runs are also similar at M∗ = 1.70 × 109 M and 1.76 × 109 M for

Models F10P20 and F10P20d, respectively. This demonstrates that the results on the

arm-enhanced SFR presented in this work is insensitive to the choice of ∆τSN as long

as it is within a reasonable range.


We have presented the results of two-dimensional grid-based hydrodynamic simulations

to study star formation in nuclear rings of barred-spiral galaxies. The gaseous medium is

taken to be isothermal, self-gravitating, unmagnetized, and limited to an infinitesimally-

thin disk. We incorporate a prescription for star formation and delayed feedback via

SNe in the form of momentum injection. We handle the bar and spiral patterns using

rigidly-rotating, fixed gravitational potentials, and do not consider the back reaction

of the gas to the underlying patterns. To study various situations, we vary only the

strength and pattern speed of the arms, while fixing other arm and bar parameters.

The main results and corresponding discussions are as follows.

1. – Arm-enhanced SFR: Spiral arms located in the outer disks can drive gas toward

the bar region, enhancing star formation in nuclear rings considerably, only if the arm

pattern speed is smaller than that of the bar. This is because only the gas located

between the bar ends and the CR of the arms can lose angular momentum by passing

through spiral arms and move inward to the bar region, while the gas outside the CR

of the arms move radially outward. The gas entering the bar region is first gathered

to the bar ends where it loses its angular momentum additionally to move further in

along the dust-lane shocks to the nuclear ring. The inflow to the ring occurs in an

intermittent fashion, making the ring star formation episodic that lasts long until the

end of the simulations. Ignoring the star formation induced at early time by the bar,

the enhanced SFR by spiral arms is ∼ (0.6−3.5) M yr−1, which is about 3 to 20 times

larger than that in the no-arm counterpart. This has an important implication that the

SFR histories in nuclear rings can be significantly affected by spiral arms that are slow

and strong. Some galaxies are observed to have undergone several episodic bursts of


star formation in the rings over a few Gyrs (Allard et al. 2006; Sarzi et al. 2007; van der

Laan et al. 2013), which appear difficult to be explained by gas inflows driven solely by

the bar potential (see Chapter 2). We suggest that these sustained starburst activities

might have resulted from additional gas feeding due to spiral arms.

2. – Dust Lane Strength: The mass inflows driven by arms also help make the

dust lanes stronger. By analyzing Sloan Digital Sky Survey DR7 data, Comeron et al.

(2009) found that about 20% of 266 galaxies with measured bar strength host dust

lanes with appreciable strength. On the other hand, numerical simulations with only

a bar potential show that dust lanes remain strong only for ∼ 0.2 Gyr around the

time when the bar potential achieves its full strength (e.g, Chapter 2; Kim & Stone

2012). Given that bars persist for several Gyrs (e.g., Gadotti & de Souza 2005; Perez

et al. 2009; Berentzen et al. 2007; Athanassoula et al. 2013), this is not compatible

with the results of Comeron et al. (2009). Barred galaxies with strong dust lanes may

be either dynamically young or supplied with fresh gas. Our numerical results in this

chapter suggest that spiral arms with large RCR,arm can be efficient to transport the

gas from outside to the bar region. The typical density of dust lanes in our standard

model F10P20 is about 70 M pc−2. This corresponds to the extinction magnitude of

AV ∼ 5 in the visual band assuming a standard value of ∼ 3 for the ratio of total to

selective extinction, readily visible against the background stellar light.

3. – Age Gradients of Star Clusters: The gas driven in from the arms to the ring is

found to have larger angular momentum than the gas already in the ring. In addition,

feedback from active star formation at late time tends to reduce the rate of angular

momentum removal, making the rings in models with spiral arms larger by ∼ 45% than

those in bar-only models. Consequently, star clusters formed in bar-only models retain

an age gradient in the radial direction, while they do not in models with slow-rotating

spiral arms. On the other hand, the arm-enhanced star formation exhibits an azimuthal

age gradient of star clusters such that younger clusters are located closer to the contact

points when SFR is small, while clusters with different ages are well mixed when SFR

is large. The critical SFR that determines the absence/presence of the azimuthal age


gradient is set by the maximum gas consumption rate M∗,CP at the contact points,

which is ∼ 1 M yr−1 in our current models, although it may depends sensitively on

the gas sound speed and the ring size as M∗,CP ∝ c3sr

2NR (Chapter 2; see also Kim et al.

2014b). This appears consistent with the observational data of Mazzuca et al. (2008)

in that nuclear rings with an azimuthal age gradient have, on average, a smaller SFR

than those without age gradient.

Many uncertainties surround the observational determinations of the SFR in nu-

clear rings and the arm pattern speed. The derived SFRs from different methods do

not always agree. For the nuclear ring of NGC 6951, for instance, Mazzuca et al. (2008)

derived the current SFR ∼ 1.4 M yr−1 based on the old Hα–SFR relation of Ken-

nicutt (1998), which is known to yield a larger SFR by a factor of ∼ 1.47 than the

newly calibrated relation (e.g., Hao et al. 2011; Kennicutt & Evans 2012). Also, local

variations in the stellar-age mix, initial mass function, gas/dust geometry, etc. are likely

to contaminate the derived SFRs to some extent (Kennicutt & Evans 2012). On the

other hand, van der Laan et al. (2013) measured the ages and masses of stellar clus-

ters directly to obtain a temporal history of SFR in the ring of the same galaxy. They

found SFR ∼ 0− 0.15 M yr−1 during the past 1 Gyr, with the current SFR less than

0.03 M yr−1, much smaller than value reported by Mazzuca et al. (2008). Regarding

the pattern speeds, a model-independent kinematic method proposed by Tremaine &

Weinberg (1984) has been widely used to measure the angular velocities of arms and

bars (e.g., Zimmer et al. 2004; Rand & Wallin 2004; Merrifield et al. 2006; Meidt et al.

2008; Fathi et al. 2009; Speights & Westpfahl 2011). This method relies critically on a

few assumptions, notably that a galactic disk is in a steady state and that there is a

well-defined pattern, the validity of which is not always guaranteed. For instance, star

formation and ensuing feedback make the density and velocity fields non-steady in a

galactic disk. When arms are not corotating with a bar, there are significant non-steady

motions in the gas flows in the region where the bar joins the arms, which is likely to

compromise the derived pattern speeds based on gas tracers.

In addition to pattern speeds, there are many other factors such as the ring size,


bar strength, magnetic fields that may affect the ring SFR. Therefore, it is not yet

viable to make a definitive comparison of SFRs between our numerical predictions and

observations. Nevertheless, the observational data tabulated in Table 1 of Mazzuca et al.

(2008) show that the averaged ring SFR of SAB and SB galaxies are 2.90±2.00 M yr−1

and 2.00 ± 1.71 M yr−1, respectively. Since the relative importance of spiral arms is

larger for SAB than SB galaxies, these observations are not inconsistent with the idea

of arm-enhanced SFR in the nuclear rings. In addition, the spiral arms of NGC 4314

seem to corotate with the bar (Buta & Zhang 2009), and the ring SFR in this galaxy

is quite low at ∼ 0.1 M yr−1 (Mazzuca et al. 2008). On the other hand, NGC 4321

known to undergo starburst activities in the ring (Ryder & Knapen 1999) have spiral

arms rotating slower than the bar (Hernandez et al. 2005). These are consistent with

our result that the arm pattern speed affects the ring SFR.

To explore star formation in nuclear rings, we have adopted a very simplified model

of gas in barred-spiral galaxies. First of all, we treated the gas as being isothermal

and unmagnetized, whereas the interstellar gas in real disk galaxies is multi-phase,

magnetized, and turbulent (e.g., Wolfire et al. 2003; McKee & Ostriker 2007). This

required us to handle star formation feedback in the form of momentum injection

rather than thermal energy injection (e.g., Thacker & Couchman 2001; Agertz et al.

2011; Kimm & Cen 2014). We considered an infinitesimally-thin disk, which precludes

a potential effect of fluid motions that involve the vertical direction. Most importantly,

we here adopted a simple bar potential with fixed strength and pattern speed. Recent

N -body simulations for bar formation show that not only the bar strength but also

the bar size and pattern speed vary with time over a few Gyrs (e.g., Minchev et al.

2012; Manos & Machado 2014). The parameters of spiral arms also appear to change

as a bar evolves (e.g., Athanassoula 2012; Roca-Fabrega 2013). Therefore, it would be

interesting to study how star formation in nuclear rings studied in this work would

change in a more realistic environment where a bar, consisting of live stellar particles,

is self-generated and interacts with the gaseous component under radiative cooling and

heating, which would be an important direction of future research.

Chapter 4

Effects of Gas on Formation and

Evolution of Stellar Bars and

Nuclear Rings in Disk Galaxies

4.1 Introduction

More than 30% of disk galaxies in the local universe have a well-developed stellar

bar (e.g., Sellwood & Wilkinson 1993; Lee et al. 2012a; Gavazzi et al. 2015). Stellar

bars exert non-axisymmetric gravitational torque to gas to create substructures such as

dust lanes and nuclear rings (e.g., Sanders & Huntley 1976; Athanassoula 1992; Buta &

Combes 1996; Martini et al. 2003a,a; Kim et al. 2012a). Gas in orbital motions hits dust

lanes and loses angular momentum to infall toward the galaxy center. The inflowing

gas is gathered to form nuclear rings where intense star formation takes place (e.g.,

Burbidge & Burbidge 1960; Phillips 1996; Buta & Combes 1996; Knapen et al. 2006;

Mazzuca et al. 2008; Comeron et al. 2010; Sandstrom et al. 2010; Mazzuca et al. 2011;

Hsieh et al. 2011). Some galaxies possess filamentary dust spurs that are connected

perpendicularly to dust lanes (e.g., Sheth et al. 2000, 2002; Zurita & Perez 2008),

although their origin has been unidentified so far.

To explain the formation of gaseous substructures and understand what controls

81

82 3D Self-Consistent Simulations

their physical properties, a number of previous studies used hydrodynamic simulations

with a fixed gravitational potential to represent a stellar bar (e.g., Athanassoula 1992;

Englmaier & Gerhard 1997; Maciejewski et al. 2002; Maciejewski 2004; Regan & Teuben

2003, 2004; Ann & Thakur 2005; Kim et al. 2012a,b; Kim & Stone 2012; Li et al. 2015;

Shin et al. 2017). These studies found that dust lanes are shocks (Athanassoula 1992)

lying almost parallel to the trajectories of x1-orbits in a steady state, while the shape

of a nuclear ring is well described by x2-orbits (e.g., Athanassoula 1992; Englmaier &

Gerhard 1997; Patsis & Athanassoula 2000; Kim et al. 2012b). Nuclear rings form by

the centrifugal barrier that the inflowing gas cannot overcome, rather than resonances,

and are smaller in galaxies with stronger bars (Kim et al. 2012a), consistent with the

observations of Comeron et al. (2010). Although these models with fixed bar potentials

are useful to explore the parameter space in detail, they are unrealistic in that stellar

bars in real galaxies form and evolve so that their properties such as strength, size, and

pattern speed can vary considerably with time.

There have been numerous studies on how stellar bars form and evolve, based

mostly on N -body simulations. These work found that bars form due to dynamical

instabilities of self-gravitating stellar disks (Miller et al. 1970; Hohl 1971; Kalnajs 1972;

Goldreich & Tremaine 1979; Combes & Sanders 1981; Sellwood & Wilkinson 1993;

Polyachenko 2013; Saha & Elmegreen 2018). Recent pure N -body simulations showed

that not only the bar strength and length, but also the pattern speed continuously

changes with time on the course of disk evolution (Minchev et al. 2012; Manos &

Machado 2014). Sometimes, when the vertical velocity dispersion becomes very small,

bars can undergo buckling instability which in turn makes the bars weaker and shorter

(Combes & Sanders 1981; Combes et al. 1990; Raha et al. 1991; Merritt & Sellwood

1994; Martinez-Valpuesta et al. 2006; Kwak et al 2017). The bar properties and their

temporal evolution appear to be quite sensitive to the initial galaxy models. For in-

stance, Saha & Elmegreen (2018) very recently showed that the bar strength at late

time can differ, by more than a factor of two, depending on the bulge mass and density

structure in the initial galaxy models. The bar growth time as well as its strength are

3D Self-Consistent Simulations 83

also dependent upon the halo spin parameter (Collier et al. 2018). While these results

are informative, they are based on models with no gaseous component, and thus cannot

tell how gas responds to the bars to form substructures and thus how star formation

occurs in real barred galaxies.

In recent years, several studies adopted Smoothed Particle Hydrodynamics (SPH)

simulations to include the effects of the gaseous component on stellar bars (e.g., Bour-

naud et al. 2005; Berentzen et al. 2007; Athanassoula et al. 2013; Renaud et al. 2013;

Carles et al. 2016). Since gas is dynamically highly responsive, it can readily change

the density distribution of the whole disk to affect the bar formation and evolution.

However, the results of the various studies mentioned above differ, both quantitatively

and qualitatively, in the effects of gas on the bar formation and evolution. For instance,

Berentzen et al. (2007) found no significant relation between the gas fraction fgas and

the bar formation time, while Athanassoula et al. (2013) reported that disks with larger

fgas stay longer in a near-axisymmetric state and form a bar more slowly. Robichaud et

al. (2017) showed that disks with larger fgas form bars earlier when feedback from active

galactic nuclei (AGN) is considered, while the bar formation without AGN feedback is

almost independent of the fgas.

The presence of gas appears to make stellar bars weakened or destroyed to some

extent. Bournaud et al. (2005) argued that gas can completely dissolve a bar within

2 Gyr by exerting gravitational torque, while Berentzen et al. (2007) and Athanassoula

et al. (2013) found that bars are not completely destroyed even with the gaseous com-

ponent. In particular, Berentzen et al. (2007) found that the bar weakening in gas-poor

disks is caused by buckling instability, whereas the central mass concentration (CMC)

due to gas infall in gas-rich galaxies heats the disks and weakens the bars. They further

showed that the bar strength after the weakening does not differ much in disks with

different fgas. Athanassoula et al. (2013) showed that bars, albeit not completely de-

stroyed, weaken more strongly in galaxies with larger fgas. These results suggest that

the role of gas on a stellar bar is not yet clearly understood.

In this chapter, we run high-resolution simulations of Milky Way-sized, isolated,


disk galaxies consisting of a live halo, a stellar disk, and a gaseous disk. These three

components interact with each other through mutual gravity, while the gaseous compo-

nent suffers radiative heating and cooling and is subject to star formation and related

feedback. Our main objectives are two-folds. First, we want to understand how the

gaseous disk affects the formation and evolution of a stellar bar. Second, we want to

study how a nuclear ring evolves under the situation where the bar properties vary self-

consistently with time. The high-resolution models presented in this work improve the

previous self-consistent simulations mentioned above that did not have sufficient reso-

lution to investigate gaseous structures in detail. These models also extend our previous

studies with fixed bar potentials by allowing stellar bars to grow self-consistently. To

explore how bars and nuclear rings develop in various situations, we vary the velocity

anisotropy parameter (or Toomre stability parameter) as well as the gas fraction by

fixing the total disk mass.

The remainder of the chapter is organized as follows. In Section 4.2, we describe

our galaxy models and numerical methods that we adopt. In Section 4.3, we present

how stellar bars form and evolve in the presence of the gaseous component. In Section

4.4, we give evolution of gaseous structures that form and star formation rates in the

nuclear, bar, and outer disk regions. In Section 4.5, we summarize and discuss the

astronomical implications of the current work.

4.2 Models and Methods

4.2.1 Galaxy Models

To study formation and evolution of a stellar bar and its gravitational interactions with

the gaseous component, we consider galaxy models with physical properties similar to

those of the Milky Way. Our initial galaxy models consist of a stellar disk, a gaseous disk,

a dark matter halo, and a central supermassive black hole. For the density distribution

of the dark halo, we adopt the Hernquist (1990) profile

ρDM(r) =MDM

2π

rhr(r + rh)3

, (4.1)


where r is the radial distance, and MDM and rh denote the total mass and the scale

radius of the halo, respectively. The scale radius is often specified in terms of the

concentration parameter c and the virial radius r200 through

rh =r200

c

[2 ln(1 + c)− 2c

1 + c

], (4.2)

(Springel et al. 2005). For all models, we fix MDM = 3.1 × 1011 M, c = 24, and

r200 = 110 kpc, corresponding to rh = 10.7 kpc. Initially, we place a supermassive

black with mass MBH = 5 × 107 M at the galaxy center, which is allowed to accrete

surrounding gas without feedback.

For the stellar disk, we initially adopt the following density distribution

ρs(R, z) =Ms

4πzsRsexp

(− R

Rs

)sech2

(z

zs

), (4.3)

where R and z are the radial and vertical distances in the cylindrical coordinates, while

Ms, Rs = 3 kpc, and zs = 0.3 kpc refer to the mass, the radial scale length, and

the vertical scale height of the stellar disk, respectively. For models with gas included,

we adopt the same form as Equation (4.3) for the initial density distribution ρg of a

gaseous disk, but with the gas mass Mg, the scale radius Rg = 3 kpc, and the scale

height zg = 0.1 kpc. To study the effect of gas on the bar formation, we vary the gas

fraction fgas ≡Mg/(Mg+Ms) in the range of 0 to 10%, while fixing the total disk mass

to Mdisk = Ms +Mg = 5× 1010 M.

To construct a stellar disk by distributing particles, one needs to specify the velocity

anisotropy parameter

fR =σ2R

σ2z

, (4.4)

where σR and σz are the velocity dispersions in the radial and vertical directions,

respectively (Yurin & Springel 2014). For fixed σz, varying fR corresponds to changing

Toomre (1966) stability parameter

QT =κσR

3.36GΣtot= f

1/2R

κσz3.36GΣtot

, (4.5)

where κ is the epicycle frequency and Σtot =∫

(ρs + ρg)dz is the surface density of

the combined (stellar plus gaseous) disk. To study the effect of the velocity anisotropy


Figure 4.1. (a) Radial profile of the Toomre stability parameter QT and (b) rotation

curves in models with a cold disk (solid line) or a warm disk (dashed line).


Table 4.1. Model Parameters and Simulation Outcomes

Model fgas (t = 0) fR 〈A2〉〈R〉〈Rring〉〈SFRring〉 fgas (t = 5 Gyr)

(1) (2) (3) (4) (5) (6) (7) (8)

C00 - 1.0 0.63 1.65 - -

C05 5 1.0 0.49 1.50 400 0.19 3.2

C07 7 1.0 0.47 2.10 350 0.13 3.7

C10 10 1.0 0.22 4.50 500 0.04 4.6

W00 - 1.44 0.59 1.47 - -

W05 5 1.44 0.52 1.55 270 0.20 3.5

W07 7 1.44 0.46 1.45 310 0.31 5.2

W10 10 1.44 0.36 1.56 200 0.19 6.5

Note. — Note. the brackets 〈〉 denotes the temporal average over t = 4.5–5.0 Gyr.

Column 1: model name. Column 2: initial gas fraction (%). Column 3: initial ratio

of the radial and vertical velocity dispersions. Column 4: time-averaged bar strength;

Column 5: time-averaged ratio of the corotation radius to the bar length. Column

6: time-averaged nuclear ring size (pc). Column 7: time-averaged SFR in the ring

(M pc−1). Column 8: final gas fraction at t = 5 Gyr


(or QT ) on the bar formation, we in this chapter consider two sets of disk models:

relatively cold disks with fR = 1.0 and relatively warm disks with fR = 1.44. Figure

4.1(a) plots the radial distributions of QT for the cold and warm disks. Note that QT

is minimized at R = 5 kpc, with the minimum values of QT,min = 1.0 for the cold disks

and QT,min = 1.2 for the warm disks. Table 4.1 lists the names and parameters of all

models together with some numerical outcomes. The models with postfix “C” and “W”

have a cold and warm disk, respectively, and the number after the postfix represents

the gas fraction fgas in each model.

Our initial galaxy models with no gas are realized by making use of the publicly

available GALIC code (Yurin & Springel 2014). GALIC is very flexible in generating an

equilibrium configuration. It adjusts particle velocities iteratively to obtain a desired

density distribution. For models with gas, we reduce the mass ms of each stellar particle

to ms(1 − fgas), while keeping their number and positions intact. We then insert an

isothermal gas disk with mass fgasMtot and the vertical scale height zg = 0.1 kpc. Since

the conversion of a part of the stellar disk to the gaseous disk effectively reduces the

scale height and velocity dispersion, the new hybrid disk is slightly out of equilibrium.

We thus evolve the whole system over 0.1 Gyr by imposing an isothermal condition and

no star formation. The system gradually relaxes to a quasi-equilibrium state in which

the stellar disk remains almost unchanged with zs ' 0.3 kpc, while the gaseous disk,

being more dynamically responsive, becomes thinner to zg ≈ 50 and 80 pc at R = 3

and 5 kpc, respectively. While zg at the relaxed state tends to be smaller for larger

fgas and smaller fR, the differenes are only within a few percents. Figure 4.1(b) plots

the rotational velocities vrot at the relaxed state for the cold- and warm-disk models,

insensitive to fgas.

Each model is constructed by distributing a total of 1.1×107 particles: Nh = 5×106,

Ns = 5×106, and Ng = 1×106 for the halo, stellar disk, and gaseous disk, respectively.

The mass of each halo particle is mh = 6.2×104 M, while stellar and gaseous particles

each have mass of ms = 9.5×103 M and mg = 2.5×103 M for models with fgas = 5%,

and ms = 9.0× 103 M and mg = 5.0× 103 M for models with fgas = 10%.


4.2.2 Numerical Method

We evolve our galaxy models using the GIZMO code (Hopkins 2015), which is a second-

order accurate magnetohydrodynamics code based on a kernel discretization of the

volume coupled to a high-order matrix gradient estimator and a Riemann solver. It

thus conserves mass, momentum, and energy almost exactly. Gravity is solved by an

improved version of the Tree-PM method with an opening angle of θ = 0.7. Softening

lengths for stellar and halo particles are set to 10 and 50 pc, respectively, corresponding

to the mean particle separations. The gaseous particles have a fully adaptive softening

length with the minimum value of 1 pc.

For the evolution of gaseous particles, we use a second-order accurate meshless

finite-mass method that conserves angular momentum very accurately (Hopkins 2015).

The gaseous particles are subject to radiative cooling due to various line emissions and

heating mostly via photoionization and photoelectric effects (e.g., Katz et al. 1996;

Hopkins et al. 2014, 2018).

We implement a stochastic prescription for star formation and feedback. Star for-

mation is allowed to occur only in dense, self-gravitating regions where the velocity field

is converging and the local gas density exceeds the critical value ncrit = 10 cm−3. For

a gaseous particle satisfying the above criteria, the star formation probability over the

time interval ∆t is given by p = 1− exp(−εff∆t/tff), where εff ≈ 1% is the star forma-

tion efficiency (Hopkins et al. 2011; Seo & Kim 2013). In each time step, we generate a

uniform random number N and turn a gaseous particle into a new stellar particle with

the same mass only when N > p.

We handle star formation feedback using simple momentum input as well as mass

return to the neighboring gaseous particles in the form of continuous stellar winds and

Type Ia and II SNe events. In our models, each stellar particle (with mass approximaly

∼ 104 M) corresponds to an unresolved star cluster. Assuming the Kroupa (2001)

initial mass function and using the lifetime of Type II SNe progenitors (Lejeune &

Schaerer 2001) and the rate of Type Ia SNe (Mannucci et al. 2006), we calculate the

number of SN events NSN expected from a stellar particle in each time step ∆t. Type


II SNe occur only from newly formed clusters younger than 10 Myr, the lifetime of

8 M stars, while Type Ia SNe can explode from not only newly-created stars older

than 10 Myr but also the pre-exisitng particles comprising of the initial stellar disk.

Each SN injects momentum and mass to the surrounding gas particles inside the shell

radius

rsh = 0.025× N1/4SN

( n

1 cm−3

)−1/2kpc, (4.6)

corresponding to the shock radius at the shell formation stage for NSN almost simulta-

neou SNe. The amount of the total radial momentum depositied is given by

PSN = 2.8× 105 N7/8SN

( n

0.1cm−3

)−0.17M km s−1 (4.7)

(e.g., Chevalier 1974; Shull 1980; Cioffi et al. 1988; Seo & Kim 2013; Kim & Ostriker

2015). Stellar winds from O/B-stars and AGB stars are set to contributes only to mass

return at a rate M = 10−6 M yr−1 and 10−5 M yr−1, respetively: we do not consider

momentum injection from the winds. The integrated returned mass of a stellar particle

is about ∼ 20% in 1 Gyr after its creation.

4.3 Stellar Bars

In this section, we focus on the formation and evolution of stellar bars and the effects

of the gaseous component on them. Evolution of gaseous structures including nuclear

rings and star formation therein will be presented in Section 4.4.

4.3.1 Bar Formation

Since finite disk thickness reduces self-gravity at the disk midplane, all combined disks

with QT,min ≥ 1 are stable to axisymmetric gravitational perturbations. However, non-

axisymmetric perturbations are still able to grow as they swing from leading to trailing

configurations (e.g., Binney & Tremaine 2007; Kim & Ostriker 2007; Kwak et al 2017),

eventually organizing into bars. We find that the effect of gas on bar formation is

different between models with a cold disk and a warm disk, as described below.


Figure 4.2. Snapshots of logarithm of the stellar surface density Σs for the cold-disk

models at t = 0.1, 0.3, 0.4, and 0.6 Gyr from left to right. Each row corresponds to

model with fgas = 0, 5, 7, and 10% from top to bottom. Discrete bright spots represent

newly formed stars from the gas disk, while smooth color distributions display the

stellar particles in the initial disk. Colorbar labels log(Σs/[ M pc−2]).


Figure 4.3. Radial distributions of the normalized Fourier amplitudes am/a0 for mod-

els C00, C05, and C10 at t = 0.1 Gyr (left) and t = 0.3 Gyr (right).


4.3.1.1 Cold-disk Models

Figure 4.2 plots the stellar surface density in logarithmic scale in the 10 kpc regions of

the cold-disk models at t = 0.1, 0.3, 0.4, and 0.6 Gyr. It is apparent that the disks at

early time are subject to swing amplification and produce spiral structures that extend

from the galaxy center almost to the outer edge. At t = 0.1 Gyr, the disks harbor various

spiral modes with high azimuthal mode numbers (m ≥ 6). At t = 0.3 Gyr, they become

dominated by m = 3 and 4 modes, insensitive to fgas, while their amplitudes depend on

fgas. In model C10, newly formed stars indicated by bright spots are distributed along

the spiral arms at t = 0.3 Gyr. The early dominance of the three-arm spiral modes

in the process of bar formation is also seen in the QT,min = 1 models of Fanali et al.

(2015) Model C10 has strongest spirals since its disk is coldest effectively.

Figure 4.3 plots the radial distributions of the various Fourier amplitudes am relative

to a0 in the stellar disks of models C00, C05, and C10 at t = 0.1 and 0.3 Gyr. At

t = 0.1 Gyr, the modes with m = 3, 4 and m = 6, 7 have largest amplitudes in the

R ∼ 2–4 kpc and R ∼ 4–6 kpc regions of the disks, respectively. The strength of

swing amplification is measured roughly by the instantaneous growth rate multiplied

by the duration of amplification (e.g., Julian & Toomre 1966; Kim & Ostriker 2001).

While modes with high m may have a large instantaneous growth rate, they usually

have a limited time for amplification because they quickly wind out kinematically due

to background shear (e.g., Goldreich & Lynden-Bell 1965; Julian & Toomre 1966). It

turns out that the m = 3 mode grows most strongly at R . 3 kpc in all the cold-disk

models, although modes with m = 4 or 5 also contribute to the perturbed density in

the outer regions.

Swing amplification in the cold-disk models is so virulent that the spirals rapidly

become nonlinear. For instance, the m = 3 spirals in model C10 at t = 0.3 Gyr have

an amplitude δΣs/Σs ∼ 0.5 at R ∼ 1–3 kpc. These spirals interact nonlinearly with

other spirals with higher m that propagate radially inward. As a consequence, one arm

of the m = 3 spirals becomes loose and merge with the other two arms, eventually

transforming into an m = 2 bar mode that is supported by stable x1-orbit families


(Contopoulos & Grosbøl 1989). Since gaseous particles are colder than stellar particles,

the swing amplification and ensuing bar formation occurs faster in models with larger

fgas. Figure 4.2 shows that model C10 already possesses a well-developed bar by t =

0.4 Gyr, while it takes model C00 about ∼ 0.5 Gyr longer to form a bar.

4.3.1.2 Warm-disk Models

Warm disks form a bar slower than cold disks due to lager QT . Unlike in the cold

disks, the presence of gas delays the bar formation in the warm disks. Figure 4.4 plots

the stellar surface density in logarithmic scale in the 10 kpc regions of the warm-disk

models at t = 0.8, 1.5, 2.0, and 3.0 Gyr. Figure 4.5 plots the radial distributions of

the amplitudes of the m = 2–4 azimuthal modes that dominate in models W00, W05,

and W10 at t = 0.8 Gyr. At this time, by which all cold disks already form a bar, the

warm disks still exhibit only weak spiral structures. Similarly to the cold disks, the

warm disks are subject to swing amplification, but the amplification factor is less than

10% and the resulting spiral waves after the initial swing amplification are in the linear

regime.

Still, a warm disk with larger fgas is more unstable especially at R ∼ 4–6 kpc

where QT is smallest. Since random velocity dispersions (or acoustic waves) tend to

stabilize small-scale modes, disks with larger fgas favor larger-m modes for growth.

This expectation is consistent with Figure 4.5 that shows that the modes with m = 2,

3, and 4 have grown most strongly by t = 0.8 Gyr in models W00, W05, and W10,

respectively. This in turn indicates that the amplitude of the m = 2 mode that will

seed the bar formation is larger in models with smaller fgas. Note that these mode

numbers favored in the warm disks are smaller than those dominant in the early phase

of swing amplification in the cold disks.

Because of relatively large QT , the initial swing amplification in the warm disks

is too mild to form a bar instantly. Without a bulge, these trailing waves are well

positioned to propagate right through the center and then emerge as leading waves

in the opposite side, amplifying further as they unwind again from leading to trailing


Figure 4.4. Snapshots of logarithmic stellar surface density for the warm models at

t = 0.8, 1.5, 2.0, and 5 Gyr from left to right. All panels show the 20× 20 kpc regions.

Colorbar labels log(Σs/[ M pc−2]).


Figure 4.5. Normalized amplitude am/a0 of m = 2 to 4 as function of R for the models

W00, W05, and W10 at t = 0.8 Gyr.


configurations (e.g., Binney & Tremaine 2007). An eventual bar formation requires

several cycles of swing amplifications and feedback loops, which takes longer than ∼

1 Gyr. A disk with larger fgas takes longer to form a bar since the bar-forming m = 2

perturbations are weaker at the end of the initial swing amplification.

4.3.2 Physical Properties

4.3.2.1 Bar Strength

In our models, the strength and size of bars vary with time considerably, which is

related to the CMC. We define the bar strength A2 using the Fourier amplitude of the

m = 2 mode as

A2 = max

a2(R)

a0(R)

. (4.8)

To measure the CMC, we use the total (star plus gas) mass inside the central regions at

r ≤ 0.5 kpc. Figure 4.6 plots the temporal changes of the bar strength and the CMC for

the cold-disk (left) and the warm-disk (right) models. Figure 4.7 plots the distribution

of the stellar surface density at t = 5 Gyr in all models. Clearly, a bar forms earlier in

the cold disks than in the warm disks. The presence of gas causes a bar to form faster

and more strongly in the cold-disk models, while it makes the bar formation delayed

in the warm-disk models. Bar formation necessarily involves mass relocation and thus

changes the CMC even in the gas-free models, although the CMC is more significant in

models with gas since the bar potential can induce strong gas inflows. Changes in the

central mass affect orbits of stars, notably in the vertical direction, when they pass close

to the galaxy center, leading to bar thickening and weakening (e.g. Martinez-Valpuesta

et al. 2006; Berentzen et al. 2007; Kwak et al 2017).

In the cold-disk models, the rapid decay of the bar strength after reaching a peak

is caused by the rapid increase in the CMC. The bar weakening in model C10 is so

dramatic that it quickly turns into an oval shape, as illustrated in Figure 4.7. With a

relatively slow increase of the CMC, the bar in model C00 does not experience such

weakening: it keeps longer and stronger secularly, and attains a peanut-like shape at

t = 5 Gyr, a common late-time feature of N -body bars (e.g., Manos & Machado 2014).


Figure 4.6. Temporal variations of the bar strength (upper panels) and the central

mass concentration within the r = 0.5 kpc regions for the cold-disk models (left) and

the warm-disk models (right).


Figure 4.7. Snapshots of logarithm of the stellar surface density in the 10 kpc regions

for all models at t = 5 Gyr. Colorbar labels log(Σs/[ M pc−2]).

The central mass increases more rapidly as the bar grows faster and stronger, resulting

in stronger bar weakening in the cold-disk models with larger fgas. This collectively

makes a bar stronger in disks with smaller fgas at the end of the runs.

In the warm-disk models, on the other hand, the bar growth time is relatively long

and the CMC growth is accordingly quite slow. Therefore, the bar weakening after the

peak strength is not so severe as in the cold disks. As a result, the bars at t = 5 Gyr in

the warm disks are stronger for smaller fgas. Column (4) of Table 4.1 lists the values

of A2 averaged over t = 4.5–5.0 Gyr. Putting together all the results for both cold and

warm disks, we conclude that the bar strength in the late phase is inversely proportional

to fgas, independent of QT , notwithstanding temporal evolution in the early phase. A

mild drop in A2 of model W00 at t = 3.6–3.7 Gyr is due to the buckling instability,

which will be discussed in Section 4.3.3.

4.3.2.2 Bar Length and Pattern Speed

Since a bar is smoothly connected to the disk in which it is embedded, it is quite

ambiguous to determine the bar ends. We empirically find that the stellar surface


density of Σs = 80 M pc−2 traces the bar boundaries reasonably well, which allows

us to measure the bar semi-major axis Rb. We also calculate the bar pattern speed Ωb

based on the normalized cross-correlation of the surface densities at R = 3 kpc (e.g., Oh

et al. 2015). Figure 4.8 plots temporal variations of Rb and Ωb for all models. Overall,

the increasing and decreasing trend of Rb with time is similar to that of the bar strength,

such that a bar becomes long (short) when it is strong (weak). Bars in the early phase

of models C07 and C10 are short compared to their strength since they form by rapidly

redistributing the disk mass. Note that the bar pattern speed decreases continuously

after the formation in all models, and a stronger bar slows down at a faster rate. This

is because the angular momentum transfer between the bar and halo is more active for

a stronger bar (e.g., Athanassoula et al. 2013; Martinez-Valpuesta et al. 2006). Note

also that all bars except an oval in model C00 grow in size as they slow down over time

(e.g., Athanassoula 2002; Athanassoula et al. 2003).

Bars are usually classified as being “fast” or “slow” when R = RCR/Rb is less or

greater than 1.4, respectively, where RCR is the corotation radius. It is interesting to

see whether the bars formed in our models are fast or slow rotators. At t = 2.0 Gyr,

R = RCR/Rb in all models except models C10 and W10 with fgas = 10% are in the

range of 1 < R < 1.4, corresponding to fast bars. The bars even in models C10 and

W10 are rotating fast immediately after the formation (see Column 5 of Table 4.1).

However, R in all models increases gradually with time and exceeds 1.4 at the end of

the simulations, suggesting that fast bars eventually turn to slow bars. The transition

from fast to slow bars is consistent with the results of recent simulations of Pettitt &

Wadsley (2018) who showed that bars end up being slow under a massive dark matter

halo (see also Chemin & Hernandez 2009).

4.3.3 Buckling Instability

We find that the bar only in model W00 undergoes a vertical buckling instability, while

the other models remain stable to it. Figure 4.9 plots the contours of the stellar den-

sity in the x–z plane at some selected epochs of model W00, where x and z denotes


Figure 4.8. Temporal changes of (a) the bar semi-major axis and (b) the pattern

speed. Solid and dotted lines correspond to the cases of the warm disks and cold disks,

respectively.


Figure 4.9. Contour of logarithmic slice density in x-z plane along the bar semi-major

axis for Model W00.

the directions along the bar semi-major axis and perpendicular to the galactic plane,

respectively. At t . 3 Gyr, the disk thickens gradually increases over time as the CMC

excites stellar motions along the vertical direction. The buckling instability starts to

occur at t = 3.6 Gyr, promptly increasing σz and breaking the reflection symmetry in

the stellar disk about the z = 0 plane (Martinez-Valpuesta et al. 2006). It also causes

a temporal drop in the bar strength (see Figure 4.6). The disk subsequently becomes

more or less symmetric at t = 4 Gyr, and the bar becomes boxy or peanut-shaped when

viewed edge-on.


Figure 4.10. (a) Temporal variations of the radial (thin lines) and vertical (thick lines)

velocity dispersion, and (b) the ratio of velocity dispersion for the models C00, W00,

and W05 at R = 2 kpc.


The operation of buckling instability requires the ratio σz/σR of the velocity dis-

persions to be smaller than a critical value (e.g., Binney & Tremaine 2007). Toomre

(1966) and Araki (1987) found that infinitesimally-thin, non-rotating slabs are unstable

if σz/σR < 0.3, while Merritt & Sellwood (1994) suggested an instability criterion of

σz/σR < 0.6 for axisymmetric rotating disks.For realistic disks with spatially varying

σz/σR, Martinez-Valpuesta et al. (2006) and Kwak et al (2017) used N -body simula-

tions to show that the critical values for buckling are at σz/σr ∼ 0.6 for their models,

suggesting that the critical value may depend on the density and velocity distributions

inside the disk.

Figure 4.10 plots temporal changes of σz/σR for models C00, W00, and W05 at R =

2 kpc. The bar formation in itself increases σR, while the CMC that also results from

the bar formation tends to increase σz. Therefore, the value of σz/σR is decided by

the competition between the bar strength and the CMC. It turns out that all of our

models with gas included suffer a large increase in the CMC to always have σz/σR >

0.5 throughout their entire evolution, and thus remain stable to buckling instability.

Although model C00 has no gas, its bar is strong enough to enhance the CMC very

rapidly, resulting in σz/σR > 0.5 for all time. The bar in model W00 grows strong but

relatively slowly, and incurs only a mild increase in the central mass. As a consequence,

it has σz/σR as low as ∼ 0.5 and undergoes buckling instability. These results suggest

that the critical value for the buckling instability is σz/σr ∼ 0.5 for our models, and

that the presence of gas tends to suppress buckling instability by increasing the CMC.

4.4 Gaseous Structures and Star Formation

4.4.1 Nuclear Ring

Figure 4.11 plots snapshots of the gas surface density in logarithmic scale of model

W05 at six different epochs. The upper panel show the 10 kpc regions, while the lower

panels zoom in the central 1 kpc regions to display dust lanes and a nuclear ring in

more detail. Both gas and bar are rotating in the counterclockwise direction. Evolution


Figure 4.11. Snapshots of logarithmic gaseous surface density for the standard model

W05 at six different time. Upper panels show the 20 × 20 kpc regions and each lower

panels show the 2× 2 kpc regions at same time.


of other models are qualitatively similar.

At t = 1 Gyr, a weak stellar bar produces a pair of dust lanes that are relatively

straight at R & 0.5 kpc and take a shape of trailing spirals toward the center (e.g.,

Maciejewski 2004). As the bar grows further, the dust lanes become quite straight and

are located very close to the semi-major axis of the bar (e.g., Kim et al. 2012b). A small

nuclear ring with radius ∼ 40 pc is beginning to form at t = 1.5 Gyr as the material

driven inward by the bar potential accumulates near the galaxy center. To explore

how the ring size varies with time in our models, we measure the density-weighted,

angle-averaged ring radius as

Rring ≡∫R〈Σg(R)〉dR∫〈Σg(R)〉dR

, (4.9)

where 〈Σg(R)〉 = (1/2π)∫ 2π

0 Σg(R,φ)dφ, and plot it in Figure 4.12 as functions of

time for all models with gas. In model W05, the ring remains small but exhibits some

fluctuations until the bar strength reaches its peak at t = 2 Gyr, During this time,

intermittent star formation occurring in the nuclear regions disperse the ring and dust

lanes temporarily.

After the bar achieves the maximum strength, the gas in the bar regions experiences

massive infall and is added continuously to the ring. The ring in model W05 grows in size

slowly with time to Rring ∼ 0.3 kpc at t = 5 Gyr, which is caused mainly by the increase

in the bar size (Fig. 4.8(a)). As the bar becomes longer, fresh gas at larger R and higher

angular momentum infalls to be added to the ring. At the same time, the gas already in

the ring that has lower angular momentum is continually consumed to star formation

at a rate of ∼ 0.2 M yr−1. Since the ring mass is typically 4 × 107 M, the ring gas

is almost completely replaced by newly inflowing gas from outside in ∼ 0.2 Gyr. The

decrease in the bar pattern speed which tends to make the dust lanes move to far from

the bar semi-major axis (e.g., Li et al. 2015) as well as the CMC increase also help to

increase the ring size. Large fluctuations in the ring size at late time in the warm-disk

models are due to active star formation feedback (see Section 4.4.3).

That a nuclear ring forms small and becomes larger with time found in our current

models is different from the cases with a static bar potential in which a ring is large


when it forms (e.g., Kim et al. 2012a,b). When the stellar bar properties are fixed with

time, the non-axisymmetric bar torque produces dust-lane shocks downstream away

from the bar semi-major axis. In this case, a ring forming at the inner ends of the dust

lanes has quite a large radius, although it subsequently shrinks in size by 10 to 20% as

collisions of dense clumps inside the ring take away angular momentum from the ring

(Kim et al. 2012a). In our self-consistent models, the physical properties of stellar bars

keep changing with time, providing non-steady gravitational potentials to the gas. Near

the time when a nuclear ring forms (t ∼ 1.5 Gyr), the bar potential is strong enough

to induce shocks only in the innermost (R . 0.1 kpc) parts of the dust lanes, so that

the resulting nuclear ring should be much smaller than the counterpart under the fixed

bar potential where the dust-lane shocks are extended across the whole bar length.

In model W10, the nuclear ring is relatively large from the beginning and its size

does not vary much with time. This is becuase not only the bar strength grows very

slowly but also the CMC has grown to some extent before the strong bar forms (Li

et al. 2017). However, since the bar strength increases continuously until the end of

the simulation and the growth of the CMC and the decreases of the pattern speed are

slower than other models, the ring size does not increase with time in late stage.

4.4.2 Filamentary Spurs

While perpendicular filamentary interbar spurs have often been observed in association

with dust lanes in real galaxies (e.g., Sheth et al. 2002; Zurita & Perez 2008; Elmegreen

et al. 2009), previous hydrodynamic simulations with a fixed bar potential were unable

to produce such structures (e.g., Kim et al. 2012a,b; Seo & Kim 2013, 2014). In these

simulations, the bar regions quickly reach a quasi-steady state in which gas approxi-

mately follows x1-orbits that are almost parallel to the dust lanes (see, e.g., Figure 7

of Kim et al. 2012b).

Unlike in the previous simulations, we find that the current self-consistent simu-

lations with gas produce interarm spur-like structures at late time, except for model

C10 in which a bar evolves to a weak oval. Figure 4.13 plots the gas surface density


Figure 4.12. Temporal variations of the ring size for all models with gas component.

Figure 4.13. (a) Gas surface density for the model W07 at t = 4.75 Gyr. (b) The gas

surface density along the green dashed line that denote filamentary spur region, and

(c) blue dotted line that denote dust lane in (a).


in the inner 7.5 kpc regions at t = 4.75 Gyr of model W07, as well as the profiles of

the gas surface density along the slits marked as dashed line (bar semi-major axis) and

dotted line (lower-left dust lane) in the left panel. It is apparent that five or six spurs

are connected perpendicularly to the dust lane at each side. The density enhancement

associated with the spurs is only ∼ 1 M pc−2 in the interbar regions, which becomes

larger in the dust lanes by up to an order of magnitude. High-density peaks formed by

collisions of the spurs with the dust lanes sometimes undergo star formation.

Spur-like structures in our models are originated from star formation feedback as

well as non-steady gas streamlines. Shells produced by feedback in the low-density

interbar regions are stretched by shear in the background flows, creating filamentary

structures there. Since the bars in our simulations change with time, there are no

well-defined x1-orbit families that gas can follow. As the dust lanes slowly move away

from the bar semi-major axis since off-axis shocks becomes stronger due to the bar

slowdown (Li et al. 2015), the gas streamlines that turn their directions near the semi-

major axis become almost perpendicular to the dust lanes. The sheared filaments also

turn directions near the bar semi-major axis and hit the dust lanes perpendicularly to

enhance the local density.

4.4.3 Ring Star Formation

Star formation in our cold-disk models is widely distributed across the entire disk,

while the cold-disk models actively form stars only inside the bar and nuclear regions.

Figure 4.14 plots temporal changes of the SFR in the warm-disk (left) and cold-disk

(right) models. The top, middle, and bottom panels give the SFR occurring in the

central regions (R < 0.5 kpc), the bar regions (R = 0.5–5 kpc), and the outer disk

(R > 5 kpc), respectively.

In the outer disk, star formation occurs mostly inside spiral arms and is stronger in

the cold disks with stronger arms. The presence of strong spirals before the development

of a bar is responsible for a sharp increase in the SFR in the outer disks of the cold-

disk models. As the spirals become weaker due to heating, the SFR in the outer disk


Figure 4.14. Temporal variation of the SFR in nuclear ring region (top panels), the

SFR in bar region (R = 0.5−5 kpc) (middle panels), and the SFR in outer spiral region

(R > 5 kpc) (bottom panels). Left panels plot the SFR of the cold models and right

panels plot that of the warm models.


decrease rapidly with time.

The formation of a bar certainly triggers star formation in the bar and central

regions. As the bar grows, the non-axisymmetric potential produces a pair of dense

ridges and a nuclear ring in which most of the disk star formation takes place at late

time. The early increasing trend of the ring SFR is similar to that of the bar strength.

After the bar achieves a full strength (t ∼ 0.5 Gyr in the cold disks and t ∼ 2 Gyr

in the warm disks), the bar SFR experiences a dramatic drop except for models with

fgas = 10% , while the decrease in the ring SFR is only mild. This is in contrast to the

cases with a fixed bar potential where fast gas exhaustion caused by a fast bar growth

on the timescale of ∼ 0.2 Gyr makes the ring SFR declines very rapidly afterward

(e.g., Seo & Kim 2013, 2014). The relatively slow decrease of the ring SFR results

from the fact that bars in our models not only grow slowly but also become longer in

size over time. This makes the duration of star formation extended and expands the

regions influenced by the bar potential, allowing sustained gas inflows tho the rings.

In addition, mass ejections in the form of winds and SNe from star particles help to

increase the gas mass in the bar regions.

Overall, the ring SFR is larger when a bar is stronger. The ring SFR is highly

episodic and bursty caused by star formation feedback. Sometimes, especially at early

time, feedback is so strong that the rings are completely destroyed and reform multiple

times. In model W10, the bar is weak and grows very slowly, and the resulting SFR

also exhibits a slow growth with intermittent bursts. In model C10, on the other hand,

the bar size remains almost unchanged after t ∼ 1 Gyr, so that the ring SFR keeps

decreasing as the bar region becomes devoid of gas.

The conversion of gas to stars decreases the gas fraction in the disks, while mass

return via stellar winds and SN feedback increases the gas mass. Figure 4.15 plots the

gas fraction as a function of time in our models with gas. When fgas = 5% initially,

the gas fraction increases before the bar formation since the gas surface density is not

high enough to induce active star formation, while the initial disk stars emit winds

continuously. This is also the reason for the increase of the gas fraction in model C10


Figure 4.15. Temporal variation of the gas fraction fgas for all gas models.


after t = 1 Gyr. In the cold-disk models, star formation is more active in the outer disk

than in the rings, resulting in a rapid drop in the gas fraction at t = 0.2–1 Gyr. Star

formation in the warm-disk models occurs more gradually, and is more active in the

rings than in the outer disk.


4.5.1 Summary

We have presented the results of self-consistent three-dimensional simulations of barred

galaxies that possess both stellar and gaseous disks. Our primary goals are to under-

stand the effects of the gaseous component on the bar formation and to explore how

nuclear rings form and evolve in galaxies where the physical properties of bars vary

self-consistently with time. We consider radiative heating and cooling of the gas and

allow for star formation and related feedback, but do not include magnetic fields in the

present work. We consider two sets of models, similar to the Milky Way, which differ in

the velocity anisotropy parameter fR (or, equivalently, the minimum Toomre stability

parameter QT,min). The models with fR = 1 and 1.44 have a disk with QT,min = 1.0

and 1.2, and are thus referred to as “cold-disk” or “warm-disk” models, respectively.

In each set, we vary the mass of gaseous disk, fgas = Mg/Mdisk, in the range between

0 and 10%, while fixing the total disk mass to Mdisk = 5× 1010 M. The main results

of our work can be summarized as as follows.

1. Effects of Gas on the Bar Formation: Perturbations in the initial disks are swing

amplified to form spiral structures. In the cold-disk models, the initial swing

amplification is strong enough to make spirals highly nonlinear, and the disks

soon become dominated by the spirals with m = 3 that have a long duration for

growth. These m = 3 spirals interact nonlinearly with other modes with different

m and consequently transform rapidly to an m = 2 bar mode supported by closed

x1-orbit families. Since the gaseous component is effectively colder than the stellar

component, a bar in a disk with larger fgas forms faster and more strongly.


In warm-disk models, however, the initial swing amplification is only moderate

and the resulting spirals are in the linear regime. Thus, m = 2 spirals that will

eventually become bars should be amplified further via successive swing ampli-

fications and multiple loops of feedback. Modes with larger m are favored in a

disk with larger fgas due to lower effective velocity dispersions, indicating that

the amplitude of the bar-seeding m = 2 spirals is lower for larger fgas. Unlike

in the cold disks, therefore, a warm disk with larger fgas forms a bar slower and

weake and weaker.

2. Bar Evolution:

Bar formation necessarily involves the mass re-distribution as well as gas inflows

toward the center, increasing the CMC. The CMC in turns weakens the bar by

exciting stellar motions in the vertical direction. As a bar forms more rapidly,

the CMC grows at a faster rate and the bar decays more strongly after the

peak. For example, a bar in model C10 grows very rapidly (∼ 0.1 Gyr) and then

becomes weaker by a factor of three in ∼ 1 Gyr and turns to an oval, while model

C00 with a slow increase of the CMC does not experience such bar weakening.

Consequently, bars in both cold and warm disks become stronger in disks with

smaller fgas at the end of the runs. After a temporal weakening due to the CMC,

bars grow again in size secularly with time. The bar length Rb is correlated with

the bar strength such that a stronger bar is usually longer. The bars are fast

rotators with R = RCR/Rb < 1.4 when they form, and then gradually slow down

by transferring angular momentum to their surrounding halos and become slow

rotators.

We find that only the gas-free, warm-disk model W00 undergoes buckling instabil-

ity when σz/σR . 0.45, where σz and σR refer to the velocity dispersions in the

vertical and radial directions, respectively. The presence of gas tends to stabilize

the buckling instability by enhancing the CMC and σz. Although model C00 does

not have gas, its bar growth and the CMC increase are strong enough to quench

the buckling instability.


3. Nuclear Ring and Spur :

The non-axisymmetric bar torque induces shocks in the gas flows and form dust

lanes. The gas experiences infall along the dust lanes to form a nuclear ring. At

early time when the bar grows, only a gas close to the galaxy center responds to

the bar potential, leading to a small nuclear ring with a radius of Rring ∼ 40 pc.

Due to strong feedback from explosive star formation inside the ring, the tiny ring

is repeatedly disrupted and reforms. As the bar grows in size, gas at larger radii

starts to infall and be added to the nuclear ring. Since the gas at larger R has

increasingly larger angular momentum, this makes the nuclear ring larger with

time, up to Rring ∼ 0.2 − 0.5 kpc at the end of the runs. Except for model C10

that has an oval rather than a bar, a model with a stronger bar has a smaller

ring, consistent with the results of Comeron et al. (2010) who found that stronger

bars host smaller nuclear rings (see also Kim et al. 2012a). The decrease in the

bar pattern speed and increase in the CMC are also likely to increase the ring

size (Li et al. 2015).

Unlike the previous simulations with a fixed bar potential, we find that our self-

consistent simulations form filamentary interbar spurs that are connected perpen-

dicularly to dust lanes at late time. The origin of filaments is SN shells produced

by star formation feedback that are sheared out in the low-density bar regions.

Since the bars becomes stronger and longer over time, the dust lanes move grad-

ually away from the bar semi-major axis. When the filaments hit the dust lanes

perpendicularly, the local density is enhanced by an other of magnitude in the

dust lanes, sometimes enough to form stars.

4. Star Formation:

4. Star Formation:

The cold-disk models form stars both in the outer disks with spiral arms and in

the inner disk with a bar, while star formation in the warm-disk models with weak

spirals is concentrated in the inner disk. Bar formation triggers star formation in


the bar regions (mostly inside dust lanes) as well as in the nuclear rings. Overall,

the ring SFR is stronger for a stronger bar. The ring star formation is highly

episodic and bursty due to feedback that can sometimes disrupt the rings. The

SFR in the bar regions rapidly declines after the bar attains the peak strength.

However, the ring SFR decreases quite mildy due to a slow bar growth as well as

a temporal increase in the bar length, the latter of which can continuously supply

the gas to the ring at late time. Mass return via winds and SNe also help the ring

SFR persist longer than the cases with a fixed bar potential (see Chapter 2 and

Chapter 3).

4.5.2 Discussion

In this chapter, we consider galaxy models similar to the Milky Way to study bar

formation in disks with gas. The properties of the bars and nuclear rings formed in

our simulations are very close to those in the Milky Way. The Milky Way is known to

have a bar with the semi-major axis of Rb ∼ 3–5 kpc (Morris & Serabyn 1996; Dame

et al. 2001; Ferriere et al. 2007; Kruijssen et al. 2015), similarly to those in models

with fgas = 5 or 7% shown in Figure 4.8. The central molecular zone (CMZ, Morris &

Serabyn 1996), a nuclear ring in the Milky Way, has a radius of Rring ∼ 0.2–0.5 kpc,

consistent with the ring sizes displayed in Figure 4.12. The CMZ is observed to have a

total gas mass of ∼ 2− 5× 107 M (Immer et al. 2012; Longmore et al. 2013), similar

to gas mass Mg ∼ 3 − 5 × 107 M in the rings in model W05 at t ∼ 2 Gyr. The SFR

in the CMZ is estimated to be ∼ 0.1 M yr−1(Yusef-Zadeh et al. 2009; Immer et al.

2012; Longmore et al. 2013; Koepferl et al. 2015), about 10 times smaller than the

value inferred from the CMZ mass (Tsuboi & Ukita 1999; Longmore et al. 2013). This

is probably because star formation in the CMZ is episodic and currently in the low

state (Kruijssen et al. 2014), consistent with the results of our simulations.

We note that all nuclear rings formed in our models have radii less than ∼ 0.6 kpc

at the end of the runs. Although these are more-or-less comparable to the ring sizes

in galaxies like the Milky Way, they are certainly smaller than typical nuclear rings in


normal barred-spiral galaxies such as NGC 1097 (e.g., Comeron et al. 2010) and those

formed under fixed bar potentials (e.g., Kim et al. 2012a,b; Li et al. 2015). Relatively

small nuclear rings are presumably due to the absence of a bulge in our initial galaxy

models. Recently, Li et al. (2017) used hydrodynamic simulations with static stellar

potentials to show that a nuclear ring forms only in models with a central object

exceeding ∼ 1% of the total disk mass, and that the ring size increases almost linearly

with the mass of the central object. This opens the possibility that the presence of a

massive compact bulge would make a ring large when it first forms. The ring can be

even larger as it grows due to an addition of gas with larger angular momentum.

We find that the effects of gas to the bar formation depends rather sensitively on

the velocity anisotropy parameter or QT,min such that gas makes a bar form faster and

stronger in cold disks with QT,min = 1.0, while tending to suppress the bar formation in

warm disks with QT,min = 1.2. This is different from the results of Athanassoula et al.

(2013) who found that the gaseous component with fgas ≤ 50% always prevents the bar

formation, similarly to our warm-disk models. Robichaud et al. (2017) ran simulations

of bar formation with or without AGN feedback, and showed that gas in models with

AGN feedback promotes bar formation, similarly to our cold-disk models, whereas the

bar formation without AGN feedback is independent of fgas. These discrepancies in

the results of various simulations with different parameters suggest that bar formation

involves highly nonlinear processes especially with gas and is thus very sensitive to the

initial galaxy models as well as the gas fraction.

After the bar formation, as with the results of Athanassoula et al. (2013), our

models also shows that the bar strength becomes weaker as the gas fraction increases.

Some previous studies have shown that bars in gas-free or gas-poor models experience

weakening due to the vertical buckling instability, while in our models the pure N-body

models Berentzen et al. (2007) has shown that bars in gas-free or gas-poor models

experience weakening by the vertical buckling instability (see also Martinez-Valpuesta

et al. 2006; Kwak et al 2017), while the bar weakening in gas-rich models is due to

the growth of the CMC. In our models, however, the pure N-body models were also


weakened by the growth of the CMC. Although the model C10 experienced the buckling

instability, it appeared after the first weakening by the CMC. This difference also can

be attributed to the fact that the bar evolution is very sensitive to the initial model as

well as the bar formation. However, Gajda et al. (2018) showed that the tidally induced

bars in dwarf galaxies become weaker as the gas fraction increases, despite the absence

of gas inflow to the central region. Although the size of the galaxy is very small and

the bar formation mechanism is completely different from our models, this result shows

that the presence of the gas as well as the gravitational potential change by a mass

redistribution affects the bar weakening. Therefore, numerical simulations with more

various initial conditions are required to further analyze the role of gas components.

In addition to the gas fraction and velocity anisotropy parameter, the properties of

a DM halo also appear to affect dynamical evolution of bars via angular momentum

exchanges with the disks (e.g. Sellwood 1980; Debattista & Sellwood 2000; Athanassoula

et al. 2003). Recent numerical simulations showed the bar evolution is influenced by the

shape and the spin parameter of the DM halo (e.g., Athanassoula et al. 2013; Collier et

al. 2018). In particular, Athanassoula et al. (2013) found that bars under a triaxial halo

form earlier and experience stronger decay than in galaxies with a spherical halo. On

the other hand, Collier et al. (2018) found that bars in both prolate and oblate haloes

start to form later. They further showed that a halo with faster spin is less efficient in

absorbing angular momentum and thus results in a weaker and smaller bar. It will be

interesting to see how the presence of gas conspires with the halo spin to guide the bar

formation and evolution.

In this work, we did not consider feedback from the central black hole that was

allowed to accrete the surrounding gas passively. Some barred galaxies host AGN at

their centers, but the physical connection between a bar and AGN activities is not

clear. Some observational studies suggest that the bar fraction in AGN-host galaxies is

higher than in galaxies without AGN (e.g., Arsenault 1989; Laine et al. 2002; Galloway

et al. 2015), while other studies do not find any specific correlations between them (e.g.,

Bang & Ann 2009; Lee et al. 2012b; Cheung et al. 2014 ). Recently, Robichaud et al.


(2017) used numerical simulations to find that AGN feedback suppresses star formation

in the vicinity of a black hole, while forming a dense ring in which star formation is

enhanced. They found that such positive and negative effects are almost equal, making

no overall quenching or enhancement of star formation in barred galaxies. We note that

these results were based on models with a static (rather than live) halo and a not-well

resolved gas disk with Ng = 1.2 × 105 particles. It is desirable to run self-consistent

models with high resolution to accurately assess the effects of AGN feedback on star

formation in barred galaxies.

Chapter 5

Conclusion

5.1 Summary

In this thesis, we presented three studies on the formation and evolution of the gas

structures and the stellar bar in barred-spiral galaxies.

In Chapter 2, we studied the star formation in nuclear ring of barred galaxies. To

study what determines the SFR in nuclear rings, we have used the CMHOG2 code to run

hydrodynamic simulations under two-dimensional cylindrical geometry. We treated the

bar potential as being fixed and employed a logarithmic radial grid to resolve the central

parts very well. We included a simple prescription for star formation and feedback in

the form of mass and momentum injections via type II supernovae. We found that the

SFR in nuclear rings is directly proportional to the mass inflow rate to the ring, but

almost independent of the gas mass in the rings. The SFR exhibits a strong primary

burst and a few weak secondary bursts before declining to small values. The primary

burst is associated with the rapid gas infall due to the bar growth in the early stage,

while the secondary bursts are caused by the re-infall of the gas ejected during the

primary burst. After these bursts, the SFR decreases to very small values as the gas

in the bar regions becomes almost evacuated. We also found that the existence of an

azimuthal age gradient of young star clusters is deeply related to the SFR. When the

SFR is low, most star formation occurs near the contact points between the ring and

121

122 Conclusion

the dust lanes, naturally leading to the presence of an azimuthal age gradient. When

the SFR is large, on the other hand, star formation is randomly distributed over the

length of a ring and star clusters do not have an apparent age gradient. In our models, a

nuclear ring shrinks in size over time, although self-gravity of the ring tends to maintain

gas orbits relatively intact, reducing the ring shrinking rate.

In Chapter 3, we presented the effects of spiral arms on the ring star formation

of barred-spiral galaxies. Our models with only a bar potential show that ring star

formation lasts only about 0.3 Gyr, of order of the bar evolution time, over which the

gas in the bar region experiences infall. However, observations suggest star formation

histories in nuclear rings exhibit episodic behavior lasting a few Gyr. To explain these

long-lived and episodic star formation, we considered spiral arms which can transport

the gas from outside to the bar region. We found that spiral arms located outside the

bar can indeed induce gas inflows toward the bar regions, which eventually enhances

the ring SFR at the late stage of evolution provided that the CR of the spiral arms is

located far from the bar ends. After entering the bar region, the inflowing gas piles up

at the bar ends. Mutual collisions of gas orbits and interactions between the spirals and

the bar take away angular momentum from the gas, intermittently making it move in

to the nuclear ring along the dust lanes. This intermittent inflow of the gas makes the

ring star formation episodic over time. The arm-enhanced SFR is larger by a factor of

3−20 than that in the no-arm counterpart, with larger values corresponding to stronger

and slower arms. The rings in models with slow arms are larger in size by about 45%

than those in the no-arm models.

In Chapter 4, we presented the effects of gas on the formation and evolution of

stellar bars and central gaseous structures in disk galaxies. To study how the stellar

and gaseous components interact with each other to form a bar and a nuclear ring as well

as ring star formation, we have run fully self-consistent three-dimensional simulations

of isolated disk galaxies similar to the Milky Way using GIZMO. We found that bars form

earlier and stronger as the gas fraction increases in the cold disk (Q = 1.0), while the

bar formation is delayed in relatively warm disk (Q = 1.2). The bar formation causes a

Conclusion 123

growth of the CMC, which in turn weakens the bar by exciting vertical motions of stellar

components. In cases of gas rich models with cold disks, the CMC also increases sharply

as the bars are formed rapidly, resulting in strong weakening. Consequently, at the ends

of the simulations, bars in both cold and warm disks become weaker with higher gas

fraction. The inward gas inflow by the bar formation produces a dense nuclear ring.

The nuclear ring is very small when it forms with the bar formation. And after the

bar formation, unlike hydrodynamic simulations wiht a fixed bar potential in Chapter

2 and 3, the ring is enlarged by the addition of gas with larger angular momentum.

We also found that our self-consistent simulations form filamentary spur structures in

interbar region which could not seen in hydrodynamic simulations. The star formation

in the ring is higher for a stronger bar and episodic and bursty due to the strong star

formation feedback.

5.2 Future Research Plans

5.2.1 Various Galaxy Models

The mass and size of disk, bulge, and halo differ widely from galaxy to galaxy (Bosma

1998; Widrow et al. 2003; Sofue 2012; Pettitt & Wadsley 2017). While the results of

this thesis were able to explain some properties of real barred spiral galaxies, but the

galaxy models were limited to Milky Way-sized galaxies. Therefore, it is desirable to

run various galaxy models to deeply understand how the bar properties and resulting

gaseous structures depend on the size and shape of each galaxy component.

• Parameters of Dark Matter Halo - DM halo is very important for the

evolution of the bar because it not only contributes to the rotation curve but also

interact with the disk to absorb angular momentum of a bar (e.g., Sellwood 1980;

Debattista & Sellwood 2000; Athanassoula 2003). Recent simulations showed the

bar evolution is affected by the shape and the cosmological spin parameter of the

DM halo. Bars in galaxies with both triaxial haloes (Athanassoula et al. 2013) and

prolate/oblate haloes (Collier et al. 2017) evolve differently from those in galaxies

124 Conclusion

with spherical haloes. Athanassoula et al. (2013) found that bars in triaxial haloes

start to form earlier, but are generally weaker when fully developed than those

with spherical halos. Collier et al. (2017) found that the halo spin parameter

λ ≡ Jh/√

2MvirRvirvc greatly changes the bar size and growth time. They further

showed that the time scale of bar instability and the final bar length are smaller

as λ increases. Since they employed gas-free models, however, it is not yet known

how the presence of gas can change their results.

• Initial Central Bulge - Even if there is no bulge component in the initial

galaxy model, a disk is unstable to a bar that secularly evolve to a pseudo bulge.

The presence of an initial spherical bulge can affect the formation of a nuclear

ring in the very early stage of the bar evolution. Using hydrodynamic simulations

with a static stellar potential, Li et al. (2017) found that a ring forms only in the

models with a central object of a few percent of the total disk mass, while it does

not form in the models with no central object. Without a nuclear ring, the gas

can move in directly to the galaxy center. Li et al. (2017) also found that a more

massive central object results in a larger ring size. The difference in the ring size

at early time in fully self-consistent simulations will not only lead to subsequent

evolutionary changes, but will also result in changes in the mass accretion rate

to the central massive black hole. In our ongoing work, models with strong bar

have a relatively small nuclear ring. However, some galaxies like NGC 1097 show

a large nuclear ring despite having a strong bar. Since the presence of an initial

central bulge can help to form a large nuclear ring.

• AGN Feedback - Some barred galaxies host AGN at their center, but the

link between a galactic bar and the presence of an AGN is not yet clear. Some

observations suggest that the bar fraction in AGN-host galaxies is higher than

in galaxies without AGN (Arsenault 1989; Galloway et al. 2015), while other

studies did not find significant differences between the AGN and bars (Moles et

al. 1995; Mulchaey & Regan 1997; Cheung et al. 2015). In early stage of the

evolution without an initial stellar bulge, the gas can flow directly to the nucleus,

Conclusion 125

but the gas inflow can be suppressed if an AGN feedback is added. Moreover,

strong outflows by AGN feedback can have a great influence on the evolution

of gaseous structures and star formation in the central regions. Robichaud et

al. (2017) reported that bars reach peak strength earlier when AGN exists and

that AGN feedback affects star formation in the very central regions, but they

did not focus on detailed gaseous structures.

5.2.2 Interaction with other galaxies

In our works, each galaxy models assumed to be isolated, however, many real galaxies

often have other objects close enough to affect them. In isolated galaxy models, the

strength of spiral arms in the late phase of evolution becomes very weak, even though

it was strongly formed in the early stage. This means that it is difficult for the steady

bar and the strong spiral arms to exist together in isolated galaxy. Using numerical

simulations, Pettitt & Wadsley (2017) showed that the properties of spiral arms and

bars are heavily influenced by the presence of companions, however they use isothermal

equation of states and do not include star formation and feedback. And, since closest

approach for their all models with companion occurs in very early stage (∼ 0.2 Gyr), it

is not well known how the interaction that occurs after the bar is fully formed affects

the evolution of the bar and spiral arms.

5.2.3 Effects of Magnetic Fields

Magnetic fields are one of the most important ingredient of the interstellar medium.

Observations found that magnetic fields are strong in gaseous structures in central

regions of barred galaxies (Beck et al. 2005), and some barred galaxies show magnetic

arms which have strong field with low density (Beck et al. 2002, 2005). There have

been a number of numerical studies of gas dynamics in barred galaxies with magnetic

fields. Kulesza-Zydzik et al.(2010) found that the presence of magnetic fields lead to

the formation of magnetic arms. In addition, Kim & Stone(2012) found that nuclear

rings become smaller as the field strength increases. However, these previous works

126 Conclusion

adopted a fixed stellar gravitational potential, and thus were unable to consider the

evolution of galactic bars in disks with magnetic fields. Magnetic fields mainly affect

the gas component, but it can also play an important role in the formation and the

evolution of the bar since changes in gaseous structures can have a direct impact on

radial mass transfer. Moreover, if magnetic fields change the SFR, they can also have

a significant effect on the bar formation.

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요 약

막대 구조는 나선 은하에 일반적으로 존 그렇기 때문에 막대 구조의 진화에 대한 이해

는 원반은하의 전반적인 이해에 매우 중요한 요소이다. 많은 기존 연구들이 막대 은하의

진화에 대한 연구를 수행했지만, 핵고리의 형성 및 진화, 핵고리에서의 별형성의 분포와

시간에 따른 변화, 막대의 형성 및 진화에 기체가 미치는 영향 등의 이해가 부족한 부분이

여전히 존재한다. 본 학위 논문에서는 별 형성과 되먹임 (feedback) 효과가 포함된 기체역

학 수치실험과 자기 일관성이 있는 (self-consistent) 수치실험을 이용하여 막대나선은하의

전반적인 진화에 대해 연구하였다.

먼저 나선팔이 없이 막대만 존재하는 은하의 핵고리에서의 별형성에 대해 연구하였다.

이연구에서는막대포텐셜의영향으로만들어지는핵고리에서의시간적,공간적별형성의

변화에 대해 이해하기 위해 기체역학 수치실험을 이용하였다. 별형성 알고리듬은 별형성

발생을 제어하는 밀도 한계와 별형성률을 고려하였고 기체가 별 입자로 전환이 된 후 시

간차이를 두고 초신성 폭발을 통해 되먹임 효과가 발생하도록 설정하였다. 핵고리에서의

별형성률은핵고리자체의질량보다는핵고리로유입되는기체의유입률에의해결정된다.

일반적으로 폭발적인 첫 번째 별 형성과 뒤따르는 상대적으로 규모가 작은 폭발적 별형성

이 발생한 이후 별형성률은 매우 낮아진다. 최초의 폭발적 별 형성은 막대의 성장에 의해

기체가 급격히 중심영역으로 유입되기 때문에 발생하는 것이고, 뒤따르는 작은 폭발들은

첫 번째 별 형성으로 인해 발생하는 되먹임 효과에 의해서 원반 바깥쪽으로 밀려났던 기

체가 다시 중심부로 유입되면서 발생하는 것이다. 하지만 몇몇 은하들에서 수십억년 동안

지속적으로, 혹은 주기적으로 핵고리에서 별 형성이 발생하는 것을 보여주는 관측 결과와

달리 우리의 수치실험 결과에서는 수억년이 지난 후에는 별형성률이 매우 낮아진다. 이

는 막대영역 내부의 기체가 별형성에 의해 대부분 소진되기 때문이고, 별형성률이 긴시간

유지되기 위해서는 막대의 영향 이외에 추가적인 기체 유입을 일으키는 기작이 필요함을

시사한다. 별형성률이 작을 때는 대부분의 별 형성이 핵고리와 먼지띠가 연결되는 영역에

서 발생하고 새롭게 형성되는 별들이 고리를 따라 회전하기 때문에 방위각 방향을 따라

젊은성단들의나이가증가하는현상을보인다.반면에별형성률이클때는별형성이고리

전체영역에서폭발적으로발생하기때문에젊은성단들의나이가특별한경향성을보이지

않는다. 고리의 크기가 시간이 흐르며 작아지기 때문에 성단들은 방사방향으로도 나이가

많아지는 경향성을 보인다. 성단 밀집 함수는 멱법칙을 잘 따르고 멱법칙의 기울기는 SFR

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에 영향을 받는다.

다음으로 나선팔의 존재가 핵고리의 별형성에 미치는 영향을 연구하였다. 이를 위해

막대만 존재하는 모형과 동일한 수치실험코드를 이용했고 막대의 바깥쪽 영역에 나선팔

포텐셜을 추가하였다. 나선팔의 회전속도가 은하의 회전속도와 일치하는 동시회전(co-

rotation) 반경 안쪽에서는 기체가 나선팔을 지난 후 각운동량을 잃고 안쪽으로 움직이기

때문에 나선팔의 회전속도가 막대의 회전속도보다 느린 모형에서는 나선팔의 효과에 의

해 막대영역으로 추가적인 기체가 유입된다. 나선팔이 강하고 느릴수록 나선팔의 효과에

의해 별형성률이 더 많이 증가하고, 나선팔이 없는 경우에 비해 3–20배 커진다. 추가적인

기체의 유입은 먼지띠의 밀도 또한 증가시킨다. 고리의 크기 또한 나선팔의 존재에 영향을

받는다. 느린 나선팔을 가진 모형의 경우 나선팔이 없는 모형에 비해 고리의 크기가 적게

줄어들어서 크기가 약 45% 정도 더 크다. 나선팔이 존재하는 모형의 경우에도 별형성률

이 작은 경우에만 젊은 성단들의 나이가 방위각 방향을 따라 커지지만 고리 크기가 덜

작아지기 때문에 방사방향의 나이변화는 특별한 경향성을 보이지 않는다.

마지막으로 별 입자와 기체입자가 모두 포함된 자기 일관성이 있는 수치실험을 이용

하여 기체의 존재가 막대 구조의 형성과 진화에 어떤 영향을 미치고, 동시에 막대 구조의

변화가 기체구조의 진화에 어떤 영향을 미치는지 연구하였다. 우리은하와 비슷한 크기와

질량을 가진 은하 모형을 고려했고, 기체의 냉각과 가열 및 별형성 되먹임 효과를 포함시

켰다. 툼레 (Toomre) 안정성을 변화시켜서 두 집단의 은하 모형을 설정했고, 전체 원반을

질량을 고정시키고 각각의 집단의 기체함량률을 변화시켰다. 차가운 원반을 가진 모형들

은 기체함량이 높아질수록 막대가 더 빠르고 강하게 형성된다. 이와 반대로 따뜻한 원반을

가진 모형들은 막대의 형성이 기체의 함량이 커질수록 느려진다. 막대가 형성되면서 중심

영역의 질량 집중도가 높아지게 되고 이는 막대를 약화시킨다. 이후 막대는 다시 세기

와 크기 모두 천천히 증가한다. 따뜻한 원반을 가진 기체가 없는 모형에서만 휨 불안정

(buckling instibility)이 나타나는데 이는 다른 모형들의 경우 중심 영역의 질량 집중도가

상대적으로 크게 증가해서 별들의 수직방향의 움직임을 활발하게 만들기 때문이다. 막대

형성에의해중심영역으로이동한기체는핵고리를형성한다.핵고리는형성단계에서매우

작은 크기로 만들어지지만 이후에 상대적으로 각운동량이 큰 기체가 지속적으로 유입되

면서 시간에 따라 크기가 커진다. 고리에서의 별형성률은 되먹임 효과에 의해 주기적으로

변하고 전반적인 양상은 막대의 세기와 연관성을 보인다. 우리의 모형은 막대와 핵고리

142

양쪽 모두 우리은하와 비슷한 물리량을 보인다.

주요어: 은하:성간물질,은하:운동학과역학,은하:막대,은하:나선팔,은하:진화,은하:

별형성 및 되먹임, 방법: 수치실험

학 번: 2011-30128

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