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 Permanent Link:
 http://ufdc.ufl.edu/AA00058066/00001
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 Title:
 Numerical modeling of large N galactic disk systems
 Portion of title:
 Numerical modeling of galactic disk systems
 Creator:
 Davies, Chad Leslie, 1966
 Publication Date:
 1995
 Language:
 English
 Physical Description:
 200 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Angular momentum ( jstor )
Density ( jstor ) Galaxies ( jstor ) Galaxy rotation curves ( jstor ) Mass ( jstor ) Particle mass ( jstor ) Simulations ( jstor ) Star formation ( jstor ) Stellar disks ( jstor ) Velocity ( jstor ) Disks (Astrophysics) ( lcsh ) Dissertations, Academic  Physics  UF Physics thesis, Ph. D
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1995.
 Bibliography:
 Includes bibliographical references (leaves 187199).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Chad Leslie Davies.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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NUMERICAL MODELING OF LARGE N GALACTIC DISK SYSTEMS
BY
CHAD LESLIE DAVIES
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995
To my wife,
who stood by my side.
ACKNOWLEDGMENTS
I would like to begin by acknowledging a number of dedicated educators and researchers who gave of their valuable time to mentor me and develop within me qualities as a scholar and as a man: James A. Savard, who by his life and his encouragement taught me the principles of scouting that carry through to this day; Richard McReynolds, who kindled in me the appetite to ask questions and search for the answers; Dr. Gordon Wolfe, who taught me how to think like a physicist and see the beauty of the subject and Dr. James Hunter Jr., who patiently showed me how to conduct research and critically evaluate my own work. Without the time and patience shown by these men, I would never have reached this point.
Additionally, I thank the past and present members of my committee who have supervised my progress and encouraged me along the way. In the department of Physics: Dr. J. Robert Buchler, who, through his teaching me the fundamentals of electromagnetic theory, impressed upon me the level at which graduate work is done and challenged me to rise to meet to the standard. Dr. James Ipser, who showed me early on that I didn't really want to do relativity but encouraged me with many kind words nevertheless and Dr. Gary Ihas, who has shared with me his humor and relaxed perspective. In the Department of Astronomy: Dr. Henry Kandrup, who has provided a great deal of positive input on the subject of the inclusion of counterrotating angular momentum and reminded me of the limitations of numerical modeling; Dr. Haywood Smith, who has assisted me greatly in helping me understanding some of the numerical aspects of this study and Dr. Steve Gottesman who kept me from forgetting that there really are galaxies out their and who iii
asked the right questions at good times. It is said that one of the wisest decisions a Ph.D. student can make is to choose his committee well. Through no fault of my own, I believe I could not have done a better job
I would also acknowledge the assistance of several other individuals who have helped me along the way. From Greece; Dr. Nikos Hiotelis, for helping me, during the summer of 1991, understand the fundamentals of Smoothed Particle Hydrodynamics and Dr. George Contopoulos who took an interest in me and my work during his visits and provided an unparalleled example as to what a researcher is to be. Also, I acknowledge the discussions carried on with Drs. Dimitris Christodoulou, Clayton Heller, and Isaac Shlosman. I thank Nikos and Clayton for providing codes to the galaxy program at the University of Florida upon which this modeling effort is based and Dimitris for sharing with me an analytic framework in which to view much of what is happening in my models. Also, I wish to thank Ronald Drimmel for his invaluable help and fellowship along the way. His patient teaching and explanation has made the code understandable and accessible. More importantly, he has been a stalwart friend and a great encouragement to me though all stages of this work. Finally, I thank Keith Kerle and Donald Haynes for their friendship and willingness to listen to my various diatribes on the injustices of graduate education when the days were long.
Most of all, I thank my advisor, mentor and supervisor, Dr. James Hunter, who has invested an enormous amount of time and energy to bring me along to the level of respectable scientist from the state in which I began working for him. His example of critical analysis and skeptical thought has informed me of the responsibility of a scientist iv
to do and report work that is reproducible. As importantly, he has taught me the skill of looking at my own work with an eye to additional detail. He has passed along to me the essence of what it is to be a scientist and a researcher. He has also passed along to me the incalculable wisdom that, "Gravity, the pathological beast, by fiat, does the trick, fair enough."
I wish to express my gratitude to my parents, Henry and Toni Davies, who provided their love, moral support and endless encouragement when the going was slow and difficult and I doubted my ability to finish what I had begun. I also acknowledge the patience, love and support of my wife who has stood by my side and helped me in so many ways through my course of study. Kathy has shown me what unconditional love and dedication are by her patience with and support of this project that has stolen so much of our time. She has shown me that there is so much more to life than learning, knowledge and research.
I wish to thank the Department of Physics for providing teaching and logistical support over several periods of time during which a portion of this research was done. Also, I thank the Department of Astronomy for providing access to their exceptional computing facilities and an office with a dedicated workstation to complete this dissertation and its reported research. Additionally, I acknowledge the support of NSF grant AST9022827 and the Division of Sponsored Research at the University of Florida. Computational time for the numerical simulations reported herein was made available, free of charge, by the Research Computing Initiative, a cooperative venture of the University of Florida, the Northeast Regional Data Center and International Business Machines.
V
Lastly, I acknowledge the role of God Almighty in this endeavor, without whose strength I would never have finished this undertaking, and His Son, Christ Jesus, whose sacrifice at Calvary has redeemed mankind. '...ovr...r7t tXcr t...aAAa r7Nc01r 6
aairrc ~evpyovytvr' (Gal. 5:6)
vi
TABLE OF CONTENTS
ACKNOWLEDGMENTS ........... ...................... iii
LIST OF TABLES ........................... .. ........... x
LIST OF FIGURES .................................... xi
ABSTRACT ............................... ......... xiv
CHAPTERS
1. PHYSICAL MOTIVATION FOR THE STUDY OF GALACTIC DISK
SY STEM S . . . . . . . . . . . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . . . 1
Interacting System s ........................ ......... 3
Observations ...... ........ .. ..................... 3
Work on Galaxy/M31 Interactions ............ ............... 4
Previous Numerical Work on Encounters ...................... 6
Counterrotating Systems ............................... 8
O bservations . . .. ... . .. .. . .. .. . .. . .. . . 8
Previous Analytical W ork .............................10
K alnajs . . . . . . . . . . . . . . . . . . . 10
A raki . . . . . . . . . . . . . . . . . . . 11
Christodoulou et al ................................12
Previous Numerical Work .............................13
2.THE CODES .......................................16
Introduction . . . . . . . . . . . . . . . . . . . 16
Hierarchical Tree Algorithm .............................17
Smoothed Particle Hydrodynamics ......................... 21
Tim e Integration ........................ .. ....... ...35
M iscellaneous ... . .... . . .. .. . .. . .. .. . . .39
Tests of the Algorithm ................................41
Tests in One Dimension ..............................41
Tests in Two Dimensions .............................42
Tests in Three Dimensions ............................45
Specifics of the Codes Used .............................46
TN D SPH ......................... .............46
FT M . . . . . . . . . . . . . . . . . . . . .46
vii
3. INITIAL CONDITIONS ................................ 48
Introduction . . . . . . . . . . . . . . . . . ... 48
Observation of Isolated Galactic Disks ..... . . . . . . 49
A nalytical D isks.. .................. ..... ........ 51
General Discussion.................... .............51
Kalnajs/Hohl Disks ...................... ........ 54
Toom re D isks .. . .. .. .. .. .. . .. .. . .. .. .. . ..55
Other Initial Condition Formalisms ........................ 66
H ernquist . . . . . . . . . . . . . . . . . . .66
Fall and Efstathiou ...............................67
Calculation of Velocity Dispersion . . . . . . . . . . . . 67
Stability C riteria . . . . . . . . . . . . . . . . . .71
Toomre's Local Stability Criterion . . . . ..... ... . . . 71
OstrikerPeebles Parameter . . . . . . . . . . . . . . 76
Christodoulou's Parameter ............................78
G EN STD . . . . . . . . . .. . . . . . . . . . .79
4. NUMERICAL MODELING OF DISK SYSTEMS ................. 81
Introduction . . . . . . . . . . . . . . . . . . .81
Relaxation Effects ...................................81
Star Form ation ....................................85
Numerical M odels ..................................89
U nits . . . . . . . . . . . . . . . . . . . . 89
Toomre n=0 Disks .................................90
Toomre n=1 Disks .................................95
5. ENCOUNTER SIMULATIONS ........................... 103
Introduction ... .. . ... .. . .. .. . .. .. . ... . 103
Stellar M odels .. . .. .. .. . .. .. . .. . . . .. . . 105
Massive Particle Encounters . . . . . . . . . . . . . 105
Dwarf/Disk Encounters ............................. 110
Disk/Disk Encounters .............................. 115
Stellar/Gas M odels ................................. 120
Massive Particle Encounter ........................... 120
Dwarf/Disk Encounter .............................. 125
Models with Star Formation ............................ 130
M assive Particle Encounter . . . . . . . . . . . . . 130
Dwarf/Disk Encounter .............................. 133
viii
6. COUNTERROTATING ANGULAR MOMENTUM ............... 137
Introduction . . . . . . . . . . . . . . . . . . 137
CR Angular Momentum Inclusion in the Initial Conditions .......... 138 Models for the Development of Systems with CR Angular Momentum . 140 Numerical Simulations ............................... 143
Fully M ixed Case Results ............ ............... 143
Step/Slope Function Case Results ....................... 145
Toomre n=0 disks .................. ............ 145
Toom re n=l disks ............................... 148
7. DISCUSSION OF RESULTS AND FUTURE WORK .............. 153
General Conclusions ................................ 153
Introduction . . . . . . . .. . . . . . . . . . 153
Numerical Integration of Toomre Disks .................... 154
Encounters and Local Group Dynamics .................... 155
Counterrotating (CR) Systems ......................... 157
Future W ork to be Done .............................. 158
Im provem ents ................................ .. 158
Continuing W ork ................................. 160
Questions Still to be Answered ......................... 161
Questions Raised ................................. 162
APPENDIX. ........................................ 164
BIBLIOGRAPHY .................................... 187
BIOGRAPHICAL SKETCH .............................. 200
ix
LIST OF TABLES
2.1 Period of oscillation for two dimensional Maclaurin disks . . . . 44 2.2 Properties of the Codes used in the Present Modeling Effort . . . . 47 5.1 Encounter models run ..................... ....... 105
6.1 The mode strength and pattern speed of the m=2 mode for Toomre n=0
and n=1 disks as a function of percent CR angular momentum in the fully
m ixed case . . . ... . . ... . . .. . . . . . .. 144
6.2 Results of the Christodoulou stability criterion check for Toomre n=0 and
n= 1 disks . . . . . . . . . . . . . : . . . . .145
6.3 Toomre n=0 models step function models investigated as a function of k
and JCR . . . . . . . . . . . . . . . . . . .146
6.4 Step/Slope function models for a Toomre n=l disk with JCR = O.1JT. 148
LIST OF FIGURES
2.1 An ordering diagram to determine the advancement of particles with
m ultiple tim e steps ............................... 38
3.1 A schematic diagram illustrating the mass calculation using homoeoids 59
3.2 The cold rotation curve and surface density for a consistently truncated
Toomre n=O disk ................................. 64
3.3 The cold rotation curve and surface density for a consistently truncated
Toomre n= 1 disk ................................ 65
4.1 The time evolution of a Toomre n=0 disk constructed to be stable to
nonaxisymmetric modes. ............................ 91
4.2 Plots of the rotation curve and surface density versus radius for a Toomre
n=0 disk at t = 40 tdyn .. ........................... 92
4.3 The time evolution of a dynamically unstable Toomre n=0 disk . . . 94 4.4 A plot of bar strength versus time for the simulation shown in Figure 4.4 95 4.5 The time evolution of a dynamically stable Toomre n=1 disk . . . 97
4.6 The time evolution of the gas component in the model displayed in Figure
4.5 without star formation ... . . . . .. 98
4.7 The time evolution of the gas component in the model displayed in Figure
4.5 with star formation ............................. 99
4.8 The star formation maps for the model displayed in Figure 4.7 . . 100
4.9 Plots of the rotation curve and surface density versus radius for a stable
Toomre n=1 disk at t = 40tdyn. ........................101
4.10 The time evolution of a dynamically unstable Toomre n=1 disk . . 102
5.1 Plot of particle positions for a positive interaction angular momentum,
purely stellar massive particle encounter . . . . . . . . . 108
5.2 Plot of particle positions for a negative interaction angular momentum,
purely stellar massive particle encounter . . . . . . . . . 109
5.3 A plot of particle positions for a purely stellar dwarf/disk encounter.
Interaction angular momentum is positive . . . . . . . . . 111
xi
5.4 A detailed plot of stellar particle positions for each disk individually for
the simulation shown in Figure 5.3 ...................... 112
5.5 A plot of particle positions for a purely stellar dwarf/disk encounter.
Interaction angular momentum is negative . . . . . . . . . 113
5.6 A detailed plot of stellar article positions for each disk individually for
the simulation shown in Figure 5.5 . ..................... 114
5.7 A plot of stellar particle positions for a purely stellar disk/disk encounter.
Interaction angular momentum is positive . . . . . . . . . 116
5.8 A detailed plot of stellar particles for each disk individually for the
simulation shown in Figure 5.7 ...................... 117
5.9 A plot of stellar particle positions for a purely stellar disk/disk encounter.
Interaction angular momentum is negative . . . . . . . . . 118
5.10 A detailed plot of stellar particles for each disk individually for the
simulation shown in Figure 5.9 . . . . . . . . . . . . 119
5.11 A plot of stellar particle positions for a massive particle encounter . 122
5.12 A plot of gas particle positions for a massive particle encounter prior to
perigalaxion .......... ........................ 123
5.13 A plot of gas particle positions for a massive particle encounter after
perigalaxion . . . ... . . ... . . .. . . . . . ... 124
5.14 A plot of stellar particle positions for a dwarf/disk encounter . . . 127 5.15 A plot of gas particle positions for a dwarf/disk encounter . . . . 128
5.16 A detailed plot of gas particle positions for each disk in the dwarf/disk
encounter shown in Figure 5.15 . . . . . . . . . . . . 129
5.17 A plot of gas particle positions for a massive particle encounter where star
formation is allowed ..............................131
5.18 A star formation map for the simulation shown in Figure 5.17 . . . 132
5.19 A plot of gas particle positions for a dwarf/disk encounter where star
formation is allowed ..............................134
5.20 A star formation map for the simulation shown in Figure 5.19 . . . 135 xii
5.21 A detailed plot of gas particle positions for each disk for the dwarf/disk
encounter shown in Figure 5.19 . . . . . . . . . . . 136
6.1 Schematic diagram illustrating the distribution of angular momentum in
the step function case .. ............................ 139
6.2 A graphic showing the function of percent CR angular momentum versus
radius in a slope function model . . . . . . . . . . 139
6.3 Evolution of an unstable Toomre n=0 step function case with 50% of the
angular momentum counterrotating . . . . . . . . . . . 147
6.4 Isodensity plots of Toomre n=l step/slope function model #1 prior to inner
bar reversal . . . . . . . . . . . . . . . . .... 150
6.5 Isodensity plots of Toomre n=1 step/slope function model #1 following
inner bar reversal ............................. ...151
6.6 Isodensity plots of Toomre n=l step/slope function model #3 . . . 152
XIii
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy.
NUMERICAL MODELING OF GALACTIC DISK SYSTEMS By
Chad Leslie Davies
August 1995
Chairman: James H. Hunter Jr.
Major Department: Physics
In this study, hierarchal tree numerical methods with a smoothed particle hydrodynamics (SPH) formalism are used to simulate galactic disk systems in two dimensions. An identification of a new axisymmetric evolution of Toomre family disks to exponential disks is made. Using two dimensional disks with gas and stellar particles to model nonmerging encounters, several scenarios are explored. By investigating the resultant gas inflows and star formation patterns, we suggest that interacting gas rich systems will have elevated gas inflow and star formation rates. In addition, a numerical study of the inclusion of counterrotating angular momentum in disk systems reveals the dynamical importance of such systems. Galaxies with significant portions of counterrotating angular momentum have been observed and, as such, a study of the properties of such systems may provide valuable information regarding the global parameters of such systems. By including counterrotating orbits, physical systems are achieved that are not possible to realize using fully direct initial conditions.
xiv
CHAPTER 1
PHYSICAL MOTIVATION FOR THE STUDY OF GALACTIC DISK SYSTEMS Introduction
The aim of numerical modeling of galactic disk systems is to investigate the time evolution of systems that, for practical reasons, can only be observed at one specific time. Such modeling must take into account the known physical principles that are relevant to the particular system and must handle gas components with a fair approximation of the continuum limit. If these conditions are met, such modeling can be used to gain insight into the effects of less well understood phenomena and to investigate systems that are not tractable through analytical study. For the techniques of numerical modeling to have any meaning in this context, a study involving such considerations must be guided by analytical understanding of simpler but related systems and by observational input. To neglect the former may produce physically unrealistic results and to ignore the latter may result in models that may have very little relation to the systems which occur in nature. This study attempts to meet both criteria by using modeling codes that are founded upon those physical processes that are understood and tested on systems that have analytical solutions in closed form. Additionally, these codes must use as input, initial conditions which are, in some sense, based upon observations.
It is well understood that galaxies exist, not in isolation, but rather in groups that may be gravitationally bound. As such, it is likely that such systems will undergo encounters, which will fundamentally alter their global evolution. This is true not only for close
I
2
encounters, but also for encounters that are more long range in nature. As the Galaxy is a member of the Local Group, with at least one gravitationally bound companion of comparable size, a relevant question to pose is whether one can deduce from the present state of the constituent systems anything about encounters between our Galaxy and other members of the group. One can hope to place constraints on the recent interactions by observing whether certain effects predicted by a numerical model of an encounter are consistent with observations. While it may seem that such a study may be restricted to attempting to reproduce only those processes that may have given rise to the present configuration of the Local Group, any insight into the general dynamical consequences of these processes will provide astronomers with a better understanding of similar processes taking place throughout the universe.
Furthermore, one may inquire as to whether stellar disk systems all evolve from the same types of initial conditions, thereby leading to unidirectional disks or whether these systems, by reason of initial influences during the formation process or by the effects of the above mentioned interactions, may have a significant portion of their angular momentum reversed. If such systems can form, then a very interesting question arises as to what dynamical effects this counterrotating angular momentum will have on the subsequent global evolution of the system. Also of interest is the question of what effect the initial configuration of the counterrotating angular momentum will have on stability. Given that such systems have been observed to exist, the relevance of such a study is of more than strictly academic interest. Important in this study is also the question of the role of mixing of the direct and counterrotating components of the system.
3
Interacting Systems
Observations
When one looks for galaxies, one finds that they are usually found in aggregations, or groups, of other galaxies of differing types and kinds. It can be argued that many of these groups are gravitationally bound and therefore are in association for dynamically significant times. As such, it can be shown with a simple mean path calculation that galaxies in even fairly low number density groups will be relatively likely to have encounters with one or more members over the lifetime of the group. These interactions are varied in nature and depend on a number of factors, such as the types of galaxies that are interacting. Therefore, one must consider the influence of such interactions on the structure and evolution of individual galaxies and on the structure and evolution of the group itself. This is especially important when one considers systems in very high number density groups such as the Coma cluster.
When a galaxy is observed, one might ask if there are any evidences of any such interactions. It is thought that one class of observed galaxies, those referred to as Irregulars, may be chiefly due to such interactions. In addition, several observed giant elliptical galaxies, found mainly near the centers of rich clusters, exhibit dense stellar regions that may very well be the nuclei of former cannibalized smaller galaxies. An interesting type of galaxy, the polarring galaxy, shows a mainly stellar disk, known as an SO disk, with a ring of gaseous material rotating in a plane tilted at an angle that is sometimes as great as 900 from the disk plane. Such systems are now widely thought to be due to an interaction between the SO galaxy and a gas rich system. Lastly, disk
4
galaxies in nearby interactions with other systems oftentimes show significant warping of their disk components.
In the Local Group, the question of interactions is an important one because both of the dominant, large spiral disk galaxies, the Milky Way and Andromeda (M31), are known to have several companions. In addition, The Milky Way and M31 are known to be travelling towards each other and may be gravitationally bound. Thus, the two systems may play a dynamically significant role in each other's global evolution. Additionally, these large systems may have a dramatic impact on the evolution of the smaller satellite systems, such as the Small and Large Magellanic Clouds (SMC and LMC, respectively) and M33 and vice versa. Given this possibility, one asks what observable evidence may be left behind to point to such interactions having occurred, as the interactions discussed above are not obvious within the local group (such as a warp in the disk of the Milky Way).
Work on Galaxy/M31 Interactions
When it was determined that the Galaxy and M31 are moving towards each other at approximately 100 km/s the question arose as to whether these two systems were actually not just simply gravitationally bound within the context of the local group, but rather, orbiting each other. The problem was originally considered by Kahn and Woltjer (1959), who were able to put an upper limits on the mass of the M31/Milky Way system as a function of the Hubble constant. An important conclusion of this work was the possibility that the M31/Milky Way system had passed close to each other at an earlier time, thereby effecting the evolution of both systems. In addition, it brought into question what other
5
Local Group interactions may have occurred that might have left their mark on these two main constituents.
More recently, a group consisting of M. J. Valtonen, G. Byrd and their collaborators (Byrd et al. 1994, Byrd and Valtonen 1985, Mikkola et al. 1984, Valtonen and Byrd 1986, Valtonen and Mikkola 1991, Valtonen and Wiren 1994, Zheng et al. 1991, Valtonen et al. 1985, 1992, 1993) have done a detailed study of Local Group interactions using a reduced mass numerical technique. While these studies suffer from the unfortunate deficiency of not including the very recently observed spiral galaxy in our "zone of avoidance," they raise several very interesting points. One of these is that M31 may have arisen from a merger of two multiple (binary) systems. Valtonen points out that observational evidence may support such a scenario. He notes that such an encounter changes the orbital dynamics of a number of the Population II objects in an early stellar system. Due to the interaction, many of the Population II objects in the halo component of the system would be have their orbits altered towards plunging trajectories. This change would result in the destruction of a number of the globular clusters in the systems due to tidal effects near the core of the system and leave the resulting merger with a deficiency of clusters near the center of the system. Observationally, there is a deficiency of clusters near the center of M31 (Racine 1991). In addition, there should also be a notable spreading of the cluster distribution beyond the edge of the merger. The central deficiency noted here with the globular clusters should not occur with other Population II objects due to the fact that they will not be disrupted by tidal interactions. Observations by Ciardullo et al. (1989) show there to be no deficiency in planetary nebulae in the center of the M31
6
system. Lastly, Valtonen investigated how Population II metallicity would be affected by a merger. He argues that if the metallicity is determined by the initial mean distance of a cluster from its host galaxy, then plotting initial distance versus mean final distance for a cluster in the simulation should reflect the final radial metallicity distribution. When this is done, Valtonen finds that although the metallicity gradient remains, at any given distance the spread in metallicity is wider due to the new distribution of clusters via the encounter. Observations by Huchra et al. (1991) of globular clusters in M31 seem to resemble the numerical results found in Valtonen's study.
An important result of this work is that there may be ways to trace the effects of an encounter through the global and local dynamics of the systems involved, as well as through the characteristics of the populations involved. Hence, a question that might be addressed by numerical simulations is whether the effects of an encounter can be modeled in detail. If so, the results could be used to establish criteria for testing whether a disk system has undergone a recent encounter. One important limitation of the work of Valtonen et al. is that the fundamentally important effects of gas dynamics and star formation were neglected. On this topic, more will be said below. Previous Numerical Work on Encounters
As the subject of encounters and mergers is of great interest, a plethora of numerical work has been done to realistically model these phenomena. The initial work on the subject was done by Toomre and Toomre (1972) using a restricted three body scheme. They were able to produce bridges and tails in systems undergoing close encounters that closely resemble those observed in obviously interacting disk galaxies (e.g., NGC
7
4038/NGC 4039). More recently, Hernquist and Barnes (Barnes 1986, 1988, 1992, Barnes and Hernquist 1991, 1992, Hernquist 1992, Hernquist and Barnes 1991, Hernquist and Quinn 1989, Mihos and Hernquist 1994a,b) have investigated a number of disk/disk system mergers. Their work has shown that the luminosity profiles of merger remnants resemble elliptical galaxies (i.e., they follow a de Vaucoulers' r4 profile) but suffer from the difficulty that they are not as tightly bound as are observed giant elliptical systems. Nevertheless, this work has strengthened the hypothesis that giant elliptical systems are the results of mergers of several smaller systems. An important physical aspect of this work is that the models studied included particles representing gas as well as stellar particles. However, no attempt was made to model the process of star formation or to allow for energy loss due to cooling. Cooling must be of paramount importance, as the characteristic cooling times in molecular clouds are very short ( te 104 105 years). Given this fact, it seems likely that the systems investigated by Hernquist and Barnes would behave differently if cooling effects and star formation were included.
The subject of nonmerging interactions has been studied extensively by Byrd and his collaborators (Howard et al. 1993, Howard and Byrd 1990, Byrd and Howard 1990, Byrd and Klaric 1990, Byrd 1976, 1977, 1979a,b, 1983, Sundelius et al. 1987, Byrd et al. 1984, 1986, 1987, 1993, 1994, Byrd and Valtonen 1987) who have successfully modeled several interacting nonmerging disk/disk systems. Byrd has had success in matching detailed features observed in such systems by using a multiple encounter scenario for a captured lower mass companion. Additionally, some work has been done on the capture and orbital decay of low mass spherical dwarf galaxies by Lin and Tremaine (1983),
8
Quinn and Goodman (1986), Quinn, Hernquist and Fullager (1993), Mihos and Hernquist (1994a) and Mihos et al. (1995). These studies show that the capture of such systems and their eventual decay and destruction has the initial effect of generating a strong two armed response in the capturing disk. Over a longer time scale, the disk either spreads radially and inflates vertically and settles into a new axisymmetric equilibrium or can form a triaxial bulge or bar. The orbit of the satellite's core slowly decays into the disk plane where it is destroyed as it merges into the center of the disk. Here again, a notable deficiency of the initial conditions used for these simulations.is a lack of particles representing gas. The authors observe a significant thickening of the disk by the merger that may not occur if the gas cools efficiently.
Counterrotating Systems
Observations
The subject of counterrotating angular momentum in stellar systems and models was regarded as being of only academic interest until the observations of Rubin, Graham and Kenney (1992) showed that NGC 4550, an E7/SO galaxy, has a definite segregation of direct and retrograde angular momentum in its stellar disk. This system showed two cospatial systems of stars that are counterrotating. In addition, observations of the regular Sab barred spiral NGC 7217, along with deconvolution of the full lineofsight velocity distribution, show that thirty percent of the disk stars orbit retrograde (Kuijken 1993, Merrifield and Kuijken 1994). This population of stars shows little dispersion in the velocities and as such may be considered a kinematically discrete population within the system. Similar analysis applied to several other systems show that stellar line profiles
9
deviate from the assumed Gaussian by ten percent in M31, M32 and NGC 3115 (Rubin 1994b) and by twenty percent in NGC 4594 (van der Marel and Franx 1993) due to an excess of stars at low rotational velocities. While these systems have not been shown to possess counterrotating populations, the interpretation of the observational data (which has been cited as evidence for nuclear black holes) is called into question.
An even more bizarre and complicated system is found in NGC 4826. This Sab barred spiral (also known as M64, the Sleeping Beauty, etc.) has several distinct kinematic components. Data from the VLA (Braun et al. 1992, 1994) and optical observations (Rubin 1994a,b) present the following picture. The disk is comprised of inner and outer gas disks that are essentially coplanar. In the inner disk, gas and stars rotate in the same sense with trailing spiral arms. This inner disk includes a prominent dusty lane. For slightly larger radii, observations show that the ionized gas shows a rapid, orderly fall from 180 km/s prograde to 200 km/s retrograde. In this region, which is approximately 500 pc wide, the gas shows a radial velocity component of over 100 km/s towards the nucleus. However, the stars in this region continue their prograde rotation. Beyond this transition region, there does not seem to be spiral structure, but rather weak luminosity enhancements that appear to be more like circular arcs. Here the neutral hydrogen continues to rotate retrograde while the stellar velocities are prograde with respect to the inner disk.
Finally, an interesting set of observations have been made by Wozniak and Friedli (Wozniak et al. 1995, Friedli et al. 1995). that show several barred spirals with secondary or nuclear bars. These nuclear bars seem to have a different pattern speeds and kinematical
10
properties than the larger bars they lie inside of. Work by Friedli & Martinet (1993), Shaw et al (1993) and Davies and Hunter (1995b) have shown that it is possible to produce these numerically. In addition, Zasov (1995) has reported that his observations of NGC 497 show an inner counterrotating region with isophotal structure indicative of an nuclear bar. This system has a constantly rising rotation curve, but is lacking a large bar. Previous Analytical Work
While this subject has been, until recently, regarded of only academic interest, some previous analytical work has been done. The earliest general work that has been done is that by Kalnajs (1976). Very recently, a great deal of work has been done by Christodoulou and his coworkers (Christodoulou et al. 1995a,b) regarding the determination of a global stability parameter for rotating systems that has direct bearing on the problem. Also of interest is the work of Araki (1987) who examined the twostream gravitational problem.
Kalnajs
In the third of a series of papers devoted to discussing the determination of a distribution function for a family of flat, rigidly rotating disks, Kalnajs (1976) discusses the dynamical effects of the inclusion of a significant amount of counterrotating angular momentum. In this work, models with large eccentricity and/or retrograde orbits within the central disk regions were examined to determine if bar (m=2) instabilities could be suppressed. The models examined include the Toomre n=l disk (Toomre 1963) which have had their distribution functions parameterized through the use of both a Fourier
11
transform and Mellin convolution via inverse Laplace transform. Thus, the amounts of both ordered rotational kinetic energy and dispersional ("heating") energy can be varied to produce models of differing distribution functions, which produce the desired surface densities. In the noncircular cases, Kalnajs found a nonvanishing mean rotational velocity near the center of the models. He argues that this is to be expected as high eccentricity orbits will be at perigalactica in this region and, consequentially, their tangential velocity components will be the largest and in the same direction. As this is a general effect, the only way to force to be zero is to admit retrograde orbits. In earlier work, (Kalnajs and Athanassoula 1974) it was suggested that to avoid the formation of bar instabilities the mean angular rotation rate in the central portions of a model must be reduced to approximately one half of the circular rate. One way to do this is to reverse the angular momentum beginning at the radius where rcirc=l and increasing the fraction of counterrotating angular momentum to 1/2 at r=0.
Araki
Of interest in this discussion is the work of S. Araki (1987) who investigated a stellar twostream system that is analogous to that studied in plasma physics. He found, in agreement with other investigators (LyndenBell 1967, Harrison 1970), that there is no twostream instability for an infinite homogeneous stellar system with Maxwellian velocity distributions. He notes that LyndenBell (1967) concluded that such instabilities are possible if the velocity distribution is nonMaxwellian. For uniformly rotating stellar disks, Araki analyzes the Kalnajs disk using a weighted distribution function in order to determine whether a twostream instability is present. By using both a linear analysis of
12
the dispersion relation and a shortwavelength perturbation analysis in the WKB limit, he shows that counterstreaming Kalnajs disks exhibit twostream instabilities for only five low order modes over a small range of an order parameter defined as:
V
k = (1.1) where V is the ordered rotational velocity of the cool disk and Vo is the rotational velocity of the disk supported entirely by rotation. This instability is strongest for the m=l mode and sets in for k > 0.707. The WKB analysis showed that only density waves for the m=0 and 1 modes would be supported by counterstreaming. For differentially rotating disks, Araki applies kinematic arguments to again assert that only the m=0 and 1 modes will support density waves. As such, he notes the results of Zang and Hohl (see below) (1978) wherein the inclusion of a greater percentage of counterrotating angular momentum leads to an increasingly prominent onearmed feature in the later stages of evolution.
Christodoulou et al.
In more recent work, D. Christodoulou, along with a number of collaborators, has begun investigating the possibility of deriving a global stability parameter of flat and three dimensional stellar (noncollisional) and gaseous systems. Prior to these studies, it had been common to use the semiempirical global energy stability parameter proposed by Ostriker and Peebles (1973). The OstrikerPeebles (OP) criterion (see chapter 3 for further description) is based only on numerical experimentation with little analytical basis. Moreover, as several counterexamples have been found, there has been a strong need for an analytically derived criterion. The OP criterion is based only on one integral of motion,
13
namely the energy of the system. As such, it is clear that a more complete consideration of the problem of global stability (as opposed to local stability as discussed by Toomre (1964, 1972)) is needed. Such is the approach of Christodoulou and his collaborators. By including both the energy and the angular momentum as conserved quantities in the derivation of a stability parameter, a, for uniformly rotating systems, they are able to analytically arrive at a criterion. By generalizing these results, Christodoulou is able to calculate a for systems that rotate differentially. In numerical experiments, it has been found that this parameter predicts global stability for those systems that violate the OP criterion as well as those that do not. As the counterrotating systems in the present research violate the OP parameter, it is important to ascertain whether they are accurately described using the Christodoulou criterion. The derivation of this parameter will be discussed in greater detail in chapter 3 and applied to the systems of interest in chapter 6.
Previous Numerical Work
Prior to the observations of Rubin et al., the subject of counterrotating angular momentum was largely regarded as academic and therefore, what little numerical work has been done is mainly an afterthought included in other studies. The earliest discussion of the topic was in a paper by Zang and Hohl (1978). In this work, they found that the inclusion of counterrotating angular momentum tended to stabilize a disk that would have been otherwise unstable to the formation of an m=2 (bar) mode. They noted, however, that they were unable to completely suppress the formation of a nonaxisymmetric global mode. They found that a strong m=l, lopsided mode formed in many of their models
14
(as was later explained by Araki (1987)). This effect was exacerbated as the number of counterrotating orbits increased, especially if those orbits were added in the outer portions of the initial disk. In their study the largest fraction of counterrotating to direct angular momentum was onefourth. The authors were numerically able to confirm many of the conclusions of Kalnajs that were based upon linear theory. In addition, Hohl and Zang found that when a disk was initially composed of entirely direct angular momentum, the formation and subsequent dynamical action of the bar mode reversed the angular momentum of approximately 5% of the orbits.
Since the announcement of the discovery of the peculiar dynamical arrangement of NGC 4550, there has been a flurry of work by several authors on this topic (Byrd 1992, Sellwood and Merritt 1994). All of these studies have been done concurrently with the work reported in this dissertation. The most notable of these is the work of Sellwood and Merritt (1994). Using a modified version of the Miller cylindrical grid code, the authors studied the inclusion of counterrotating angular momentum in several ToomreKuzmin (19??) three dimensional disks. This inclusion was done in a fully mixed fashion and the authors reported that they were able to completely stabilize an otherwise unstable disk and that they were able to generate disks with two barlike patterns rotating in opposite directions. However, a reported drawback of their study seems to be that the growth rates are dependant on their grid and, therefore, they have expressed caution in the interpretation of their specific results.
Finally, Byrd and his collaborators have investigated the effects of including counterrotating angular momentum in their encounter simulations (Byrd 1992). It has been
15
shown that the close encounter of a massive object and a stable disk galaxy will excite the bar mode in the system. Bryd has found that while the inclusion of counterrotating angular momentum will damp the growth of the bar mode somewhat, it will not suppress it entirely, regardless of the amount placed in the disk (up to onehalf of the original angular momentum). From this he concludes that he is not able to stabilize an unstable disk with a constantly rising rotation curve with the reversal of any number of orbits, thereby contradicting Sellwood and Merritt (1994) and Davies and Hunter (1995a,b). As will be discussed in chapter 7, this author is of the opinion that Byrd's results may be flawed due to the choice of his initial conditions.
CHAPTER 2
THE CODES
Introduction
To model systems such as those described in chapter 1, it is important to be able to approach, on some level, the complexity of a real system. To do this, a code must be able to incorporate the actual physical processes as well as model those processes with a large enough number of particles to separate discreteness effects from actual dynamical effects. In addition, it is desirable for the code to be implementable on available platforms, easy to modify to allow for the incorporation of greater detail and accept and produce input and output that is compact and meaningful.
The two codes used to model the systems of interest are both based on the heirarchical tree algorithm. The algorithm incorporates the physical considerations of Newton's law of gravitational attraction and the laws of fluid dynamics to evolve the systems of investigation. Selfgravity is built into the algorithmic structure of the code while fluid dynamic effects are simulated through the formalism of Smoothed Particle Hydrodynamics (SPH). These two concepts, the heirarchical tree algorithm and SPH, form the backbone of the modeling effort and will be discussed in some detail below.
It should be noted that the present author has done little of the coding in the realization of the algorithms in their present form. However, extensive testing has been performed on both codes to provide a strong level of confidence in the results produced. These tests will be described in below. Lastly, these codes were used to evolve the same sets 16
17
of initial conditions, resulting in very similar global evolutions of the initial conditions as well as their conserved quantities. Therefore, we are certain that the results produced are believable and reportable.
Hierarchical Tree Algorithm
At the heart of the computational scheme used herein is the need to, in a physically consistent way, treat the selfgravity of a large system of particles (2 x 104, say). To do this, a fully Lagrangian method has been devised which incorporates the advantages of previous particleparticle (PP) and mean field expansion methods.
Early studies of finite particle systems relied on particleparticle methods (sometimes referred to direct summation Nbody methods), wherein a straightforward sum of all other particles in the system was done incorporate forces due to selfgravity. While no approximations were made in these calculations, thereby enabling such codes to model systems over a great dynamical range, the time required by the CPU to integrate the model scales as O (N2) in the simplest implementations of the algorithms (von Hoerner 1960, Aarseth 1963). While refinements made on these simple schemes (Aarseth 1971via higher order integrator, Ahmad and Cohen 1973via multiple time scales) have reduced the dependance on N somewhat ( O(N16) for the Ahmad and Cohen scheme) it is still infeasible to integrate models of N Z 10, 000.
More recently (Aarseth 1967), an independent type of code was developed through the use of multipole expansions to solve Poisson's equation, thereby eliminating the need for extensive summation of individual particleparticle forces. Further work (van
18
Albada and van Gorkum 1977, van Albada 1982 and Villumsen 1982) has shown that if the expansions are truncated at a low order, the technique is highly efficient, scaling as ~ O(nN), where n is the number of terms in the expansion. Additionally, systems evolved using this method will be less collisional owing to the mean field nature of the solution to Poisson's equation. However, for a number of reasons, this approach is rather limited in the cases to which it may be applied. In order to use a low order expansion accurately, the system and the basis functions chosen for the expansion must share the same symmetry. Also, due to the above stated low collisionality, collisional systems (i.e., those including gas) cannot be modeled and twobody effects are suppressed. Finally, due to the anisotropy introduced by the truncated multipole expansion (McGlynn 1984), it is not a trivial exercise to establish a rigorous estimate of the error of such an algorithm.
The codes used in the present study use an even more recent combination of these two methods (Appel 1981, 1985; Jernigan 1985; Porter 1985; Barnes and Hut 1986; Barnes 1986; Hernquist 1987; Hernquist 1988; Hernquist and Katz 1989; Hernquist 1990; Heller 1991). The algorithm follows that of Barnes and Hut (1986). An n dimensional box is drawn around the system, n being the dimensionality of the system to be evolved. The center of mass and quadrupole moment of the mass distribution of the box or node are then calculated. For reasons to be discussed shortly, the dipole term is not calculated. This acts as the parent node for either an octal or quadratic tree for three or two dimensions respectively. (Hereafter, the discussion of the algorithm will be treated as if the model to be evolved is a three dimensional one. To convert to two dimensions, all octal terms should be replaced by quadratic terms.) This parent box is then divided into eight equal
19
boxes that then become subnodes to the parent node. For each subnode, the center of mass and quadrupole moment are then calculated and stored. This process is recursively continued until each mth level sub cell either has 0 or 1 particles in it. A force calculation may then be done by walking through the octal tree that has been constructed. For each node, a comparison is made between the separation of the mass distribution (henceforth referred to as a cluster) represented by the node and the point of calculation, d, and the size of the cluster, s. If s/d < 0, where 0 is a fixed tolerance parameter, the walk down the tree beyond the node is terminated. All particles in the tree below the specific node that meets the tolerance parameter are now included in a single term (or set of terms) in the force calculation. This process is done for each branch of the parent node and each subnode until the entire tree has been pruned. The force between two individual particles is calculated using a Keplerian potential
5 = mim2 (2.1)
r
where ml and m2 are the masses of the particles. Here and throughout this communication, G has been set to 1 for simplicity. For the calculation of the pair force between a single particle and a node, used when the fixed tolerance parameter is met, the potential is calculated using a multipole expansion about the center of mass of the cluster. It can be shown (Jackson 1975) that, in such an expansion, the dipole term vanishes and as such, the lowest order correction to the monopole term is given by the quadrupole moment tensor. Hence the potential will be given by M 11 (2.2)
 r 2 r2
20
where M is the mass of the cluster, r is the distance from the particle to the center of mass of the cluster, and Q is the traceless quadrupole tensor defined by Qij = mk(3xkiXkj rkSij), (2.3)
k
when evaluated with respect to the center of mass (Jackson 1975, Goldstein 1980). The codes used in this study allow the user to specify as an input compilation parameter whether quadrupole terms will be included. The merits and costs of doing so will be discussed in the test section of this chapter.
To avoid singularities in the force calculation at small separation distances, it is common to introduce a softening parameter into the Keplerian potential in an ad hoc manner. In most cases this modification has been along lines corresponding to a
1
Plummer density profile of the form 0 Oc (r2 + 2)2. This, however, may not be the best prescription for the solution to the problem. This is because of the fact that the acceleration derived from this function converges only slowly to the Keplerian value, compromising local spatial resolution. A better choice for the softening of gravitational interactions is to use a spline kernel. The splinesoftened form of the potential, 4 = mf(r), and acceleration, a=mrg(r), where f(r) and g(r) are given by
S2 1 ( 2 (I 4 +(I )5 70
[( ()u2 u + (k) ] + j, O
(r) = [( u 3 + ()U )5] + 1 < u < 2, (2.4)
r, u > 2,
1[3 5. )U + (1)U < n < 1, g(r) = + ( )u 3u~4 )U6 1 2, where u = (Gingold and Monaghan 1977, Hernquist and Katz 1989), f being the gravitational softening length. The advantages of this form are that the acceleration and
21
potential are Keplerian for r > 2e and that the kernel has compact support. This means that the kernel and its first derivative are continuous everywhere and that for r > 2e, the Keplerian limit is recovered and no trace of the modifying effect of the kernel is present.
Additionally, it has been shown (Romeo 1994) that if the gravitational softening is too large, artificial suppression of fluid dynamical instabilities may occur. As this is the case, we allow for the value of e to vary with each type of particle (i.e.gas, stellar, halo), and it is possible to allow the softening length to vary in time. This procedure is similar to that of varying the smoothing length parameter in the Smoothed Particle Hydrodynamics formalism and will be described in some detail below. An additional refinement to the hierarchical tree computation would be to allow for all particles within 2e of the point of interest to be resolved into individual particles. The drawback to this is that the code performance degrades in cases of high clustering, due to the increasingly large number of particles in the interaction lists.
Smoothed Particle Hydrodynamics
While it is assumed in our modeling that the stars may be treated as collisionless particles, any gas dynamic effects must be dealt with assuming collisionality. Consequently, representation of the laws of fluid dynamics in the code must be implemented such a way as to accurately model the continuum limit. We choose to do this using the formalism known as Smoothed Particle Hydrodynamics (SPH). Developed by Lucy (1977) and Gingold and Monaghan (1977) to circumvent the limitations of gridbased systems (namely that of the requirement of a high degree of symmetry), SPH represents
22
the fluid elements constituting a system with particles which are evolved according to the dynamical equations obtained from the hydrodynamic conservation laws in their Lagrangian form. As such, SPH is a fully Langrangian formalism that is easy to implement in three dimensions as there is no grid to constrain the global geometry of the system. Additionally, studies by Lucy (1977), Gingold and Monaghan (1980), Nolthenius and Katz (1982) and Durisen et al. (1986) have shown that SPH is less diffusive than most simple finitedifferencing schemes. The disadvantages of SPH are that it handles shocks using an artificial viscosity, thereby imposing a limited spatial resolution similar to that of grid schemes and it is not possible to represent an arbitrarily large density gradient with a finite number of particles. Still, as this method is Lagrangian in nature it complements the hierarchical tree algorithm described above.
This formalism must use a finite number of particles to approximate the continuum limit. SPH does this by assuming that the particle mass density is proportional to the mass density of the fluid, p. As such, the algorithm is able to estimate p as it is evolved according to the laws of hydrodynamics by keeping track of the local density of particles. Inherent in this assumption is the understanding that local averages of the pertinent physical quantities must be performed over nonzero volumes. In SPH this is done through a systematic smoothing of the local statistical fluctuations of the particle number. Within this assumption, the mean value of a physical quantity, f(r), can be determined through a kernel estimation according to
(f(r)) = W(rr';h)f (r')dr', (2.6)
23
where W(r) is known as the smoothing kernel, h is the smoothing length which specifies the extent of the averaging volume and the integration is over all space (e.g., Lucy 1977, Gingold and Monaghan 1977, 1982, Monaghan 1982, 1985). The kernel is normalized such that
W(r; h)dr = 1, (2.7) and
lim W(r; h) = 6(r), (2.8) h0
due to the fact that we require that as h 0, (f(r)) + f(r).
While a number of different kernels may be selected it is important to keep in mind what Monaghan (1992) calls the "Golden Rule" of SPH. This is: "... if you want to find a physical interpretation of an SPH equation, it is always best to assume the kernel is a Gaussian." The kernel used herein is based on splines (Monaghan and Lattanzio 1986, Hernquist and Katz 1989, Heller 1991), as they have proven to be computationally efficient. Since the numerical system is not continuous in distribution, Equation (2.6) is approximated by a summation,
(f (r)) = (j)W (rr'; h), (2.9) j=l
where the summation is over the number of collisional particles in the simulation. The error in estimating equation (2.6) using equation (2.9) depends on the disorder of the particles (Monaghan 1982) and is given by, for a spherically symmetric kernel, h2
(f(r)) = f(r) + c h2V2f + 0(h3), (2.10)
6
24
where c f u2h3W(u)du is independent of h (Monaghan and Gingold 1982). As such, f(r) can be replaced by its smoothed equivalent to within the error of the smoothing process, O(h2). The kernel employed in the codes used in this study is that proposed by Monaghan and Lattanzio (1985) defined by
1 1+()() ()()3, 0< L <1,
W(r,h) = rh3 () [2 ()] 1 < 2, (2.11) 0, > 2.
The advantages of using this kernel are that it has compact support, uses only those particles within 2h for the smoothed estimates of physical quantities, has continuous first and second derivatives and is, as stated above, accurate to second order. In two 10
dimensions, the normalizing constant is 77.h2
If we then assign a mass mj to each fluid element, equation (2.9) may be written
N
(p(r)) = mW (rr'; h). (2.12) j=1
According to Hernquist and Katz (1989), equation (2.12) may be interpreted in two distinct ways that may be thought of as being analogous to the meanings of computational gather and scatter operations. In the "scatter" interpretation each particle has mass that is smeared out over space according to W and h. The density at any point is found by summing the different contributions from the density profiles of the neighboring particles. This is the more traditional point of view. Another way of regarding this is with the "gather" approach where the particles are regarded as point markers in the fluid. Local physical quantities are then arrived at by sampling all particles within 2h of the point of interest and weighting the contribution of each by W. While these two different views
25
are indistinguishable if the smoothing length, h, is the same for all particles, the codes used for this study allow for each particle to have a differing value for h. The reason for this is that the local statistical fluctuations resulting from the kernel estimates are determined by the number of particles within 2h of a given point in space. For example, if a constant value of h were used, the estimates of the physical quantities of the system would be more accurate in regions of higher density than similar estimates in regions of lower density. Moreover, an algorithm employing a constant h would not take full advantage of the distribution of SPH particles to resolve local structures.
Given the above interpretations, we may generalize the SPH formalism to allow for a variable smoothing length. If we consider the scatter interpretation, equation (2.6) becomes
(f (r)) = W [rr', h (r) f(r) dr' (2.13) which leads to equation (2.9) being rewritten as
N
(p(ri)) = Z mjW(rij, hj), (2.14) j=1
where rij = rirjl (e.g., Gingold and Monaghan 1982, Nagasawa and Miyama 1987). For the kernel stated above, it can be shown that the interpolation errors are dependent upon h (Monaghan 1987, Hernquist and Katz 1989). In the gather interpretation, equation (2.6) may now be written as
(f (r)) = JW rr'; h(r)] f(r) dr, (2.15) from which we can shown that equation (2.9) becomes
N
(p(ri)) = E mjW(rij; hi), (2.16) j=l
26
where the summation now depends on hi rather than hj as in the scatter interpretation (e.g., Wood 1981,1982, Benz 1984, Loewenstein and Mathews 1986, Evrard 1988). Here, the interpolation errors are independent of h. However, as pointed out by Monaghan (1985), the total mass of the system will no longer be conserved when using the gather approach. The size of the resulting errors can be shown to be O0(h2) and, therefore, are consistent with the other errors in the system. Hernquist and Katz (1989) have shown that similar difficulties are present in the scatter interpretation and these are of the same order as those for the gather method. A more important drawback to either scheme is that the kernel is no longer symmetrical with regards to the contributions of one element to another. When this asymmetry occurs, as it will for all systems with a dynamical h, it manifests as a violation of Newton's Third Law for the fluid elements of the system. Therefore, to conserve momentum in the equations of motion, the formalism must be symmetrized in hi and hi.
There are two approaches used to symmetrize the formalism. The first involves using +h in place of hi or hj in the kernel, while leaving the kernel in its former form. Hernquist and Katz (1989), R. Drimmel and this author have experimented with this approach and have found it unable to handle systems in which large gradients or shocks are present locally. Additionally, Hernquist and Katz have found that such a correction is likely to introduce errors into the integration of the system. Alternatively, the codes used herein use a symmetrization of the smoothing procedure itself. This method, which may be thought of as a hybrid of both the gather and scatter interpretations, is given by
(f(r)) = f(r W rr h(r)] + W [rr', h(r)] }dr (2.17)
27
in analogy to equation (2.6). This can then be shown to give, in discrete form,
N
(p(ri)) = Z mi[W(rj, hi) + W(rij, hi)], (2.18) j=1
which will reduce to equation (2.9) if hi = hi. As this equation is merely a linear combination of the two interpretation, the errors introduced by the formalism will again be ~ O(h2). Steinmetz and Miller (1991) note that in their tests, this method of symmetrization shows a somewhat better damping of post shock oscillations in the fluid in the case of strong shocks. As can be seen, a final advantage to this method is that each particle will have its own well defined size, which simplifies the implementation of both tree construction and force evaluation.
When h is allowed to change with respect to time for each individual particle, there are two common ways the algorithmic implementation has been done. Hernquist and Katz (1989) note that the smoothing length should be proportional to the local interparticle separation, scaling as h oc n3 in three dimensions. Hence, a twostep approach is used to update h such that each particle interacts with a roughly similar and constant number of neighbors. If h' is the smoothing length at step nl, and the number of neighbors at that time is AJVn1, then h7 may be predicted by
1[1
h = A 1 (2.19) where NJV is an input parameter. Then the current number of neighbors NA" is found using the above calculated value of h?. If N" differs from AfV by more than some predetermined tolerance, then the smoothing length is corrected so that the number of neighbors falls within the tolerance. While this method is stable and compels all of the
28
smoothed estimates to be of roughly the same accuracy at all points, it suffers from the deficiency that it is rather computationally intensive.
A second method, proposed by Benz (1990), is to write
h(r) = (2.20) p(r)
where ( is a parameter of order unity. By differentiating this equation with respect to time, and then using the equation of continuity to rewrite the righthand side, we arrive at dh 1
t hV v. (2.21) di3
Using this equation, the smoothing length is then evolved dynamically as would be any other hydrodynamical quantity, thereby saving a great deal of computational time.
A drawback to both of these approaches is noted by Steinmetz and MUller (1991), who point out that neither scheme guarantees the smoothness of h and stability against large amplitude density fluctuations. As h is dependent on the density, an amplification cycle can then occur wherein the small density fluctuations then amplify the formerly small fluctuations in the smoothing length and so on. They suggest a somewhat different way to evolve h. As the smoothing length can be too strongly dependant on the local density in simulations wherein the distribution is very lumpy, Steinmetz and MUller suggest taking a ratio between the local density and some average density and coupling h to that quantity. By doing this, they report that density amplification does not occur and the ancillary benefit of greater numerical stability for hydrostatic configurations. This is especially true in those cases when artificial viscosity (see below) is not used. Finally, their scheme reduces the numerical diffusion of SPH and requires no further computing
29
time than that already required to run with the varying smoothing length algorithms of Hernquist and Katz or Benz.
The process of estimating gradients in SPH is fairly straightforward. By definition, (V f(r)) = ] V f(r )dr (2.22) where W = W rr, h(r)] + W [rr', h(r)] }. Integrating equation (222) by parts, we obtain
(Vf (r)) = f(r) VWdr, (2.23) if surface terms are ignored. For discrete systems this may be rewritten as
N
f (rj)
(V f(ri)) = f(r , (2.24) j=a (n(rj))
where W = l[W(rij, hi) + W(rij, hi)] by analogy to W. From this point onwards, we will assume that index j denotes a particle label and that the summation is over all SPH particles. Therefore, we will also say that particle j has mass mj, position rj, density pj and velocity vj. It is important, at this point, to use what Monaghan (1992) refers to as the second "Golden Rule" of SPH. This rule is that it is better to write formulae with the density placed inside of the operators. Thus, we write (p(r)Vf(r)) = (V(p(r)f(r))) (f(r)Vp(r)) (2.25) which becomes, in the discrete case,
N
(p(r)Vf (r)) = mj [f(r) f(r)] VW. (2.26) j=1
30
Divergences may be written, in discrete formalism, as
N
(p(r)(V f(r))) = Imi [f(r) f(r) VW, (2.27) j=1
where the gradient of the kernel is taken with respect of the coordinates of the unprimed particle. Curls may be taken in a similar way.
Given this procedure, it is now possible to derive smoothed forms for the hydrodynamical conservation laws which may then serve as equations of motions for the particles. If equation (2.18) is used to compute the density, then to terms of O (h2) the continuity equation will automatically be satisfied and, as such, will not need to be integrated forward in time. To evolve a particle in phase space, we use the traditional form given by Euler's equation,
dri
dr= vi, (2.28) di
dvi = lVP + atisc VPi, (2.29) dt pi
where Pi is the gravitational potential, Pi is the pressure and aysc is an artificial viscosity term, used allow for the simulation of shock waves in the medium. A number of forms may be used for the smoothed estimate of V; however, it should be noted (Monaghan 1992, Hernquist and Katz 1989) that whichever form is used should be symmetrized to allow for the construction of a consistent energy equation and to conserve linear and angular momentum. Monaghan (1992) suggests an arithmetic mean be used, i.e.,
= , + (2.30) P P Pi
31
While this solves the difficulties mentioned, Hernquist and Katz (1989) note that this method can lead to instability in the integration of the thermal energy equation when a leapfrog integrator is used. Consequently, they suggest a symmetrization based on the identity,
VP
2 \P(V,) (2.31)
p
which, via the formalism, yields
N
= mj2 VW. (2.32) Pi PiPj
j=1
The introduction of an artificial viscosity is necessary for the code to accurately treat shocks in the fluid medium (Monaghan and Gingold 1983, Monaghan and Pongracic 1985). A form suggested by Monaghan (1988) is
N
a iSc mjIIijVW, (2.33) j=1
where IIij is the viscous contribution to the pressure gradient. While a number of different expressions have been proposed for IIij, one that gives an accurate description of the fluid flow near a shock is given by II'j = IJj (2.34) Pij
i='r, +r] V rij < 0, ii = hi, + (2.35) 0, v4 rij > 0, where v =vi v, ci=(c+c,) the average speed of sound particles i and j, hij = (h+hand ,P7 = The first term in equation (2.34) is analogous to a bulk viscosity, whereas the second term, which is intended to suppress particle interpenetration,
32
is similar to a von NeumannRichtmyer artificial viscosity (Evrard 1988, Monaghan 1988). Typically, a 0.5, 13 1.0 and 7 ~ 0.01 to prevent numerical divergences (Hernquist and Katz 1989). Due to the drawback that this description of the viscosity introduces a large effective shear viscosity, we have included a switch (Benz 1990, Drimmel 1995b) by multiplying pij, (ij ijfij). The factor fij = (f+fj) where
2
f (V.v)si (2.36) (V. v)i + I(V x vli + 0.001j' and the curl can be evaluated, as described above,
N
(Vx v(r)) = mj(vi vj) x VW. (2.37) j=1
While this factor reduces the shear viscosity, it does so at the expense of not being able to vary the effective shear. Thus, this factor is not applied to the first term in the viscosity equation.
To close the system of fluid dynamical equations, an equation of state must be added. Three different equations of state may be selected in the codes. The first is the familiar ideal gas law. Written in a convenient form, P = (Y 1)pu (2.38) where u is the specific thermal energy and y is the ratio of the specific heats. A second choice is the polytropic equation of state, given by P = Kp'. (2.39) If 7  1 and K= c? then this equation becomes the isothermal equation of state, which is the third option available in the codes. It should be noted here that the polytropic
33
equation assumes that the system is adiabatic and, therefore, requires no description of the evolution of u. To evolve u, we use a thermal energy equation derived from the first law of thermodynamics,
du = PdV + Tds. (2.40) In this equation, V = is the specific volume and all nonadiabatic effects are included in the change in the specific entropy. The smoothed version of the thermal energy equation may be written (Hernquist and Katz 1989) as
dui = mj + Ii v, VW + , (2.41) dt PiP + 1 VV + j=1
where A accounts for the heating and cooling terms not associated with artificial
p
viscosity (Field 1965, Spitzer 1978).
One additional note involving the dynamical evolution of the fluid in SPH. Monaghan (1992) has noted that equation (2.28) may be rewritten in a variant of SPH known as XSPH (Monaghan 1989). In doing this, equation (2.28) becomes
N
dri
vi = Vi + e m W, (2.42) j=1
where e is a constant over the range, 0 < e < 1. While no dissipation is introduced using this variant, XSPH does increase the dissipation already present. However, it has proven very useful in the simulation of nearly incompressible fluids as it keeps the particles orderly in the absence of viscosity.
Finally, before we leave the subject of SPH, it should be noted that by introducing a smoothing length that is allowed to vary in both space and time, the formal form of the
34
SPH equations change. The equations will now include terms proportional to Vh and dh In this case, the correct form of the SPH equations is obtained when the continuum equations are multiplied by W and integrated over all space. These results must then be integrated by parts. This procedure can be simplified through the use of the following relations. In the scatter formulation, (dA d(A) W A dh(r') A =f dr d dt Oh dt + A W(v'v) Vr', hdr' (2.43) + (v'v) .Vr, A Wdr,
(VrA) = Vr(A) A OWVhr'h r)dr', (2.44)
while in the gather interpretation, (dA d(A) dh(r) f W =) Adr dt/ dt dt I O (2.45) + [(v'v) VrA] Wdr',
f OW ,
(VrA) = Vr(A) Vrh(r) A h dr (2.46) I Oh
(Hernquist and Katz 1989). For the mixed formalism used to symmetrize the kernel, the corrections to SPH will be a linear combination of these. From this point, the dynamical equations are obtained in the prescribed manner. Typically, the additional terms acquired due to these extra considerations are smaller that the dominant, physical terms by powers of N3 in three dimensions and, therefore, will be unimportant for runs of large N (Gingold and Monaghan 1982, Hernquist and Katz 1989, Monaghan 1992).
35
Time Integration
The codes used in this study use a standard leapfrog integrator to update the particle positions and velocities. This integrator is accurate to second order in the time step, At, and has the advantage that in the formal limit as At 0, it preserves the Hamiltonian character of the system. For particle i, the positions and velocities are updated according to
n+i n3
r 2 ri 2 + Atv + O(At3), (2.47) vi+1 = vn + Atan+2 + O(At3), (2.48) where the superscripts refer to the time step at which the quantities are computed. For the SPH particles in the simulation, the acceleration, avsc, is dependant upon the velocity of the particle. In this case second order accuracy is maintained by updating the velocity in two stages. Initially, a predicted estimate of the velocity is calculated using
i+= v + 0.5Ata (2.49) n+1
This predicted value is then used to calculate the timecentered acceleration, ai This is then used to update the velocity according to equation (248). The use of a higher order integrator is not appropriate here since the long range gravitational forces are computed using a multipole expansion that is only carried out to the first term. A higher order integrator would only amplify the random noise in the acceleration computation. The leapfrog does not suffer from this difficulty as it computes the acceleration only once per time step (Hernquist and Katz 1989).
36
If the thermal energy equation (2.41) is to be used in conjunction with an ideal gas law equation of state (2.38), then it is necessary to advance the thermal energy along with the positions. To avoid complications with the Courant condition (see below), the thermal energy equation is evolved semiimplicitly using the trapezoidal rule, n+1 n1 At .n+ nu2 = u + 2ui2 + unT) + O(At3), (2.50)
similar to the approaches of Lucy (1977) and Monaghan and Varnas (1988). equation n+1 n+.!
(2.50) will be, in general, nonlinear in ui depending on the form of ui 2, which must be solved iteratively. If the righthand side of the thermal energy equation depended only on the nonadiabatic terms, the solution would be relatively straightforward. However,
1n+1
since the equation depends on u 2+ through Pj, which involves the neighboring particles, a rigorous solution of the equation would require the solving of two coupled nonlinear equations for each particle neighbor pair. Instead, an approximate two step approach is used. This method, which is equivalent to a secondorder RungeKutta integrator, first makes a prediction of the thermal energy by solving the thermal energy equation, implicitly assuming that the first term on the righthand side of the equation doesn't change from step n 1 to step n + I. The predicted value is then used, along with the +
predicted velocity, to solve for ui Finally, the thermal energy equation is again solved implicitly using both the predicted and old adiabatic and viscous terms. The solution of the nonlinear equation in both the predictor and corrector portions of the algorithm is done using a hybrid of the NewtonianRaphson and bisection techniques (Hernquist and Katz 1989, Casulli and Greenspan 1984).
37
The issue of time step size in an explicit integration scheme, such as the leapfrog integrator, is one of some interest. For the SPH particles, the time step is limited by the Courant condition. If the time step is fixed, the stability of the integration can be assured by varying the smoothing length according to the Courant condition (Evrard 1988). However, as this may limit the algorithm's ability to resolve regions of high density accurately, the codes allow for each particle to have its "own" time step. These are chosen so as to maintain integrational stability, according to the modified criteria derived by Monaghan (1988a). For the artificial viscosity scheme used herein, this may be written as
hi
At = C i (2.51) hijV vi + ci + 1.2(ac + 3maxjjzij i)' (2.51) where C~0.3 is the Courant number. If r or A in the thermal energy equation are nonzero, then additional constraints must be placed on the Courant condition if the time scale associated with these quantities are smaller than the dynamical time.
In simulations where the handling of multiple time scales is necessary, the limitation of having a single system time step is very inefficient. As such, the codes used herein allow for each particle to have its own time step. This is done in a manner similar to that of Porter (1985) and Ewell (1988). Each individual time step is chosen to be a poweroftwo subdivision of the system time step, At,, such that, Atk = , k = 1,2, ... ,n; (2.52) where k refers to the time bin level and Ats and n are input parameters that remain fixed (Hemquist and Katz 1989, Whitehurst 1988, Heller 1991). Each individual time step is
38
determined via equation (2.51), wherein the particle is placed within the largest time bin (smallest value of k) so that
Atk < Ati, (2.53) for particle i. All of the particles of a particular bin are advanced together; the ordering being selected according to Figure 2.1.
0 Bin 8
1 4 12
2 2 6 10 14
3 1 3 5 7 9 11 13 15 Figure 2.1: An ordering diagram to determine the advancement of particles with multiple time steps
Particles are allowed to move to a smaller time step (larger value of k) at the end of their own time step, but are only allowed to move to a larger time step if the two bins are currently time synchronized. This insures that the system will be synchronized after each set of 2n+' 1 time steps. The time step used to advance the particle positions and thermal energy equation is
Ats
Atpos = 2kmAx+1, (2.54) 2km..+1
where kmax is the largest time bin (smallest time step) currently occupied by particles in the system. Therefore, Atpos is onehalf the smallest time step thus assuring that the pressure and gravitational acceleration is computed at the midpoint of each time step as required by the integrator. Additionally, the most recently updated value of the velocity
39
is used to update the positions. If a particle changes the time bin it is in, its position must be updated using an estimated midpoint velocity in order to preserve the accuracy of the integration scheme. This is found using (Hernquist and Katz 1989, Drimmel 1995b) ri,cor = ri,uncor (1 1 + ao (2.55) where
Ati,old (2.56) X= Atinew
The same procedure may be used to make the leapfrog integrator selfstarting (Hernquist and Katz 1989). If the positions and velocities of the particles are given upon entry into the integrator, then a secondorder estimate of the positions may be obtained from
n+/ At n At' 2.7
r. = rn + v + a.
r,~ =ri 2 V 3 8ai (2.57) These additional terms guarantee secondorder accuracy in the leapfrog integrator without need of higher order integration schemes.
Miscellaneous
As SPH is formally similar to other simple Nbody methods, it is well suited for the simulation of systems comprised of both gas and collisionless matter. When this is the case, the phase space coordinates of all the particles are evolved using equations (2.48) and (2.49). For collisionless (i.e., stellar) matter, the acceleration is comprised of only the contributions due to gravity and, as would be expected, there is no need for the thermal energy equation. Additionally, since these algorithms solve the hydrodynamic equations
40
in an ab initio way, the correct continuum limit will be approximated and those physical processes that depend on the thermodynamic state of the gas can be rigidly specified. An example of this is the inclusion of accretion of gas onto a large central body (Drimmel 1995a,b, Heller and Shlosman 1994). Another example is the inclusion of star formation processes in the simulation. It has been shown by Heller and Shlosman (1994) that such effects are of vital importance in modeling the gas dynamics of the system. However, it is not clear as to how such processes are to be included in the modeling algorithm as the physics of the actual process in the interstellar medium is not well understood. An attempt to model such effects has been made and will be discussed in greater detail in chapter 4 of this dissertation.
Optimization of the algorithms used (TNDSPH (Drimmel 1995a) and FTM (Heller 1991)) is achieved on several levels. The greatest improvement in the speed of the code is achieved through vectorizing the tree descent (Makino 1990, Hemrnquist 1990) and tree construction (Makino 1990, Hernquist 1990). The methods used result in an improvement of speed of 10 and are not specific to an SPH implementation of the hierarchical tree method.
Some minor refinements of the codes involve two differing elements of the algorithms. First, the evaluation of the kernel functions (equations (2.4), (2.5) and (2.11)) is done using lookup tables to determine the values of the smoothing and softening kernels from 0 to 2u. Second, the necessary searches for nearest neighbors in the codes are combined into a single search and listing for each particle at the beginning of each time step. Thus, the CPU cost per time step is reduced at the expense of greater memory requirements.
41
Notwithstanding, unless the number of particles is very large, the codes will be limited in performance by CPU considerations, rather than those imposed by memory availability.
Tests of the Algorithm
Tests in One Dimension
A number of one dimensional tests were applied to the codes used in this study. The reason one dimensional testing has been used is that the test problems have analytical answers that may be compared to the output of the codes so as to judge accuracy. The two most commonly run tests are the shock tube test and the colliding gas flows test. It has been found by a number of authors (Monaghan and Gingold 1983, Monaghan 1989, Hernquist and Katz 1989, Heller 1991, Drimmel 1995b) that SPH does a fairly nice job of reproducing the analytical results over a large range of Mach numbers. One cautionary remark should be made here. It has been seen (Monaghan and Latanzio 1986) that SPH can have difficulty discerning structure in small regions with high densities (Hunter et al. 1986). This is due to a lack of resolution brought about by an insufficient number of particles needed to define a shock region. While some improvement can be made in the results of a specific system be modifying a and 3 in equation (2.35) (Heller 1991), generally optimal values for these parameters do not exist (Hernquist and Katz 1989). It should also be noted that when using the SPH algorithm, there will always be some postshock oscillation in the physical quantities, though these are reduced when using the version of dynamical softening put forth by Steinmetz and Miller (1991). Finally, Drimmel (1995b) has shown that to prevent interpenetration of particles in the colliding gas flows problem, it is necessary to employ an artificial viscosity having both linear
42
(a : 0) and nonlinear (3 0) terms. When using an artificial viscosity in conjunction with a polytropic equation of state Drimmel (1995a) has also shown that an effective cooling is introduced into the system. This effect can be lessened by using the viscosity switch given in Equation 37. Tests in Two Dimensions
We have done a two dimensional test of the algorithms by analyzing the small oscillation period of a Maclaurin disk. The oscillation properties of a two dimensional nonrotating disk may be found from analyzing the system using the two dimensional virial theorem,
1 d2I
2 dt2 2KB = PV + PV Q, (2.58) where, for a two dimensional disk, GMD
Q = fi (2.59) RD
is the angular potential energy of the system and I = f2MDRD (2.60) is the moment of inertia of the system and fj and f2 are factors of order unity that are dependent on the geometry and mass distribution of the disk. The other terms in the equation can be rewritten as P P P
PV = P M M P ( MD, (2.61) p v(r)6(z) Po Vo/ using a polytropic equation of state,
1 dR)2
KB = MD t (2.62)
2 dt
43
and, assuming the surface terms to be negligible, Ps V 0. (2.63) If we now introduce a small perturbation, c, into the system so that R = Ro(1 + ), (2.64) then the virial theorem may be rewritten as P0 GMD f2MDR 2E = 2(1 + C)2(71)D flaG (2.65)
0 Vo RO (.5 when terms of second order or higher are dropped. Expanding the first term on the right side using a binomial expansion and noting that, in equilibrium, 2 MD = f1 equation (2.65) may be simplified to read E + 2' GMD= 0. (2.66) f2 2 ji 9
Finally, if we assume oscillatory solutions of the form, c = coeGet, then the solution is easily arrived at and given by
1
f GMD 3(
S 2  (2.67) f2 R3 2
As can be seen, the solution becomes physically meaningless for Y < 3. For values less than this critical 7, the system is unable to support its own weight and will collapse beyond the linear regime. Note, this derivation is true for all disks if the factors fj and f2 are known. Thus in two dimensional calculations, = has the same significance as does = in three dimensional problems.
44
For a Maclaurin disk, it can be shown that fi = and f2 = and, therefore, the period of oscillation will be given by 2ir
T = 2. (2.68)
2 R, 2
Simulations show the system undergoing regular period oscillation. The analytical values for several values of t are given in Table 2.1 along with the values produced by the codes. As can be seen, the SPH codes reproduce the analytical result to better than
Table 2.1: Period of oscillation for two dimensional Maclaurin disks
_ Tanal (yrs) Tcode (yrs) 2 2.53 x 108 2.55 x 108 1.75 3.58 x 108 3.85 x 108 1.6 5.667 x 10 5.68 x 108
1%. Given that these simulations were done with only 5000 particles, we see that with even a relatively small number of particles, SPH is able to very accurately reproduce the behavior of mildly unstable systems. It should be noted that due to the effective dissipation introduced by the artificial viscosity, this oscillation is damped out over time.
For a system that is rotating, with a the virial theorem can be written as
fiMDR 2 = 2f2D ) 2kRo (7 2)e, (2.69) 2Ro 2 where kRo = 3MDRiw. Again assuming oscillatory solutions, the solution is
2
2 =W [ 2kRo ( 2)
WR = QVR[1 3) (2.70)
45
or
WR 12t 2)] (2.71) where WNR is the oscillation frequency of the nonrotating case and t is the OstrikerPeebles (1973) ratio of rotational kinetic energy to gravitational potential energy. It should be noted that the system will only be stable over a range of y and t such that
1 (7 2)
 > 2 (2.72)
2 2)
22
Simulations have been done to test the validity of the code in this case and the analytical results were found to be reproduced as long as the system stayed within the linear regime. As is discussed below, if t :S 0.14 the system will be stable against the formation of nonlinear modes.
Tests in Three Dimensions
Tests of the code have been performed in three dimensions by investigating the radial collapse of initially isothermal gas spheres (Hernquist and Katz 1989). The results were found to be in excellent agreement with those obtained using finite differencing (Thomas 1987) and particlemesh (Evrard 1988) schemes. This is an excellent test of the code due to the large range of smoothing lengths and time steps needed in the problem. Another test that has been conducted (Heller 1991) is very similar to the two dimensional tests conducted above. Here, Heller simulated the radial oscillations of a n=1 polytrope (Chandrasekhar 1967) and found that the code was able to accurately reproduce the expected pulsation frequency. Hence, we are confident that the algorithm presented here can accurately model physical phenomena in fully three dimensional settings.
46
Specifics of the Codes Used
TNDSPH
This code was developed from the TREECOD hierarchical tree algorithm provided by L. Hernquist (1987) to our group. The SPH implementation was done by R. Drimmel (1995a) with assistance from N. Hiotelis. Written to run on an IBM VMS platform, the code is fully vectorized and has separate two and three dimensional versions. Much of the initial work in this modeling program was done with this version of the code, and it is mainly used, at this time, to confirm results produced by the FTM code. The main properties of this code are listed in Table 2.2. FTM
Provided to our group by C. Heller (1991), the FTM code was originally written as a C implementation of the hierarchical tree/SPH algorithm. The code is now configured to run on a number of UNIX platforms and is written in FORTRAN. This code is used for the bulk of the modeling program at this time due to its more flexible output and inclusion of star formation routines. The code allows for a number of switches to be set at run time, further enhancing its flexibility. The code is further detailed below in Table 2.2.
47
Table 2.2: Properties of the Codes used in the Present Modeling Effort Property TNDSPH FTM SelfGravity Always Switch 2D/3D Kernel Switch Switch Multistep Always Switch Quadrapole Switch Switch Dynamical Softening None Switch XSPH None Switch Artificial Viscosity Switch Yes No External Potential Hard Coded Switch Accretion Yes Yes Star Formation No Switch Boundaries No Yes Output ASCII IEEE/IBM Binary Black Hole No Yes
CHAPTER 3
INITIAL CONDITIONS
Introduction
While it is vitally important to use computer algorithms that are both accurate and computationally efficient, it is just as important to begin a simulation with initial conditions that are physically meaningful and analytically tractable. In practice, this requires that the initial disk must be constructed from a set of meaningful assumptions that will give insight into observed physical systems. Additionally, it is important to be able to vary certain physical parameters of the stellar system, such as the first two integrals of motion (energy and angular momentum), in an easy, systematic way in order to be able to explore a parameter space of interest and relevance. Lastly, the set of initial conditions used should resemble the properties of real, observed disk systems or, if they do not, as in the case of the Kalnajs/Hohl (KH) disk (Kalnajs 1970, 1971, 1972, 1976a,b, 1977, Hohl 1970a,b, 1971, 1972, Hohl and Hockney 1969, Hockney and Hohl 1969), they should have other physically important properties that warrant investigation.
The construction of such sets of initial conditions has been referred to, by some, as something akin to an art. Certainly, it is true that a number of subtle techniques must be used to construct initial disks that are in radial and azimuthal equilibrium, easy to generate and that mimic the possible evolution of observed disk systems. As this topic is of some interest and importance, we choose to spend some time here on the details of disk building for numerical integration. A good review of the subject has been written 48
49
by Sellwood (1987) who, in addition to the fundamental considerations of the subject, discusses the techniques used to obtain quiet starts (i.e., those initial conditions that are sufficiently close to equilibrium that they do not produce undesirable dynamical effects during adjustment to "equilibrium").
Observation of Isolated Galactic Disks
When observing disk systems, a number of useful properties can be measured. Of greatest interest to this present study are a measurement of where the mass in the system lies, i.e., the density (stellar and gas), and how the mass moves, i.e., the rotational velocity curve. Additional useful information is the stellar velocity dispersions around the mean rotational velocity. As the measurement of these quantities is a complicated business, we will state only the general results with a few important notes to be made when necessary.
In measuring the density of the system, the issue is first complicated by the fact that the observations are usually made of disks that are inclined to the plane of observation, rather than face on. Even when this effect is accounted for, the observations will still only produce a surface density, E(r, 0), where 0 is the azimuthal angle. (E(r, 4) has units of mass/area.) To produce a volume density, an assumption must be made as to the distribution of matter in the z direction. Oftentimes, it is assumed that p(r, 0, z) = E(r, 4)g(z), where g(z) is some z distribution with units of length. A final complication is that there is an assumption of a proportional relationship, that is constant in all regions of a disk, between the density and the measured surface brightness of the system. While there is observational evidence in support of this assumption in the outer portion of the disk, there is also much that calls it into question when considering
50
the innermost region of disk or barred disk systems. A great deal more work needs to be done to sort out this problem in its entirety. Nevertheless, the results reported here are based on the claim that this assumption is true. Observations in the visible spectrum, as well as A 21 cm data, suggest that the surface density in many disk like systems may be fitted by an exponential curve in the r direction (de Vaucouleurs 1959, Freeman 1970). While it is entirely possible for the data to be fitted using other function (as will be discussed below), this seems to be the simplest function to use. While data for the determination of the z distribution of the mass is sketchier, a common assumption is that it may be described using isothermal sheets distributed perpendicular to the disk plane (Spitzer 1942, Bahcall and Soneira 1980). Another possible assumption is that this distribution may also be modeled using an exponential function in the z direction. Lastly, it should be noted that it is usually found that the gas component of a disk system has a greater physical extent in the r direction and a lesser extent in the z direction than do the stellar counterparts. Consequently, it is usually assumed that the two components of the system will have differing scale lengths.
To measure the rotation curve of a disk system, there are two main techniques. The first is to infer the rotational velocity of the stellar component by using the rotationally induced Doppler shift of particular spectral lines of the system as a function of r. A determination of the velocity dispersion may also be made in a similar way through measuring the broadening of the spectral line and assuming a distribution function (usually a Gaussian) to describe the stellar motions that give rise to this feature. This technique is quite difficult and the procedures used to separate the rotational velocity from the velocity
51
dispersions are subject to a number of assumptions. The second, somewhat more reliable, method is to make a similar measurement of the A 21 cm lines of neutral hydrogen gas. By measuring the gas, one also gains a clearer picture of the underlying potential of the system. Both effects, the greater reliability and the ability to probe the spatial variability of the potential, are due to the fact that the gas component is constrained to move along hydrodynamic streamlines. Therefore, the velocity dispersions of the gas will be much lower than those of the stellar component of the disk, which means that the measurements of the rotational velocity of the gas will be more accurate.
The observations of galactic disk systems reveal the following general information regarding the rotation curve. Near the center of the disk, the velocities rise linearly at a fairly rapid rate. This is. seen mainly in the stellar component of the disk, though A 21 cm and CO observations confirm this. At some radius, the rotation curve flattens out or in some cases turns over and begins a slight decrease. These data come mainly from A 21 cm observations because usually the gas disk extends well beyond the visible stellar disk. As will be discussed below, several different analytical fits can be made to the rotation curve data. Finally, it should be noted that the rotation curve data, again for reasons to be discussed below, is perhaps the most reliable and useful tool in studying disk systems.
Analytical Disks
General Discussion
To establish a set of selfconsistent initial conditions from which we may proceed, it is important to have a theoretical background from which to start. As these analytical
52
solutions must eventually correspond, in some sense, to observable systems, it is germane to the topic at hand to review the basic observational evidence in light of the theoretical framework. From this, we may extend what is available analytically by using justifiable assumptions to produce a set of initial conditions that meet the criteria set forth at the beginning of this chapter.
When considering selfgraviting systems, we begin from three fundamental suppositions. First, we assume that Newtonian mechanics, namely the second law, is the dominant consideration in understanding the dynamics of the system and its individual components (i.e., relativistic effects may be ignored). If we assume that the particles obey the second law and are confined to circular orbits (this constraint shall be relaxed considerably), then through the use of potential theory we can write
 = (3.1) dr r
where D is the gravitational potential of the system and r the radius from the center of the system to the point of interest. Second, we assume that we may represent the gravitational interaction via a smoothed potential function and therefore, Poisson's equation may be written as
V2(r, 4, z) = 4rGp(r, 0, z), (3.2) where p(r,q,z) is the volume mass density of the system. Third, we assume that there is a distribution function, f(r, 4, z, Vr, v, z, t), that will describe the individual particle motions of the system such that equations (3.1) and (3.2) are satisfied. It should be noted that while knowledge of f (r, 0, z, vr,, v, vz, t) will uniquely determine p(r, 4, z) and, therefore, v(r, z), the converse is not true.
53
Observationally, as previously discussed, we are able to determine reasonably good estimates of p(r, 0, z) and v6(r) as well as obtain a range of velocity dispersions for both the stellar and gas components of a disk. Using equations (3.1) and (3.2), we may formally eliminate D and in doing so relate p(r, 0, z) and v,(r). As these are both observationally determined quantities, it is then possible to determine one from the other. Unfortunately, however, it is not possible to determine the exact nature of the system potential (though a great deal can be inferred from studying the gas dynamic motion of the system) or, more importantly, the distribution function, f. Consequently, it is not possible to uniquely determine the system from observational data.
Given that this is the case, we are left with three options to establish workable initial conditions. One is that we can choose an analytical set of initial conditions for which a distribution function is known. This is the approach used in constructing the KH disk, which has a unique specification of its distribution function. While this approach has many advantages, its main drawback is that the system described rotates as a rigid body and, therefore, only resembles a physical disk system in the innermost regions of real galaxies. The second available option is to assume an analytical form for the rotation curve of the initial simulation disk and from this determine the mass distribution of the system. At this point, a distribution function may be selected and the initial disk built. The major drawback with this approach is that the disk will not be exactly in equilibrium and, therefore, will adjust early in the simulation. Nevertheless, following the work of Burbidge et al. (1959), Brandt (Brandt 1960, Brandt and Belton 1962), Mestel (1963), Toomre (1963) and Hunter et al. (1984), this is the approach we have chosen. Lastly,
54
one may assume a density distribution for the initial disk and from this determine the rotation law that will determine the motion of the particles in the system. Here again, one must pick a form for the distribution function and, almost invariably, the disk will begin somewhat out of equilibrium. This is the method of Fall and Efstathiou (1980) and Hernquist (1993b).
Kalnajs/Hohl Disks
In two separate series of papers Kalnajs (1970, 1971, 1972, 1976a,b, 1977) and Hohl (1970a,b, 1971, 1972, Hohl and Hockney 1969, Hockney and Hohl, 1969) investigated a disk characterized as an infinitely flattened Maclaurin spheroid. Given a density profile of =)2 A (3.3)
0 > 1,
we can derive, via the Poisson equation, the potential, 2(r) = 2, (3.4)
2
or, relating this to the velocity we can write = (3.5)
r
showing that these disks will rotate as rigid bodies. Additionally, a distribution function for this system can be derived analytically. This distribution function can be written as
f(E, Lz) = F F[(2 02)r2 + 2(E OLz)]1 [...] > 0, (3.6) S0 [...] (3.6) ,
where
F 27rR 2 (3.7) 27rRV 0 2
55
In analytical studies of these systems, Kalnajs (1972) found that they were axisymmetrically stable for random to rotational kinetic energy ratios greater than threehalves. Numerical studies (Moore 1991) have verified this conclusion. In addition, Kalnajs was able to show that the dominant mode in the system was an m=2 (bar) mode. This phenomenon had already been observed in the work of Hohl (1971) who had shown that disks with insufficient random kinetic energy, as parameterized by the local Toomre Q parameter (1964) (see below), developed strong bar responses. While this type of disk only approximately models the inner region of real disk systems, it points to the fact that the integrals of motion play an important part in determining the global evolution of the system.
Toomre Disks
In their 1959 communication, Burbidge, Burbidge and Prendergast detail procedure by which a mass determination of an observed, three dimensional disk can be made using the rotation curve. By using the equation of motion for the gas in the system, v2 Vp
 = v4q  (3.8)
r p
and assuming that a similar equation can be derived to describe the stellar motion in the disk (the socalled "stellar hydrodynamic" method) where (generally anisotropic) velocity dispersions will appear instead of the pressure term above, the authors are able to determine the force at any point in the disk. This is done by assuming that the equidensity surfaces in the disk will be selfsimilar ellipsoids (homoeoids). Also, if circular motion is assumed, then the pressure term in equation (3.8) will be negligible.
56
Thus the gravitational force due to a uniform shell of equatorial thickness da will be
dF = p(a) OX(ra.k)da, (3.9) Ba
where p(a)y(r, a, k) will be the force per unit mass in the equatorial plane of an oblate spheroid of density p(a), semimajor axis, a, and eccentricity k a The total force due to all such spheroids will then be given by
r
F(r)= = p(a)& da, (3.10)
0
since, by Newton's theorem, shells for a > r do not contribute 'to the integral. By substituting the form of the exterior potential of a uniform spheroid into equation (3.10) and taking the derivative, we arrive at
v2(r) = 4rG(1 k2)2 a (3.11) (r2 k2a2)1
0
which must be solved for p(a) given v(r).
From this point we follow the work of Brandt (Brandt 1960, Brandt and Belton 1962) who notes that the mass of the uniform spherical shell being considered is
dm(r) = 4ra2'(1 k2) 12p(a, k)da, (3.12)
and, therefore equation (3.11) may be rewritten as
r
v(r) = G dm(a,k) (3.13) 0(2 k2a2)7
or in a more convenient form,
r
2(r)G [ dm(a,k) ada (3.14) ada [ (2 k2a2)
0
57
By noting that the term in the brackets has the units of surface density as the homoeoids are infinitely flattened, i.e., as k + 1 we may finally write
r
v2(r) = G o(a) ada (3.15) o (r2 a2)2
where o,(a) is the surface density of the now infinitely thin disk. By use of a simple substitution, this integral transforms into an Abelian form and thus may be inverted (Arfken 1985) thus giving
2 d I v2(a)ada
r(r) = .dr (r2_a2). (3.16)
0
Grr dr o(2 a I2)'
To calculate the mass that is contained by those homoeoids interior to some radius r, we use
r r
M(r) = a(a)ada = 2 f v2(a)ada(3.17) rG (r2 a2)2
0 0
If the substitution a = r sin 0 is then used, equation (3.16) may used to analyze numerical data.
To further examine the process of going towards the limiting case of a flat disk, we consider a homoeoid with an axial ratio, . When this quantity becomes very small, the
a
attraction of the spheroid of interest on an exterior point will be very close to that of a nonuniform disk in which the surface density is allowed to vary, i.e.,
1~ 1'
o, = 2cp 1 2. (3.18)
As we pass to the limit, 0, we require that the mass remain constant in each cylinder
aof unit area or c = Usin equation (3.18 alon with this restriction we may then
of unit area, or cp = o,,. Using equation (3.18) along with this restriction we may then
58
calculate the mass contained in the disk of radius r to be m a (a2 2) (3.19) By differentiation, we may also conclude the relative mass distribution within the flattened homoeoid with a circle of radius r to be
4 o' 2 3 (a2 2) 2 dm=47ro 1 (320 d' 3 2a 3(a2 22 r2 da. (3.20) Using this formalism, Mestel (1963) was able to construct the surface density distribution of an infinitely thin disk of radius RD, maintained in equilibrium solely by rotational motion. By calculating separately the contributions from those shells that lie wholly within the radius of interest and the contributing portions of those that do not, Mestel is able to arrive at the expression for the surface density. This is done by defining, MT(r) Mi(r) + M2(r), where M1(r) is the mass contributed by infinitely flattening homoeoids that lie entirely within radius r and M2(r) is the mass contribution from infinitely flattening those homoeoids that have only portions of their mass within r. This is illustrated in Figure 3.1, where the homoeoids labeled 1 and 2 lie entirely within r and thereby have all of their mass included in MI(r) and those labeled 3 and 4, which contribute only a portion of their mass when flattened to M1 (r), and thus have the remainder included via M2(r). To calculate MI(r) and M2(r), we use
2 V2(a)ada
M i (r) = (r2 a2) (3.21) irGI o2 _a2)2
0
and
OO
M2(r) = dM )1 ( )] da. (3.22)
1 a
59
z
4
3
2
1 a
Figure 3.1: A schematic diagram illustrating the mass calculation using homoeoids
Using this formalism, Mestel then calculates the surface density required to produce a disk with a uniform rotational velocity (i.e., a flat rotation curve). This surface density is found to be
cr(r) = 2 rGr sin (3.23) 27rGr 7r RD
where V is the value of the constant velocity. We will refer to this disk as the finite Mestel disk.
At this point it is useful to note the work of A. Toomre (1963). Toomre noted that while the method of Burbidge, et. al. and Brandt was quite useful, it involved the evaluation of a double integral equation. As this can be cumbersome, he developed an alternative solution to the same problem considered above using FourierBessel transforms. By assuming cylindrical geometry, Toomre writes the potential of the system as
dq(r,z) = Jo(kr)eIkIzdz. (3.24)
60
Noting that this potential satisfies Poisson's equation in three dimensions if the mass surface density is written as 1I d k de(r) = d = Jo(kr)dk, (3.25) 2rG dz __ 27rG z=0+
and requiring that the surface density be expressed as a Bessel integral,
r(r) = Jo(kr)kS(k)dk, (3.26)
0
where
00
S(k) = Jo(ku)ua(u)du, (3.27)
0
from the FourierBessel Theorem (Morse and Feshbach 1953), then the actual potential can be written as
(r, z) = 2rG J(kr)S(k)ekIzdk. (3.28)
0
Therefore, we can now write
v2 = ( ) = 27rG J(kr)kS(k)dk. (3.29) r \O)=o I
0
Following similar logic, we can also write
= Jli(kr)k v2(u)J1(ku)dudk. (3.30) k=0 u=0 Equating equations (3.29) and (3.30) and noting that the transforms must be the same, we are able to show that
00
S(k)= 2l J v2(u)Ji(ku)du, (3.31) k=0
61
and, therefore,
J(r) rG Jo(kr)k v2(u)Ji(ku)dudk. (3.32) k=0 u=0
As this is a bit unwieldy to be of direct use, we integrate the inner portion of the expression by parts (assuming that v(0) = v(oo) = 0) thereby reversing the order of integration and obtaining
2rG 2 Jo(kr)Jo(ku)dkdu. (3.33) 2xG f du
u=0 k=0
Given equation (3.33), we now have a second, somewhat simpler, method to take an analytical rotation curve and produce a density profile.
Toomre notes that it is possible to explicitly integrate equation (3.32) if one chooses as a rotation curve
C2
v2(r) = V2(b,r) = 1. (3.34)
It is then possible to obtain the corresponding surface density, C2 1 1
(b, r) = r (r2 +b2) (3.35) 27xG r (r2 + b2) 2
where b is a scale length parameter that may be thought of as determining how dense the central region of the disk is and how rapidly that density falls off. C is a constant that may be evaluated by taking the maximum of V at r2 = 2b2. If this is taken to be true, then C2 = 3,V2.
2
While this surface density has the difficulty of possessing a singularity at its center, its velocity distribution initially rises steeply as radius increases, flattening out at higher
62
radii, as is seen in many of the galactic disk observations, and Toomre reports its mass to be finite.
It may be noted that from the above pair of velocity and density distributions, it is possible, due to the linearity of equation (3.32) in v2(r), to create pairs of functions that more accurately model observed disk systems (i.e., those without singularities in their centers). By taking the derivative of equations (3.34) and (3.35) with respect to b2, we arrive at
V~(b,r) = Cr2 1 (3.36) b (b2 + r2)1
and
C2 1
al (b, r) = 1 (3.37) 27rG (b2 + r2) 2
Toomre refers to this pair of functions as "Model 1". Further models may be obtained by using
Vl2 28Vn2(b, r)
'+1(b, r) 8(b,) (3.38) and
an+l (b, r) = 2 8ba an(b, r) (3.39) 02
(Toomre 1963, Hunter, Ball and Gottesman 1984) It is also possible to generate the nlth model in the family by integrating the nth model with respect to b. Using this relationship, it is possible to produce the lowest model in the family, the n0 disk, given by ao(b, r) = [ (b,r)bdb = Co 1 (3.40) 27rG (r2 + b2)
63
and
VO2(b, r) V= 12(b, r)bdb = C b ) (3.41) (b2 + r2)
A final detail in this portion of the modeling process is that for all of the Toomre
1
disks, the fraction of the mass outside of any given radius falls off only as r r. Hunter, Ball and Gottesman (1984) have used Brandt and Mestel's method of infinitely flattened homoeoids to truncate the Toomre series at radius, RD. While the rotations laws remain the same, the density distributions have additional factors that modify the surface density. The consistently truncated Toomre n=0 model can be rewritten as
CO1 r < RD,
b) 4 (r2 +b2 )
VO2(r,b) =(3.42)
2C.' {in I(R) b _tan' ( r R D
'I __ 1 'R' b tn1RD r2+b2)7 ,W (3.42) S(r2+b2) b(r2RD) > RD, and
o o(r,b) = It(r2+b2) ;, r RD, (3.43) 0, r > RD, whereas the consistently truncated Toomre n=1 model is given by
Cr r < RD,
V,2(b,,..)= r b f ,. ,o0: ,:* ,>..+,>'),:f br2r2+) ,,)
V~7(b ~ r' = 2C? I Lr4: tan' Dr+24 RD( W232 )R2R
r r212 an b b(r2R r2(R2+b )2, r>RD,
(3.44)
and
cf(r(b, r) =22)3] t..1 + r < RD,
p(bT) R+b2)(r2+b2)
r > RD. (3.45)
0,
The successive models may be found using the procedure described in equations (3.38) and (3.39). Representative graphs for the rotation curves and density profiles of Toomre n=0 and n=l disks are shown in Figures 3.2 and 3.2.
64
Cold Rotation Curve:Toomre n0 disk
0
c
C11 .
E
0
0 5 10 15 20 25 30 35 40 Radius (kpc)
Surface Density Plot:Toomre nO disk
0 5 10 15 20 25 Radius (kpc)
Figure 3.2: The cold rotation curve and surface
density for a consistently truncated Toomre nO disk
65
Cold Rotation Curve:Toornre nI disk
o
0 5 10 15 20 25 30 35 40 Rodius (kpc)
Surface Density Plot:Toomre n1 disk
S5 10 15 20 25 Radius (kpc)
Figure 3.3: The cold rotation curve and surface
density for a consistently truncated Toomre n=1 disk
66
Other Initial Condition Formalisms
Hernquist
Hernquist (1993b) has devised a multicomponent method used by him and his collaborators to construct warm, compound galactic models based on taking the moments of the collisionless Boltzmann equation (CBE). A generalization of earlier methods (Hernquist and Quinn 1989, Hernquist 1989, Quinn, Hernquist and Fullager 1993, Barnes 1988, 1992) involving bulges, disks and haloes, he assumes a spatial density distribution for all three components, the disk being exponential in the r direction and having a sech2 distribution in the z direction (he assumes the disk to be isothermal). Using moments of the CBE, he is then able to specify the dispersions for each component. In tests, Hernquist found that the disks constructed using his approach were fairly close to equilibrium, requiring a no greater than 10% adjustment in any component to reach a final equilibrium state for models constructed to be stable
The strength of this approach is its fully three dimensional, multicomponent nature. This allows a great deal of variability in the systems to be studied. However, the shape of the rotation curve can not be affected directly (it must be changed by changing the density distributions of one or more of the components) and, as such, it seems that a systematic study of parameter space in relation to the integrals of motion would be somewhat difficult. Given this shortcoming, control over the system's global stability (local stability is taken care of by the use of the Toomre local stability criterion (Toomre 1964)) is indirect at best since there is no easy way to adjust the initial disk's "coldness".
67
Fall and Efstathiou
By assuming an exponential density profile, Fall and Efstathiou (1980) laid down the foundation for constructing a selfconsistent disk/halo system based upon observations. Building on the work of Mestel (1963) and Freeman (1970), the authors attempt to construct a disk that in profile looks much like a lenticular (SO) galaxy embedded within a dark halo. Beginning with a surface density and z distribution, OD ) = 2 MDe(r), (3.46) 27
f(z) = (1 sech2( z), (3.47) and a rotation curve,
VD = V ( 2) [ 2n (2 r2)], (3.48)
2
where a is the radial scale length, Hg = ~ rm is the radius of maximum velocity and rt is the disk cutoff radius, the authors are able to derive a halo density function that is selfconsistent with the disk. Hence, by assuming both a density distribution and a rotation curve, Fall and Efstathiou are able to deduce what the form of the halo must be. Here again, we have the strength of the model being closely matched to the observations, but difficult to vary in terms of the integrals of motion. Additionally, it should be noted that the model assumes isothermal behavior in the disk.
Calculation of Velocity Dispersion
The Toomre formalism described above assumes what is known as a "cold rotation curve". In other words, all of the energy of motion for the individual disk particles is
68
restricted to the circular motion necessary to offset the centripetal force of gravity. When observing stellar motions in disk systems, one usually finds that this is not the case, but rather that large velocity dispersions act in a manner analogous to the pressure in a fluid system. Thus, the real disks are supported through a combination of circular motion and "pressure" gradients in the radial velocity dispersions. As this is the case, it would be good to be able to easily control the how the kinetic energy is partitioned between the ordered motion of the system (i.e.circular motions) and the random motions of the system (i.e.dispersional motions). Using a method known as "stellar hydrodynamics" we are able to, by taking moments of the collisionless Boltzmann equation (CBE) quantify this process in an easy, straightforward fashion. The treatment discussed here follows the method developed by Hunter and Moore (1989) and reported by Moore (1991).
Beginning with the CBE in two dimensions, we have in polar coordinates
of + Vr f + 10 + Vr + '4, = 0, (3.49) 8t 8r rrO V O (3.49) where the distribution function f = f(r, 4, Vr V, t). Noting that V = + v, (3.50) Or r
V VVO (3.51) r ao'
and assuming azimuthal symmetry, (i.e.5= 0; = 0) we can rewrite the CBE as
Of f [VO 1 Of rVO8f
+ Vr + 0. (3.52) at ar r Or I OVr r V We may now take the moments of this equation by writing + Q f Q Of
+ Qd + (3QVr.d5
[V4O I f d VV 8f (3.53) r O r BV, I r 8VO '
69
where Q = Q(vi) and dr, is defined to be the volume element in velocity space. For the zeroth moment of the CBE, we take Q = my = m and upon integration, we obtain a continuity equation for the system, 8v(r, t) 8( V V, t + Or + r = 0, (3.54) where v f mfdrv. Taking the first moment, Q = mVr, we arrive at 8(vr) 8' 2v(, +W VO2 Or 2+ r r r (3.55) V +O + (, + 2 =0, r Or r
where we have used the identity, V2 + or being the random motion of a particle and V being its ordered motion. Using the continuity equation derived above, we may rewrite this as OV Vr
Ot )r (3.56) +(o, a) + = 1 0. r 9r r If we seek stationary solutions to this equation, Vr = 0; L= 0 and that the azimuthal velocity may be parameterized by V = kVo, where V is the cold rotational velocity, we may reduce the second moment equation to d 2) u 2 2) dO v 2 2 d (vO) + (Or 01) = Vr + k2Vo. (3.57)
Finally, by taking a cross moment, Q = mVrVO, we obtain d 2 d
d dr
 2 v O +(3.58) V V Vr 2 v V O 2 V r r 0 r r5
70
By again restricting ourselves to time independent models with no bulk radial motion allowed, the above relation becomes
2 fd InT V4o
S= 1 + dlnr, (3.59) Using this expression, we may now simplify equation (3.57), the dispersion relation, to
d 2 r ( dln d _n d VD 2V
(V + 1 = + kVo, (3.60) dr 2r dn r r r or
d 2 V2( dlnVo dlnk de v
(ve;) + I)1 = V + k2 (3.61)
r 2r dlnr dlnr dr r o At the present time, we assume that d V2
 =  (3.62) dr r
and that k is spatially invariant throughout the initial disk, which reduces the dispersion relation to its final form,
d 2 dlnVo V 2 T(V) + 2 1 d V = (1 k2) o2 (3.63)
Due to the complexity of most analytical velocity curves, it is usually necessary to solve the dispersion relation using numerical methods. As previously mentioned, knowledge of the velocity curve and through it, the density distribution and velocity dispersions do not uniquely determine the distribution function, however they do provide useful constraints as to its form. After experimentation, it has been determined that the macroscopic quantities of the system (i.e., the energy and angular momentum) are fairly
71
insensitive to the choice of initial distribution function (Moore 1991) provided that the choice is well bounded in velocity space. The simple choice that we find works well and may be close the actual distribution function is a local Gaussian form, with unequal velocity dispersions in the r and 0 directions. In practice, the high velocity tails of this function are truncated as the particles in that region of the distribution would have escaped the system. Tests have shown that the inclusion of such a truncation reduces the number of escaping particles in an axisymmetrically stable model by 90% without significantly altering the global parameters of the system.
Stability Criteria
Toomre's Local Stability Criterion
In his 1964 paper, A. Toomre discussed the problem of local stability in disk systems using hydrodynamic considerations. In presenting the derivation here, we follow the outline presented in Binney and Tremaine (1987). If the thickness of the disk is assumed to be negligible and motions of the gas are restricted to the z=0 plane, Euler's equations may be written as
OVr vVr+ rv2 2 4( 1 8p + v + (3.64)
t r r 0 r Br E 9r' 8vO v, vV 8v, rv, 1 80 1 Op
O' + vr + + = I (3.65) & r r a r r 0 rE 0' where E now represents the surface density of the disk so as to reduce notational confusion. Choosing a polytropic equation of state, p = KE, (3.66)
72
we may then write that the sound speed is given by v = KE E. (3.67) If we then define a specific enthalpy as h = K 1, (3.68) 71
Euler's equations are simplified. Assuming we may linearize these equations we say, Vr = VrO+Vrl, v4 = v0o0+v1, h = ho +hl, 4i = Oo+l where the quantities subscripted with a one are assumed to be small perturbations of the background zero subscripted quantities. First, we note that for the unperturbed disk, VrO = 0 and = = 0. This reduces the Euler's equations for the unperturbed disk to = d(,0 + ho), (3.69) r dr
as would be expected. Additionally, as the second term on the right hand side of the equation is smaller than the first term by a factor of v.) it will be neglected and therefore, we may write
(rd&bo
vOo r = rO(r), (3.70) where Q(r) is the circular frequency. Using these and keeping only terms of first order in the perturbed quantities, we write the Euler equations as vrl_._ Vrl a t+ 10 2v = r(1 + hi) (3.71)
at 8a B
and
av41 [d(r) v1v ]1 a1
at + dr + Q Vr +Q + ac9 (1 + hi). (3.72)
8it [dr 84 89 r 84T
73
At this point, we find it convenient to introduce two definitions to simplify the equations. First we define
1 [d(r) 1 dG B(r) [ + = 0 r (3.73)
2 dr 2 dr and then, using the epicyclic frequency, we have dt2
2 r 92Q) = + 4 4BQ. (3.74)
Assuming solutions of the form, 41 = ka(r)ei(mwt), we may then solve Euler's equations for Vra and va. These are given by id 2mQ Vra = (mQ w) (Da + ha) + (> + ha) (3.75)
and
1 [2 d( +h)m(mQ w Va = 2B (a + ha) + (a + ha) (3.76) A dr r where A = 2 (mQ w2) and, by using the same linearization procedure on the equation of state, ha = yKE" ~Ea = vi Using the equation of continuity in cylindrical coordinates, we may relate the perturbed surface density to the perturbed velocities by
85l 1__8 82l 10 01 + (1 orl) +  + 1 = 0, (3.77)
 (ro i
rr 84 r 84 which, if we used the forms of the solutions given above for the perturbed quantities, becomes
1 d imEo i(mt w)Ea + Id(rEovra) + R Vpa = 0. (3.78) r dr R
74
To close our system of equations we need to determine the relationship between Eaand GJa using Poisson's equation. To do this, we invoke the WKB approximation to obtain local solution to the density perturbations. The perturbed potential may be written Ga(r) = F(r)eif(r) = F(r)eifkdr, (3.79) where k = d and krl > 1. Using this, we note that Poisson's equation is approximated within O(1 kr 1) and therefore, Ea and ha also contains the factor, eif(r) As this term varies rapidly with radius, we may drop those terms that are proportional to (%+h) relative to those terms involving d(d+ha) without increasing the fractional error. If we then write
d(Q + +ha)
d( + h) ik(Ia + ha) (3.80) dr
we may simplify the velocity solutions, (mQ w)k(QOa + ha) (3.81) Vra ~ A
and
2Bik(Qa + ha) (3.82) VOa = A(3.82) Similarly, we may write
d(rYovra).
d(rEoVa)= ikrEo, (3.83) dr
which is of the same order as equations (3.81) and (3.82). Thus, this term dominates over the third term in equation (3.78) by O( lkrl) and therefore we drop the third term. Given this we may write the continuity equation as (m W)Ea + kEoyVra = 0, (3.84)
75
which allows us now to eliminate Vra, ha and Ga, leaving ( ktp 2x'GEolkj
a + k' 2G j = 0. (3.85) As we require that the quantity in parenthesis vanish, we may write the dispersion relation for a gaseous disk in the tight winding limit as (mf2 w) = 2 2rGEolkI + k2v. (3.86) Using a similar derivation, one may derive the equivalent dispersion for a stellar disk.
Assuming axisymmetry, we require m=0 and so equation (3.86) becomes
W2 2 27rGEok + k2v. (3.87) If w2 < 0, then an exponentially growing solution is allowed and the local region is unstable. Therefore, if
K 27rGEolkl + k2v = 0, (3.88) we are at the line of neutral stability and thus, by solving the quadratic we see that local stability requires
Q s >1 (3.89) 7rGEo
for a gas disk. For a stellar disk, this becomes Q > 1. (3.90)
3.36GEo
(Julian and Toomre 1966).
76
OstrikerPeebles Parameter
In their paper discussing the stability in numerical modeling of disk systems (1973), Ostriker and Peebles introduce a dimensionless parameter, t which is characterized by
T
t = IW, (3.91) where T is the rotational kinetic energy of the system and W is the system's potential energy. If the random kinetic energy of the system is represented by 1fl, then the virial theorem may be written as
1 1
T + II= W (3.92)
2 2
or
t 2. (3.93)
T
Given that n > 0, we may say that 0 < t < 1. Ostriker and Peebles found that for models in which t Z 0.14 (, < 5), the system was globally unstable to the formation of an m=2 (bar) mode. While this conclusion was merely based on a number of numerical simulations run by the authors, the conclusions reached are supported by the work of a number of other investigators (Sellwood 1981, Zang and Hohl 1978, Efstathiou et. al. 1982, Frank and Shlosman 1989). The rotational parameterization parameter, k, defined in the section describing the derivation of the dispersional velocities of our disks, can be shown to be related to the OstrikerPeebles parameter by k2
t = . (3.94)
2
This being the case, we find that we may easily vary the global stability of a disk system to the formation of a bar mode by varying k.
77
Since the publication of the stability parameter, a number of examples have been found to violate the parameter. Vandervoort (1983) found that the point of marginal stability varied between 0.1286 < t < 0.1882 for uniformly rotating stellar spheroids. While technically in agreement for the e = 1 case, the models run are in violation of the OstrikerPeebles (OP) parameter otherwise. Another exception to the empirical parameter is the case of BodenheimerOstriker gaseous spheroids (1973) where the system is stable for values of t 0.24, a clear violation of the OP parameter. Other examples of this type of violation include TohlineDurisenMcCollough gaseous spheroids (1985) and Woodward, Tohline and Hachisu gaseous tori (Tohline and Hachisu 1990, Woodward et al. 1994). Finally, Miller (1978) showed that models that are represented by a ToomreZang (Zang 1976) stellar disk (these are Mestel disks in which only a fraction of the particles are allowed to respond to perturbations in the system, while the rest are kept 'frozen' in their circular orbits) are able to violate the OP parameter by remaining dynamically stable at t=0.248. As will be shown in Chapter 6 of this dissertation, we have found further exceptions to this rule by including counterrotating angular momentum (Davies and Hunter 1995a,b).
While a number of violations of the OP parameter do exist, the criterion works as a very good empirical guide for determining the stability of a simple disk system to the development of a bar mode. The failure of this criterion to accurately predict stability in a wider range of models stems from that fact that it is only an approximate relation deduced from numerical experiment, not analytical consideration. Additionally, it takes into account only one of the integrals of motion of the system, namely the energy. A
78
more accurate description of the stability criterion for such systems may have to take into account other integrals of motion. Christodoulou's Parameter
In two papers recently submitted to the Astrophysical Journal, D. Christodoulou, I. Shlosman and J. Tohline (1995a,b) propose a new criterion to gauge the global stability of a rotating system. This stability parameter can be written as Ty
a = , (3.95) 'WI'
where
LO;j
Tj 2' (3.96)
2
L is the total angular momentum of the system, Qj is the Jeans frequency introduced via selfgravity and W, as before, is the total potential energy of the system. For stellar systems, a < 0.254 0.258 will provide stability to the development of nonaxisymmetric modes. Note that this new criterion is dependant on two integrals of motion and is therefore able to more accurately predict the behavior of those systems whose angular momentum is not included in a simple way. In can be shown that equation (3.95) may be rewritten as,
5 f L/M
5f L/M (3.97)
4 Qjaf
where fI for disks, M is the mass of the system and a, is the equatorial radius. The term fljal therefore represents the maximum angular momentum of a circular orbit in the equatorial plane of the system. When tested over a number of differentially rotating
79
systems, this parameter accurately predicts the formation of bar modes in the systems investigated, including those where the OP parameter does not.
GENSTD
GENSTD stands for GENerate Standard Toomre Disk. This is the program used to develop the initial conditions used in this modeling effort. The code is written in FORTRAN and presently there are versions that will build consistently truncated Toomre n=0 and 1 disks using the dispersion relation given in equation (3.63). The version of GENSTD that builds a two dimensional n= 1 disk is listed in the appendix.
The generation code allows the user to input as runtime parameters, the radius of the disk, RD, the mass within a unit radius, M, the total number of particle in the simulation, N, the shape parameter, b, and the heating parameter, k, as well as certain other algorithmic parameters to be described shortly. After initializing several constants, the code preliminarily calculates the mass distribution of the disk and the actual particle number and individual particle mass. As the code does not calculate the particle velocities from the analytical expression for the velocity curve (for reasons to be discussed below), but rather uses the startup routine from the hierarchical tree algorithm's leapfrog integrator, the necessary lookup tables are constructed. At this point the mass distribution of the disk is built. This is done by starting at the outer radius of the disk and calculating the mass within a thin ring. This is calculated using the analytical forms of the truncated disk. From this the number of particles in the ring is calculated and those particles are placed randomly within the ring using a MonteCarlo method. This process is done for rings inward of each other until the entire disk is laid down.
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At this point the average acceleration on each particle in a ring is calculated using a simple Nbody loop. This is done so that the effects of gravitational softening will be taken into account. If the velocities are calculated from analytical formula (which do not take softening into account) the disk will be unstable to the formation of m=1 ring modes and will spend the first several dynamical times readjusting. Once the accelerations are computed, the average circular velocities for each ring can be calculated. If the velocity of a ring is less than zero, which may occur near the center of the disk due to numerical noise, then analytical value is then substituted.
Once this is completed, the size of the dispersions for each ring are computed using the dispersion relation (equation (3.63)) and equation (3.59). The dispersion relation is solved numerically using a fourthorder RungeKutta method. The velocity is then assigned to each of the individual particles with the size of the individual dispersions selected using the truncated Gaussian described above as calculated using the GASDEV subroutine. Once done, the program writes a file containing the position, velocity and mass information in a from usable by the TNDSPH code. If the data is to be run using FTM, the file is then run through a conversion program that produces output suitable for input into the FTM code.
CHAPTER 4
NUMERICAL MODELING OF DISK SYSTEMS Introduction
The basic formalisms used in this modeling effort, namely the codes and the initial conditions, having been put forth, we shall proceed to the actual modeling of disk systems and the issues relevant therein. Given that the systems modeled are only approximations of real physical systems, due to the very large difference in particle number between our models and real galactic disk systems, we shall discuss what effects this will have on the validity of the simulations reported in this dissertation. Additionally, methods of representing of star formation effects will be considered as will be the difficulties inherent in describing processes in the interstellar medium (ISM). Finally, a discussion of runs for simple Toomre n=0 and n=1 stellar disks will be undertaken to provide a background for the work to be presented in chapters 5 and 6. It should be stressed here that the models presented herein are global in nature and focus on the formation and dynamical importance of the m=2 (bar) mode of the system.
Relaxation Effects
One of the most important questions concerning simulations of the type described here is whether these models, with a large but limited number of particles, will accurately model a physical system made up of a much larger number of particles. As is well understood (Miller 1964), such simulations will not follow the exact representation and 81
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evolution of real, physical systems in phase space due to roundoff error and discreteness effects. Nevertheless, it can be shown that in, a statistical sense, such simulations will model the global behavior a physical system. A vital consideration, however, is the effect that model particles of much greater masses than those in observed disk systems will have on the time scales for the global evolution of the system. This will be discussed for the three dimensional case and then considered in a two dimensional context.
For the derivations in chapter 3, we have relied on what is sometimes called the "smoothed potential" approximation. Namely, in our use of the collisionless Boltzmann equation we have assumed that the potential generated by the stellar component of the disk varies slowly and uniformly in both space and time. Given that this potential is created by a finite number of particles, obviously this is not exactly true. As such, an estimation of the divergence from this assumption is essential to understanding how much confidence may be placed in the results obtained by such "nonsmoothed" methods. If we consider that a model disk will be constructed of N particles, each with mass mi, we can focus on the motion of a single star across the model. To obtain an estimate of the difference between the motion of this individual particle in the model system, built from a finite number of particles, and its motion in a smoothed potential we use the derivation found in Binney and Tremaine (1987). From this derivation we find that the velocity of the particle over a crossing time of the system will differ from its smoothed potential path by
V 8 NA (4.1) v N '
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where In A in ) bmin and AvI is the deflection from the expected path due to disceteness effects. If the star is able to cross the system several times during the simulation, the velocity will change by this amount each time. Therefore, the number of crossings required for the star's velocity to change on the order of itself is
N
nr 8 In A (4.2)
The amount of time this takes is called the relaxation time and is given by trelax=nrX tcrossing, where tcrossing is the crossing time defined as tcross= R. If we take a reasonable value for A, such as A N, then we may say that the individual encounters will perturb the star from its smooth potential course on the order of
0.1N
trelax in N tcrossing. (4.3)
While the derivation used here is of an approximate, order of magnitude nature, a more detailed derivation (Spitzer and Hart 1971) finds little difference except to take a system's density into account. More accurately, the relaxation time relation is given by
0.14N r
t (4.4) relax = In (0.4N) GM' (4.4)
where rh is defined to be the system's median radius and M is the total mass of the system. As can be seen, while relaxation effects will have little impact on the evolution of a physical galaxy containing 1011 1012 particles, owing to the system's relaxation time being longer that the Hubble time; such collisional processes will have a much greater effect on systems with ~ 105 106 particles.
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For a strictly disk (i.e., two dimensional) system, Rybicki (1972) has considered the same problem. For such a two dimensional system, he finds that
k
trelax crossingn, (4.5) where k is the ratio of the ordered kinetic energy to total kinetic energy given in chapters 1 and 3. As can be seen, for a strictly two dimensional system, the relaxation time is always on the order of the crossing time and consequently, systems simulated in this way will never be able to adequately describe systems using the smoothed potential approximation.
Fortunately, there is a consideration that still needs to be factored into the two dimensional derivation. The calculations that lead to the above conclusions in both two and three dimensions assume an inverse square law force dependence. In our numerical simulations, we soften this force so as to avoid near singularities. If the calculation is redone, taking softening into account, we arrive at (1 k) ~N
trelax = 2RD crossing, (4.6) where c is the softening length. For the systems considered in this study, assuming N , 3 x 10,
trelax ~ 40tcrossing. (4.7) Therefore, we may conclude that, while collisional effects may play a role in the evolution of the systems considered, errors introduced by the secondorder integrator and numerical roundoff will be of comparable size. The validity of the above arguments must be tested by numerical experiment. From numerical experiments performed at the N ~ 105 particle
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number size, the time scales as roughly N. Given this, the time scales over which the physical processes proceed will be much shorter in models with low relative particle number than those same processes in physical systems.
While collisional relaxation may not be of great importance, we can not make similar claims regarding collective effects. Since the inverse square nature of the gravitational interaction means that such effects are long range in nature, it is likely that adjustments in the density distribution may be due to collective effects.
Star Formation
Given that the process of star formation is not well understood in the interstellar medium, any approach to model these processes numerically in the context of a much larger system will, of necessity, be very schematic. Still, however, one may gain insight into the effects of including such processes on the global evolution of the system if the implementation of these considerations are modeled in such a way as to be physically reasonable and so that they produce results in rough accord with observations. For this to be the case, star formation processes must take into account the local characteristics of the system at the point of interest. Additionally, as time the time step of the system can be on the order of 105 106 years, stellar winds and supernovae of OB stars also must be accounted for, at least schematically
The simplest approach to this problem is to use a star formation criterion based on the Jeans' length and the mass of the local region. This may be done by comparing the density of each SPH particle with some critical density, pc. If the SPH particle's density exceeds
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this amount and the particle's velocity divergence is negative, star formation is assumed to take place. Besides taking into account the Jeans' instability, pc may also represent those effects due to relative motions, magnetic fields and temperature gradients. This is the approach used by Heller and Shlosman (1994) in their investigations. While simple and easy to implement, this criterion has the shortcoming that stars form almost exclusively in the bar region of the disk. This conclusion is not supported by the observations, which show star formation occurs in both the bar and spiral arms.
Friedli and Benz (1993, 1995) have investigated three differing star forming criteria. These three criteria are the criterion used by Heller and Shlosman (1994), Toomre's local instability criterion (1964) and a negative change in the local entropy. They have found that the local Toomre stability criterion for gas, with a suitable choice for the cutoff value, reproduces the observations in the most accurate way. Their criterion is to chose a value for Q below which star formation will take place in a particle. The choice of this number is somewhat arbitrary and likely depends sensitively on the exact physics of the ISM. Nevertheless, Friedli and Benz show that choosing 1 5 Q < 1.5 leads to star formation occurring in both the bar and spiral arms. In estimating the local value for Q, the authors follow the suggestion of Elmegreen (1993) and replace K with cA, where c=2.83 and A = di is the Oort constant. Additionally, they invoke a temperature condition that requires a gas particle to have a temperature less than some critical temperature so as to have their method simulate star formation in cool regions of molecular clouds. Hence, this second selection criterion forces star formation to occur in those regions where the cooling timescale is shorter than the dynamical timescale.

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NUMERICAL MODELING OF LARGE N GALACTIC DISK SYSTEMS BY CHAD LESLIE DAVIES A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995
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To my wife, who stood by my side.
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ACKNOWLEDGMENTS I would like to begin by acknowledging a number of dedicated educator s and researchers who gave of their valuable time to mentor me and develop within me qualitie s as a scholar a nd as a man: James A. Savard who by his life and his encouragement taught me the principles of s couting that carry through to this day ; Richard McReynolds who kindled in me the appetite to a s k questions and search for the answers; Dr. Gordon Wolfe who taught me how to think like a physicist and see the beauty of the subject and Dr. James Hunter Jr. who patiently showed me how to conduct re search and critically evaluate my own work. Without the time and patience shown by these men, I would never have reached this point. Additionally I thank the past and present members of my committee who have supervised my progress and encouraged me along the way In the department of Physics: Dr. J. Robert Buchler who, through his teaching me the fundamentals of electromagnetic theory, impressed upon me the level at which graduate work is done and challenged me to rise to meet to the standard. Dr. James Ipser, who showed me early on that I didn't really want to do relativity but encouraged me with many kind words nevertheless and Dr. Gary lhas, who has shared with me his humor and relaxed perspective. In the Department of Astronomy : Dr. Henry Kandrup, who has provided a great deal of positive input on the s ubject of the inclusion of counterrotating angular momentum and reminded me of the limitations of numerical modeling; Dr. Haywood Smith, who has assisted me greatly in helping me understanding some of the numerical aspects of this study and Dr. Steve Gottesman who kept me from forgetting that there really are galaxies out their and who iii
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asked the right questions at good times. It is s aid that one of the wisest decisions a Ph.D. student can make is to choose his committee well. Through no fault of my own I believe I could not have done a better job I would also acknowledge the assistance of several other individuals who have helped me along the way. From Greece; Dr. Nikos Hiotelis, for helping me during the summer of 1991, understand the fundamentals of Smoothed Particle Hydrodynamics and Dr. George Contopoulos who took an intere s t in me and my work during his visits and provided an unparalleled example as to what a researcher is to be. Also, I acknowledge the discussions carried on with Drs. Dimitris Christodoulou, Clayton Heller and Isaac Shlosman. I thank Nikos and Clayton for providing codes to the galaxy program at the University of Florida upon which this modeling effort is based and Dimitris for sharing with me an analytic framework in which to view much of what is happening in my models Also, I wish to thank Ronald Drimmel for his invaluable help and fellowship along the way. His patient teaching and explanation has made the code understandable and accessible. More importantly, he has been a stalwart friend and a great encouragement to me though all stages of this work Finally, I thank Keith Kerle and Donald Haynes for their friendship and willingness to listen to my various diatribes on the injustices of graduate education when the days were long. Most of all, I thank my advisor, mentor and supervisor, Dr. James Hunter who has invested an enormous amount of time and energy to bring me along to the level of respectable scientist from the state in which I began working for him His example of critical analysis and skeptical thought has informed me of the responsibility of a scientist iv
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to do and report work that i s reproducible. As importantly, he has taught me the s kill of looking at my own work with an eye to additional detail. He ha s pas s ed along to me the essence of what it is to be a scientist and a researcher. He ha s also pa s sed along to me the incalculable wisdom that "Gravity the pathological bea s t by fiat, doe s the trick, fair enough. I wish to expres s my gratitude to my parents, Henry and Toni Davies who provided their love moral s upport and endless encouragement when the going was slow and difficult and I doubted my ability to finish what I had begun I also acknowledge the patience love and support of my wife who has stood by my side and helped me in so many ways through my course of study Kathy has shown me what unconditional love and dedication are by her patience with and support of this project that has stolen s o much of our time. She has shown me that there is so much more to life than learning, knowledge and research. I wish to thank the Department of Physics for providing teaching and logistical sup port over several periods of time during which a portion of this research was done Al s o, I thank the Department of Astronomy for providing access to their exceptional computing facilities and an office with a dedicated workstation to complete this dissertation and its reported research. Additionally, I acknowledge the support of NSF grant AST9022827 and the Division of Sponsored Research at the University of Florida. Computational time for the numerical simulations reported herein was made available, free of charge, by the Research Computing Initiative, a cooperative venture of the University of Florida, the Northeast Regional Data Center and International Business Machines. V
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Lastly, I acknowledge the role of God Almighty in this endeavor, without whose strength I would never have finished this undertaking, and His Son Chri s t Jesus whose sacrifice at Calvary has redeemed mankind. ... ovH ... Tl lO)(VEl .. o:>.>.o: 7rl<7TU7 tJl o:1o:7rTJ<7 wtp1ovwTJ' (Gal. 5:6) vi
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TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT. CHAPTERS 1. PHYSICAL MOTIVATION FOR THE STUDY OF GALACTIC DISK SYSTEMS ................................... Introduction . Interacting Systems . . Observations . . . . Work on Galaxy/M31 Interactions .... Previous Numerical Work on Encounters Counterrotating Systems Observations . . Previous Analytical Work Kalnajs ...... Araki .......... Christodoulou et al. Previous Numerical Work 2. THE CODES Introduction Hierarchical Tree Algorithm Smoothed Particle Hydrodynamics Time Integration . Miscellaneous . . Tests of the Algorithm . Tests in One Dimension Tests in Two Dimensions Tests in Three Dimensions Specifics of the Codes Used TNDSPH FTM ............ Vil 111 X XI XIV l 3 3 4 6 8 8 10 10 11 12 13 16 16 17 21 35 39 41 41 42 45 46 46 46
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3. INITIAL CONDITIONS Introduction . Observation of Isolated Galactic Disks Analytical Disks . General Discussion Kalnajs/Hohl Disks Toomre Disks . Other Initial Condition Formalisms Hernquist ........... Fall and Efstathiou . Calculation of Velocity Dispersion Stability Criteria . . Toomre's Local Stability Criterion OstrikerPeebles Parameter Christodoulou s Parameter GENSTD ......... .. 4. NUMERICAL MODELING OF DISK SYSTEMS Introduction .. Relaxation Effects Star Formation Numerical Models Units ... ... Toomre n=O Disks Toomre n= 1 Disks 5 ENCOUNTER SIMULATIONS Introduction . . Stellar Models . . Massive Particle Encounters Dwarf/Disk Encounters Disk/Disk Encounters . Stellar/Gas Models . Massive Particle Encounter Dwarf/Disk Encounter . Models with Star Formation Massive Particle Encounter Dwarf/Disk Encounter . v iii 48 48 49 51 51 54 55 66 66 67 67 71 71 76 78 79 81 81 81 85 89 89 90 95 103 103 105 105 110 115 120 120 125 130 130 133
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6 COUNTERROTATING ANGULAR MOMENTUM . 137 Introduction . . . . . . . 137 CR Angular Momentum Inclusion in the Initial Conditions 138 Models for the Development of Systems with CR Angular Momentum 140 Numerical Simulations . . 143 Fully Mixed Case Results . 143 Step/Slope Function Ca s e Results 145 Toomre n=O disks 145 Toomre n= 1 disk s . . 148 7. DISCUSSION OF RESULTS AND FUTURE WORK 153 General Conclusions . . . 153 Introduction . . . . 153 Numerical Integration of Toomre Disks 154 Encounters and Local Group Dynamics 155 Counterrotating (CR) Systems 157 Future Work to be Done 158 Improvements . . 158 Continuing Work . . 160 Questions Still to be Answered 161 Questions Raised 162 APPENDIX . 164 BIBLIOGRAPHY 187 BIOGRAPHICAL SKETCH 200 ix
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LIST OF TABLES 2.1 Period of oscillation for two dimensional Maclaurin disk s 2 2 Properties of the Code s u s ed in the Pre s ent Modeling Effort 5.1 Encounter model s run ..................... 6 1 The mode strength and pattern speed of the m=2 mode for Toomre n=O 44 47 105 and n= l disks as a function of percent CR angular momentum in the fully mixed case. . . . . . . . . 144 6 2 Results of the Christodoulou stability criterion check for Toomre n=O and n=l disks. ...... ..... ............. : .... ... 145 6.3 Toomre n=O models step function models investigated as a function of k and J c R ..... ........... ............. ... 146 6 4 Step/Slope function models for a Toomre n=l disk with J c R = O .lJr 148 X
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LIST OF FIGURES 2.1 An ordering diagram to determine the advancement of particles with multiple time steps . . . . . . . 38 3.1 A schematic diagram illustrating the mass calculation using homoeoids 59 3 2 The cold rotation curve and surface density for a consistently truncated Toomre n=O disk . . . . . . . . 64 3 3 The cold rotation curve and surface density for a consistently truncated Toomre n= 1 disk . . . . . . . 65 4.1 The time evolution of a Toomre n=O disk constructed to be stable to nonaxisymmetric modes. . . . . . . . 91 4.2 Plots of the rotation curve and surface density versus radius for a Toomre n=O disk at t = 40 tdyn . . . . . . . 92 4.3 The time evolution of a dynamically unstable Toomre n=O disk . 94 4.4 A plot of bar strength versus time for the simulation shown in Figure 4.4 95 4 5 The time evolution of a dynamically stable Toornre n= 1 disk . 97 4.6 The time evolution of the gas component in the model displayed in Figure 4.5 without star formation . . . . . . 98 4.7 The time evolution of the gas component in the model displayed in Figure 4.5 with star formation . . . . . . . 99 4.8 The star formation maps for the model displayed in Figure 4.7 . 100 4.9 Plots of the rotation curve and surface density versus radius for a stable Toornre n=l disk at t = 40tdyn ................ ........ 101 4.10 The time evolution of a dynamically unstable Toomre n=l disk ..... 102 5 1 Plot of particle positions for a positive interaction angular momentum, purely stellar massive particle encounter. ................ ... 108 5.2 Plot of particle positions for a negative interaction angular momentum, purely stellar massive particle encounter . . . . 109 5.3 A plot of particle positions for a purely stellar dwarf/disk encounter. Interaction angular momentum is positive . . . . 111 xi
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5.4 5.5 5.6 5 7 5 8 5 9 5.10 5.11 5.12 5.13 5.14 5 .15 5 .16 5 .17 5.18 5 .19 5.20 A detailed plot of s tellar particle positions for each disk individually for the s imul a tion s hown in Figure 5.3 ...................... 112 A plot of p a rticle po s ition s for a purely s tellar dwarf/di s k encounter. Interaction angular momentum is negative . . . . I 13 A detailed plot of s tellar particle positions for each disk individually for the s imulat10n s hown in Figure 5 5 ...................... 114 A plot of stellar particle positions for a purely stellar disk/disk encounter. Interaction angular momentum is positive ......... ......... 116 A detailed plot of s tellar particles for each disk individually for the s imulation shown in Figure 5.7 ............... ......... 117 A plot of stellar particle positions for a purely stellar disk/disk encounter. Interaction angular momentum is negative .... .... ......... 118 A detailed plot of stellar particles for each disk individually for the simulation shown in Figure 5.9 ........................ 119 A plot of stellar particle positions for a massive particle encounter 122 A plot of gas particle positions for a massive particle encounter prior to perigalaxion. . . . . . . . . 123 A I?lot of_ gas particle positions for a massive particle encounter after pengalax1on . . . . . . . . . 124 A plot of stellar particle positions for a dwarf/disk encounter A plot of gas particle positions for a dwarf/disk encounter 127 128 A detailed plot of gas particle positions for each disk in the dwarf/di s k encounter shown in Figure 5 .15 ........................ 129 A plot of gas particle positions for a massive particle encounter where s tar formation 1s allowed . . . . . . . 131 A star formation map for the simulation shown in Figure 5 .17 . 13 2 A star formation map for the simulation shown in Figure 5.19 ... 135 xii
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5.21 6. l 6.2 6.3 6.4 6 5 6 .6 A detailed plot of gas particle positions for each disk for the dwarf/disk enc o unter shown in Figure 5. l 9 . . . . . . 136 Schematic diagram illustrating the distribution of angular momentum in the step function case. . . . . . . . 139 A graphic showing the function of percent CR angular momentum versus radius in a slope function model. . . . . . 139 Evolution of an unstable Toomre n=O step function case with 50% of the angular momentum counterrotating. . . . . . 14 7 Isodensity plots of Toomre n= l step/slope function model # l prior to inner bar reversal. . . . . . . . . 150 Isodensity plots of Toomre n= l step/slope function mod~l # l following inner bar reversal. ............. ................... 151 Isodensity plots of Toomre n=l step/slope function model #3 ....... 152 xiii
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NUMERICAL MODELING OF GALACTIC DISK SYSTEMS Chairman: James H. Hunter Jr. Major Department: Physics By Chad Leslie Davies August 1995 In this study, hierarchal tree numerical methods with a smoothed particle hydrody namics (SPH) formalism are used to simulate galactic disk systems in two dimensions. An identification of a new axisymmetric evolution of Toomre family disks to exponen tial disks is made. Using two dimensional disks with gas and stellar particles to model nonmerging encounters, several scenarios are explored. By investigating the resultant gas inflows and star formation patterns, we suggest that interacting gas rich systems will have elevated gas inflow and star formation rates In addition a numerical study of the inclusion of counterrotating angular momentum in disk systems reveals the dynamical importance of such systems. Galaxies with significant portions of counterrotating angular momentum have been observed and, as such, a study of the properties of such systems may provide valuable information regarding the global parameters of such systems. By including counterrotating orbits physical systems are achieved that are not possible to realize using fully direct initial conditions. xiv
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CHAPTER 1 PHYSICAL MOTIVATION FOR THE STUDY OF GALACTIC DISK SYSTEMS Introduction The aim of numerical modeling of galactic disk systems is to investigate the time evolution of systems that for practical reasons, can only be observed at one specific time. Such modeling must taJce into account the known physical principles that are relevant to the particular system and must handle gas components with a fair approximation of the continuum limit. If these conditions are met, such modeling can be used to gain insight into the effects of less weU understood phenomena and to investigate systems that are not tractable through analytical study. For the techniques of numerical modeling to have any meaning in this context, a study involving such considerations must be guided by analytical understanding of simpler but related systems and by observational input. To neglect the former may produce physically unrealistic results and to ignore the latter may result in models that may have very little relation to the systems which occur in nature. This study attempts to meet both criteria by using modeling codes that are founded upon those physical processes that are understood and tested on systems that have analytical solutions in closed form. Additionally, these codes must use as input, initial conditions which are, in some sense, based upon observations. It is well understood that galaxies exist, not in isolation, but rather in groups that may be gravitationally bound. As such, it is likely that such systems will undergo encounters, which will fundamentally alter their global evolution. This is true not only for close
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2 encounters, but also for encounters that are more long range in nature. A s the Galaxy is a member of the Local Group with at least one gravitationally bound companion of comparable size a relevant question to pose is whether one can deduce from the present state of the constituent systems anything about encounters between our Galaxy and other members of the group. One can hope to place constraints on the recent interactions by observing whether certain effects predicted by a numerical model of an encounter are consistent with observations. While it may s eem that such a study may be restricted to attempting to reproduce only those processes that may have give n rise to the present configuration of the Local Group, any insight into the general dynamical consequences of these processes will provide astronomers with a better understanding of similar proces s es taking place throughout the universe. Furthermore one may inquire as to whether stellar disk systems all evolve from the same types of initial conditions, thereby leading to unidirectional disks or whether these systems, by reason of initial influences during the formation process or by the effects of the above mentioned interactions, may have a significant portion of their angular momentum reversed. If such systems can form, then a very interesting question arises as to what dynamical effects this counterrotating angular momentum will have on the subsequent global evolution of the system. Also of interest is the question of what effect the initial configuration of the counterrotating angular momentum will have on s tability. Given that such systems have been observed to exist, the relevance of such a study is of more than strictly academic interest. Important in this study is also the question of the role of mixing of the direct and counterrotating components of the system.
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Observations 3 Interacting Sy s tem s When one looks for galaxies one finds that they are u s ually found in aggregations, or groups of other galaxies of differing types and kinds. It can be argued that many of these groups are gravitationally bound and therefore are in association for dynamically significant times. As such, it can be shown with a simple mean path calculation that galaxies in even fairly low number density groups will be relatively likely to have encounters with one or more members over the lifetime of the group These interactions are varied in nature and depend on a number of factors, s uch as the types of galaxies that are interacting. Therefore one must consider the influence of s uch interactions on the s tructure and evolution of individual galaxies and on the structure and evolution of the group itself. This i s especially important when one considers systems in very high number density groups such as the Coma cluster. When a galaxy is observed one might ask if there are any evidences of any such interactions. It is thought that one class of observed galaxies, those referred to as Irregulars, may be chiefly due to such interactions In addition, several observed giant elliptical galaxies, found mainly near the centers of rich clusters, exhibit dense stellar regions that may very well be the nuclei of former cannibalized smaller galaxies An interesting type of galaxy, the polarring galaxy shows a mainly stellar disk, known as an SO disk with a ring of gaseous material rotating in a plane tilted at an angle that is sometimes as great as 90 from the disk plane. Such systems are now widely thought to be due to an interaction between the SO galaxy and a gas rich system. Lastly disk
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4 galaxies in nearby interactions with other s ystems oftentimes show significant w a rping of their di s k components In the Local Group, the question of interactions is an important one because both of the dominant, large spiral disk galaxies, the Milky Way and Andromeda ( M3 l ), are known to have several companions. In addition The Milky Way and M3 l are known to be travelling towards each other and may be gravitationally bound. Thus, the two systems may play a dynamically significant role in each other's global evolution Additionally these large systems may have a dramatic impact on the evolution of the smaller satellite systems such as the Small and Large Magellanic Clouds (SMC and LMC respectively) and M33 and vice versa. Given this possibility one asks what observable evidence may be left behind to point to such interactions having occurred as the interactions di s cussed above are not obvious within the local group (such as a warp in the disk of the Milky Way). Work on Galaxy/M3 l Interactions When it was determined that the Galaxy and M31 are moving towards each other at approximately 100 km/s the question arose as to whether these two systems were actually not just simply gravitationally bound within the context of the local group, but rather, orbiting each other. The problem was originally considered by Kahn and Woltjer ( 1959), who were able to put an upper limits on the mass of the M31/Milky Way system as a function of the Hubble constant. An important conclusion of this work was the possibility that the M3 l /Milky Way system had passed close to each other at an earlier time, thereby effecting the evolution of both systems. In addition it brought into question what other
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5 Local Group interactions may have occurred that might have left their mark on these two main constituents More recently a group consisting of M J. Valtonen G Byrd and their collaborators (Byrd et al. 1994, Byrd and Valtonen 1985 Mikkola et al. 1984, Valtonen and Byrd 1986, Valtonen and Mikkola 1991, Valtonen and Wiren 1994 Zheng et al. 1991, Valtonen et al. 1985, 1992, 1993) have done a detailed study of Local Group interactions using a reduced mass numerical technique While these studies s uffer from the unfortunate deficiency of not including the very recently observed spiral galaxy in our zone of avoidance," they raise several very interesting points One of these is that M3 l may have arisen from a merger of two multiple (binary) systems. Valtonen points out that observational evidence may s upport s uch a scenario. He notes that such an encounter changes the orbital dynamics of a number of the Population II objects in an early stellar system. Due to the interaction many of the Population II objects in the halo component of the system would be have their orbits altered towards plunging trajectories. This change would result in the destruction of a number of the globular clusters in the systems due to tidal effects near the core of the system and leave the resulting merger with a deficiency of clusters near the center of the system. Observationally there is a deficiency of clusters near the center of M3 l (Racine 1991 ). In addition, there should also be a notable s preading of the cluster distribution beyond the edge of the merger. The central deficiency noted here with the globular clusters should not occur with other Population II objects due to the fact that they will not be disrupted by tidal interactions. Observations by Ciardullo et al. ( 1989) show there to be no deficiency in planetary nebulae in the center of the M3 l
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6 system. Lastly Valtonen investigated how Population II metallicity would be affected by a merger. He argues that if the metallicity is determined by the initial mean distance of a cluster from its host galaxy then plotting initial distance versus mean final distance for a cluster in the simulation should reflect the final radial metallicity dis tribution When this is done, Valtonen finds that although the metallicity gradient remains at any given distance the spread in metallicity is wider due to the new distribution of clusters via the encounter. Observations by Huchra et al. (1991) of globular clusters in M31 seem to resemble the numerical results found in Valtonen's s tudy. An important result of this work is that there may be ways to trace the effects of an encounter through the global and local dynamics of the systems involved, as well as through the characte( ~ stics of the populations involved. Hence a question that might be addressed by numerical simulations is whether the effects of an encounter can be modeled in detail. If so, the results could be used to establish criteria for testing whether a disk system has undergone a recent encounter. One important limitation of the work of Valtonen et al. is that the fundamentally important effects of gas dynamics and star formation were neglected. On this topic more will be said below Previous Numerical Work on Encounters As the subject of encounters and mergers is of great interest, a plethora of numerical work has been done to realistically model these phenomena. The initial work on the subject was done by Toomre and Toomre ( 1972) using a restricted three body scheme. They were able to produce bridges and tails in systems undergoing close encounters that closely resemble those observed in obviously interacting disk galaxies (e.g., NGC
PAGE 21
7 4038/NGC 4039). More recently, Hernquist and Barnes (Barnes 1986, 1988, 1992, Barnes and Hemquist 1991, 1992 Hernquist 1992 Hemquist and Barnes 1991 Hernquist and Quinn 1989, Mihos and Hernquist l 994a,b) have investigated a number of disk/disk system mergers. Their work has shown that the luminosity profiles of merger remnants resemble elliptical galaxies (i.e., they follow a de Vaucoulers' r4 profile) but suffer from the difficulty that they are not as tightly bound as are observed giant elliptical systems. Nevertheless, this work has strengthened the hypothesis that giant elliptical systems are the results of mergers of several smaller systems. An important physical aspect of this work is that the models studied included particles representing gas as well as stellar particles. However, no attempt was made to model the process of star formation or to allow for energy loss due to cooling. Cooling must be of paramount importance, as the characteristic cooling times in molecular clouds are very short ( te ,...., 104 105 years). Given this fact, it seems likely that the systems investigated by Hernquist and Barnes would behave differently if cooling effects and star formation were included. The subject of nonmerging interactions has been studied extensively by Byrd and his collaborators (Howard et al. 1993, Howard and Byrd 1990, Byrd and Howard 1990, Byrd and Klaric 1990, Byrd 1976, 1977, 1979a,b, 1983, Sundelius et al. 1987, Byrd et al. 1984 1986, 1987, 1993, 1994, Byrd and Valtonen 1987) who have successfully modeled several interacting nonmerging disk/disk systems. Byrd has had success in matching detailed features observed in such systems by using a multiple encounter scenario for a captured lower mass companion Additionally, some work has been done on the capture and orbital decay of low mass spherical dwarf galaxies by Lin and Tremaine ( 1983),
PAGE 22
8 Quinn and Goodman ( 1986), Quinn Hemquist and Fullager ( 1993), Mihos and Hemquist ( 1994a) and Mihos et al. ( 1995) These studies show that the capture of such sy s tems and their eventual decay and destruction has the initial effect of generating a strong two armed response in the capturing disk. Over a longer time scale, the di s k either spreads radially and inflates vertically and settles into a new axisymmetric equilibrium or can form a triaxial bulge or bar. The orbit of the satellite s core slowly decays into the disk plane where it is destroyed as it merges into the center of the disk. Here again, a notable deficiency of the initial conditions used for these simulations. is a lack of particles representing gas. The authors observe a significant thickening of the disk by the merger that may not occur if the gas cools efficiently Counterrotating Systems Observations The subject of counterrotating angular momentum in stellar systems and models was regarded as being of only academic interest until the observations of Rubin, Graham and Kenney ( 1992) showed that NGC 4550, an E7 /SO galaxy, has a definite segregation of direct and retrograde angular momentum in its stellar disk. This system showed two cospatial systems of stars that are counterrotating In addition, observations of the regular Sab barred spiral NGC 7217, along with deconvolution of the full lineofsight velocity distribution, show that thirty percent of the disk stars orbit retrograde (Kuijken 1993, Merrifield and Kuijken 1994). This population of s tars shows little dispersion in the velocities and as such may be considered a kinematically discrete population within the system. Similar analysis applied to several other systems show that stellar line profiles
PAGE 23
9 deviate from the assumed Gaussian by ten percent in M3 l M32 and NGC 3115 ( Rubin 1994b) and by twenty percent in NGC 4594 ( van der Mare I and Franx 1993 ) due to an excess of stars at low rotational velocities While these systems have not been shown to possess counterrotating populations, the interpretation of the observational data (which has been cited as evidence for nuclear black holes) is called into question An even more bizarre and complicated system is found in NGC 4826. This Sab barred spiral (also known as M64, the Sleeping Beauty etc.) has several distinct kinematic components. Data from the VLA ( Braun et al. 1992, 1994) and uptical observations (Rubin 1994a,b) present the following picture. The disk is comprised of inner and outer gas disks that are essentially coplanar. In the inner disk gas and stars rotate in the sa me sense with trailing spiral arms. This inner disk includes a prominent dusty lane. For slightly larger radii observations show that the ionized gas shows a rapid orderly fall from 180 km/s prograde to 200 km/s retrograde. In this region, which is approximately 500 pc wide, the gas shows a radial velocity component of over 100 km/s towards the nucleus. However, the stars in this region continue their prograde rotation. Beyond this transition region, there does not seem to be spiral structure, but rather weak luminosity enhancements that appear to be more like circular arcs. Here the neutral hydrogen continues to rotate retrograde while the stellar velocities are prograde with respect to the inner disk. Finally, an interesting set of observations have been made by Wozniak and Friedli (Wozniak et al. 1995, Friedli et al. 1995) that show several barred spirals with secondary or nuclear bars. These nuclear bars seem to have a different pattern speeds and kinematical
PAGE 24
10 properties than the larger bars they lie inside of. Work by Friedli & Martinet ( 1993) Shaw et al (1993) and Davies and Hunter (1995b) have shown that it is possible to produce these numerically. In addition, Zasov (l 995) has reported that his observations of NGC 497 show an inner counterrotating region with isophotal structure indicative of an nuclear bar. This system has a constantly rising rotation curve, but is lacking a large bar. Previous Analytical Work While this subject has been, until recently, regarded of only academic interest, some previous analytical work has been done. The earliest general work that has been done is that by Kalnajs ( 1976) Very recently, a great deal of work has been done by Christodoulou and his coworkers (Christodoulou et al. l 995a,b) regarding the determination of a global stability parameter for rotating systems that has direct bearing on the problem. Also of interest is the work of Araki ( 1987) who examined the twostream gravitational problem. Kalnajs In the third of a senes of papers devoted to discussing the determination of a distribution function for a family of flat, rigidly rotating disks, Kalnajs ( 1976) discusses the dynamical effects of the inclusion of a significant amount of counterrotating angular momentum. In this work, models with large eccentricity and/or retrograde orbits within the central disk regions were examined to determine if bar (m=2) instabilities could be suppressed. The models examined include the Toomre n= 1 disk (Toomre 1963) which have had their distribution functions parameterized through the use of both a Fourier
PAGE 25
11 transform and Mellin convolution via inverse Laplace transform. Thus, the amounts of both ordered rotational kinetic energy and dispersional ( "heating ") energy can be varied to produce models of differing distribution functions, which produce the desired s urface densities In the noncircular cases, Kalnajs found a nonvanishing mean rotational velocity near the center of the models. He argues that this is to be expected as high eccentricity orbits will be at perigalactica in this region and consequentially their tangential velocity components will be the largest and in the same direction. As this is a general effect the only way to force to be zero is to admit retrograde orbits. In earlier work, (Kalnajs and Athanassoula 1974) it was suggested that to avoid the formation of bar instabilities the mean angular rotation rate in the central portions of a model must be reduced to approximately one half of the circular rate. One way to do this is to reverse the angular momentum beginning at the radius where r circ= 1 and increasing the fraction of counterrotating angular momentum to 1/2 at r=O. Araki Of interest in this discussion is the work of S. Araki (1987) who investigated a stellar twostream system that is analogous to that studied in plasma physics. He found, in agreement with other investigators (LyndenBell 1967, Harrison 1970), that there is no twostream instability for an infinite homogeneous stellar system with Maxwellian velocity distributions. He notes that LyndenBell (1967) concluded that such instabilities are possible if the velocity distribution is nonMaxwellian. For uniformly rotating stellar disks, Araki analyzes the Kalnajs disk using a weighted distribution function in order to determine whether a twostream instability is present. By using both a linear analysis of
PAGE 26
12 the dispersion relation and a shortwavelength perturbation analysis in the WKB limit, he shows that counterstreaming Kalnajs disks exhibit twostream instabilities for only five low order modes over a small range of an order parameter defined as: V k =Vo' (I.I) where Vis the ordered rotational velocity of the cool disk and Vo is the rotational velocity of the disk supported entirely by rotation. This instability is strongest for the m=l mode and sets in fork> 0.707 The WKB analysis showed that only density waves for the m=O and l modes would be supported by counterstreaming. For differentially rotating disks, Araki applies kinematic arguments to again assert that only the m=O and l modes will support density waves. As such, he notes the results of Zang and Hohl (see below) ( 1978) wherein the inclusion of a greater percentage of counterrotating angular momentum leads to an increasingly prominent onearmed feature in the later stages of evolution. Christodoulou et al. In more recent work, D. Christodoulou, along with a number of collaborators, has begun investigating the possibility of deriving a global stability parameter of flat and three dimensional stellar (noncollisional) and gaseous systems. Prior to these studies, it had been common to use the semiempirical global energy stability parameter proposed by Ostriker and Peebles (1973). The OstrikerPeebles (OP) criterion (see chapter 3 for further description) is based only on numerical experimentation with little analytical basis. Moreover, as several counterexamples have been found, there has been a strong need for an analytically derived criterion. The OP criterion is based only on one integral of motion,
PAGE 27
13 namely the energy of the sys tem. As such, it is clear that a more complete consideration of the problem of global stability (as opposed to local stability as discussed by Toomre ( 1964, 1972)) is needed. Such is the approach of Christodoulou and his collaborators. By including both the energy and the angular momentum as conserved quantities in the derivation of a stability parameter a, for uniformly rotating systems, they are able to analytically arrive at a criterion. By generalizing these results Christodoulou is able to calculate a for systems that rotate differentially. In numerical experiments, it has been found that this parameter predicts global stability for those ~ystems that violate the OP criterion as well as those that do not. As the counterrotating systems in the present research violate the OP parameter, it is important to ascertain whether they are accurately described using the Christodoulou criterion. The derivation of this parameter will be discussed in greater detail in chapter 3 and applied to the systems of interest in chapter 6 Previous Numerical Work Prior to the observations of Rubin et al., the subject of counterrotating angular momentum was largely regarded as academic and therefore, what little numerical work has been done is mainly an afterthought included in other studies. The earliest discussion of the topic was in a paper by Zang and Hohl ( 1978). In this work, they found that the inclusion of counterrotating angular momentum tended to stabilize a disk that would have been otherwise unstable to the formation of an m=2 (bar) mode. They noted, however, that they were unable to completely suppress the formation of a nonaxisymmetric global mode. They found that a strong m= 1, lopsided mode formed in many of their models
PAGE 28
14 (as was later explained by Araki (1987)) This effect was exacerbated as the number of counterrotating orbits increased, especially if those orbits were added in the outer portions of the initial disk. In their study the largest fraction of counterrotating to direct angular momentum was onefourth. The authors were numerically able to confirm many of the conclusions of Kalnajs that were based upon linear theory In addition, Hohl and Zang found that when a disk was initially composed of entirely direct angular momentum, the formation and subsequent dynamical action of the bar mode reversed the angular momentum of approximately 5% of the orbits. Since the announcement of the discovery of the peculiar dynamical arrangement of NGC 4550, there has been a flurry of work by several authors on this topic (Byrd 1992, Sellwood and Merritt 1994). All of these studies have been done concurrently with the work reported in this dissertation. The most notable of these is the work of Sell wood and Merritt (1994). Using a modified version of the Miller cylindrical grid code, the authors studied the inclusion of counterrotating angular momentum in several ToomreKuzmin ( 19??) three dimensional disks. This inclusion was done in a fully mixed fashion and the authors reported that they were able to completely stabilize an otherwise unstable disk and that they were able to generate disks with two barlike patterns rotating in opposite directions. However, a reported drawback of their study seems to be that the growth rates are dependant on their grid and, therefore, they have expressed caution in the interpretation of their specific results. Finally, Byrd and his collaborators have investigated the effects of including coun terrotating angular momentum in their encounter simulations (Byrd 1992). It has been
PAGE 29
15 shown that the close encounter of a massive object and a stable disk galaxy will excite the bar mode in the system Bryd has found that while the inclusion of counterrotating angular momentum will damp the growth of the bar mode somewhat it will not suppress it entirely, regardless of the amount placed in the disk (up to onehalf of the original angular momentum). From this he concludes that he is not able to stabilize an unstable disk with a constantly rising rotation curve with the reversal of any number of orbits, thereby contradicting Sell wood and Merritt ( 1994) and Davies and Hunter (l 995a,b). As will be discussed in chapter 7, this author is of the opinion that Brrd's results may be flawed due to the choice of his initial conditions.
PAGE 30
CHAPTER 2 THE CODES Introduction To model systems such as those described in chapter I, it is important to be able to approach, on some level, the complexity of a real system. To do this, a code must be able to incorporate the actual physical processes as well as model those processes with a large enough number of particles to separate discreteness effects from actu aI dynamical effects In addition it is desirable for the code to be implementable on available platforms, easy to modify to allow for the incorporation of greater detail and accept and produce input and output that is compact and meaningful. The two codes used to model the systems of interest are both based on the heirarchical tree algorithm. The algorithm incorporates the physical considerations of Newton's law of gravitational attraction and the laws of fluid dynamics to evolve the systems of investi gation. Selfgravity is built into the algorithmic structure of the code while fluid dynamic effects are simulated through the formalism of Smoothed Particle Hydrodynamics (SPH). These two concepts, the heirarchical tree algorithm and SPH, form the backbone of the modeling effort and will be discussed in some detail below. It should be noted that the present author has done little of the coding in the realization of the algorithms in their present form. However, extensive testing has been performed on both codes to provide a strong level of confidence in the results produced. These tests will be described in below. Lastly, these codes were used to evolve the same sets 16
PAGE 31
17 of initial conditions, resulting in very similar global evolutions of the initial conditions as well as their conserved quantities. Therefore, we are certain that the results produced are believable and reportable. Hierarchical Tree Algorithm At the heart of the computational scheme used herein is the need to, in a physically consistent way, treat the selfgravity of a large system of particles (2 x 104 say). To do this, a fully Lagrangian method has been devised which incorporates the advantages of previous particleparticle (PP) and mean field expansion methods. Early studies of finite particle systems relied on particleparticle methods (sometimes referred to direct sull1111.ation Nbody methods), wherein a straightforward sum of all other particles in the system was done incorporate forces due to selfgravity While no approximations were made in these calculations, thereby enabling such codes to model systems over a great dynamical range, the time required by the CPU to integrate the model scales as"' O(N2 ) in the simplest implementations of the algorithms (von Hoerner 1960, Aarseth 1963). While refinements made on these simple schemes (Aarseth 1971via higher order integrator, Ahmad and Cohen 1973via multiple time scales) have reduced the dependance on N somewhat ( "' O(Nl.6 ) for the Ahmad and Cohen scheme) it is still infeasible to integrate models of N 2: 10,000. More recently (Aarseth 1967), an independent type of code was developed through the use of multipole expansions to solve Poisson's equation, thereby eliminating the need for extensive summation of individual particleparticle forces. Further work (van
PAGE 32
18 Albada and van Gorkum 1977 van Albada 1982 and Villumsen 1982) has shown that if the expansions are truncated at a low order, the technique is highly efficient, s caling as '"'"' 0( nN), where n is the number of terms in the expansion. Additionally, systems evolved using this method will be less collisional owing to the mean field nature of the solution to Poisson's equation. However for a number of reasons, this approach is rather limited in the cases to which it may be applied. In order to use a low order expansion accurately, the system and the basis functions chosen for the expansion must share the same symmetry. Also, due to the above stated low collisionality, collisional systems (i.e., those including gas) cannot be modeled and twobody effects are suppressed. Finally, due to the anisotropy introduced by the truncated multi pole expansion (McGlynn 1984 ), it is not a trivial exercise to establish a rigorous estimate of the error of such an algorithm. The codes used in the present study use an even more recent combination of these two methods (Appel 1981, 1985; Jernigan 1985; Porter 1985; Barnes and Hut 1986; Barnes 1986; Hernquist 1987; Hernquist 1988; Hernquist and Katz 1989; Hernquist 1990; Heller 1991). The algorithm follows that of Barnes and Hut (1986). An n dimensional box is drawn around the system, n being the dimensionality of the system to be evolved. The center of mass and quadrupole moment of the mass distribution of the box or node are then calculated. For reasons to be discussed shortly, the dipole term is not calculated This acts as the parent node for either an octal or quadratic tree for three or two dimensions respectively (Hereafter, the discussion of the algorithm will be treated as if the model to be evolved is a three dimensional one. To convert to two dimensions, all octal terms should be replaced by quadratic terms.) This parent box is then divided into eight equal
PAGE 33
19 boxes that then become subnodes to the parent node. For each subnode, the center of mass and quadrupole moment are then calculated and stored. This process is recursively continued until each mth level sub cell either has O or 1 particles in it. A force calculation may then be done by walking through the octal tree that has been constructed For each node, a comparison is made between the separation of the mass distribution (henceforth referred to as a cluster) represented by the node and the point of calculation, d, and the size of the cluster, s. Ifs/ d < 8, where () is a fixed tolerance parameter, the walk down the tree beyond the node is terminated All particles in the tree below the specific node that meets the tolerance parameter are now included in a single term (or set of terms) in the force calculation This process is done for each branch of the parent node and each subnode until the entire tree has been pruned. The force between two individual particles is calculated using a Keplerian potential = m1m2' r (2. l) where m1 and m2 are the masses of the particles. Here and throughout this communication, G has been set to l for simplicity. For the calculation of the pair force between a single particle and a node, used when the fixed tolerance parameter is met, the potential is calculated using a multipole expansion about the center of mass of the cluster. It can be shown (Jackson 1975) that, in such an expansion, the dipole term vanishes and as such, the lowest order correction to the monopole term is given by the quadrupole moment tensor. Hence the potential will be given by M 11 (r) = r Q r, r 2 r2 (2.2)
PAGE 34
20 where M is the mass of the cluster, r is the distance from the particle to the center of mass of the cluster, and Q is the traceless quadrupole tensor defined by Qij = L mk(3xki Xkj r~8i1 ) k ( 2.3) when evaluated with respect to the center of mass (Jackson 1975, Goldstein 1980) The codes used in this study allow the user to specify as an input compilation parameter whether quadrupole terms will be included. The merits and costs of doing so will be discussed in the test section of this chapter. To avoid singularities in the force calculation at small separation distances, it is common to introduce a softening parameter into the Keplerian potential in an ad hoc manner. In most cases this modification has been along lines corresponding to a I Plummer density profile of the form ex (r2 + c:2 ) 2 This, however, may not be the best prescription for the solution to the problem. This is because of the fact that the acceleration derived from this function converges only slowly to the Keplerian value compromising local spatial resolution. A better choice for the softening of gravitational interactions is to use a spline kernel. The splinesoftened form of the potential,= mf(r), and acceleration, a=mrg(r), wheref(r) and g(r) are given by { _1. [(l)u2 (1...)u4 + (1)u5J + ..1... e 3 2 0 2 0 5e' f(r) = 11~r H(f)u 2 u 3 + (i3o)u4 Uo)u5 ] + le, r { ~[f(})u2 + (!)u3]' g(r) = *"[/5 + (J)u3 3u4 + (i)u5 (i)u6], ~' O:Su:Sl, 1 :S u '.S 2, u 2:: 2, 0 :S u '.S 1 1 :S u '.S 2 u 2:: 2, (2.4) (2.5) where u = (Gingold and Monaghan 1977, Hernquist and Katz 1989), E being the gravitational softening length. The advantages of this form are that the acceleration and
PAGE 35
21 potential are Keplerian for r 2:: 2c: and that the kernel has compact support. This means that the kernel and its first derivative are continuous everywhere and that for r 2:: 2c:, the Keplerian limit is recovered and no trace of the modifying effect of the kernel is present. Additionally, it has been shown (Romeo 1994) that if the gravitational softening is too large, artificial suppression of fluid dynamical instabilities may occur. As this is the case, we allow for the value of c to vary with each type of particle (i.e gas, stellar halo), and it is possible to allow the softening length to vary in time. This procedure is similar to that of varying the smoothing length parameter in the Smoothed Particle Hydrodynamics formalism and will be described in some detail below. An additional refinement to the hierarchical tree computation would be to allow for all particles within 2c: of the point of inter~st to be resolved into individual particles. The drawback to this is that the code performance degrades in cases of high clustering, due to the increasingly large number of particles in the interaction lists. Smoothed Particle Hydrodynamics While it is assumed in our modeling that the stars may be treated as collisionless particles, any gas dynamic effects must be dealt with assuming collisionality. Conse quently representation of the laws of fluid dynamics in the code must be implemented such a way as to accurately model the continuum limit. We choose to do this using the formalism known as Smoothed Particle Hydrodynamics (SPH). Developed by Lucy ( 1977) and Gingold and Monaghan ( 1977) to circumvent the limitations of gridbased systems (namely that of the requirement of a high degree of symmetry), SPH represents
PAGE 36
22 the fluid elements constituting a system with particles which are evolved according to the dynamical equations obtained from the hydrodynamic conservation laws in their La grangian form. As such, SPH is a fully Langrangian formalism that is easy to implement in three dimensions as there is no grid to constrain the global geometry of the system. Additionally, studies by Lucy (1977), Gingold and Monaghan (1980), Nolthenius and Katz ( 1982) and Durisen et al. ( 1986) have shown that SPH is less diffusive than most simple finitedifferencing schemes. The disadvantages of SPH are that it handles shocks using an artificial viscosity, thereby imposing a limited spatial resolut ion similar to that of grid schemes and it is not possible to represent an arbitrarily large density gradient with a finite number of particles. Still, as this method is Lagrangian in nature it complements the hierarchical tree algorithm described above. This formalism must use a finite number of particles to approximate the continuum limit. SPH does this by assuming that the particle mass density is proportional to the mass density of the fluid, p. As such, the algorithm is able to estimate p as it is evolved according to the laws of hydrodynamics by keeping track of the local density of particles. Inherent in this assumption is the understanding that local averages of the pertinent physical quantities must be performed over nonzero volumes. In SPH this is done through a systematic smoothing of the local statistical fluctuations of the particle number. Within this assumption, the mean value of a physical quantity, f(r ), can be determined through a kernel estimation according to (2.6)
PAGE 37
23 where W(r) is known as the smoothing kernel h is the smoothing length which specifies the extent of the averaging volume and the integration is over all space (e.g., Lucy 1977, Gingold and Monaghan 1977, 1982, Monaghan 1982, 1985). The kernel is normalized s uch that and J vV(r; h ) dr = l lim W(r; h) = S(r ) h~O due to the fact that we require that as h t O (J ( r ) ) t f ( r ) (2.7) (2.8) While a number of different kernels may be selected it is important to keep in mind what Monaghan ( 1992) calls the "Golden Rule" of SPH. This is: if you want to find a physical interpretation of an SPH equation, it is always best to assume the kernel is a Gaussian The kernel used herein is based on splines (Monaghan and Lattanzio 1986, Hernquist and Katz 1989, Heller 1991) as they have proven to be computationally efficient. Since the numerical system is not continuous in distribution Equation (2.6) is approximated by a summation, f(rj) ( I ) (J ( r)) = (n(rj)) W rr; h (2.9) where the summation is over the number of collisional particles in the simulation. The error in estimating equation (2 6) using equation (2.9) depends on the disorder of the particles (Monaghan 1982) and is given by, for a spherically symmetric kernel, (2.10)
PAGE 38
24 where c = J u2h3W ( u ) d u i s independent of h ( Monaghan and Gingold 1982) A s s uch f(r) can be replaced by its s moothed equivalent to within the error of the s moothing proce s s ,...., 0 ( h 2 ) The kernel employed in the codes used in this s tudy is that propo s ed by Monaghan a nd Lattanzio ( 1985) defined by 1 { l (!)(i / + (f)(i)3, W ( r h ) =1rh3 ( t)[2( f)]3, 0 o s; i s; 1 1 s; I s; 2 k > 2 (2 .11) The advantages of using this kernel are that it has compact support, uses only tho s e particles within 2h for the smoothed estimates of physical quantities, has continuous first and second derivatives and is, as stated above, accurate to s econd order. In two dimensions, the normalizing constant is 7;i2. If we then assign a mass m1 to each fluid element, equation ( 2.9) may be written N (p( r ) ) = L m1W(rr'; h) (2 .12) j=l According to Hemquist and Katz ( 1989), equation (2 12) may be interpreted in two distinct ways that may be thought of as being analogous to the meanings of computational gather and scatter operations. In the "scatter" interpretation each particle has mass that i s smeared out over space according to W and h. The density at any point is found by s umming the different contributions from the density profiles of the neighboring particles This is the more traditional point of view. Another way of regarding this is with the "gather" approach where the particles are regarded as point markers in the fluid. Local physical quantities are then arrived at by sampling all particles within 2h of the point of interest and weighting the contribution of each by W. While these two different view s
PAGE 39
25 are indistinguishable if the smoothing length, h is the same for all particles, the codes used for this study allow for each particle to have a differing value for h. The reason for this is that the local statistical fluctuations resulting from the kernel estimates are determined by the number of particles within 2h of a given point in space. For example, if a constant value of h were used, the estimates of the physical quantities of the system would be more accurate in regions of higher density than similar estimates in regions of lower density. Moreover, an algorithm employing a constant h would not take full advantage of the distribution of SPH particles to resolve local structures. Given the above interpretations, we may generalize the SPH formalism to allow for a variable smoothing length. If we consider the scatter interpretation equation (2 6) becomes which leads to equation (2.9) being rewritten as N (p(ri)) = Lm1W(ri1 h 1 ) J=l (2.13) (2.14) where 1'ij = lrirjl (e.g., Gingold and Monaghan 1982, Nagasawa and Miyama 1987). For the kernel stated above, it can be shown that the interpolation errors are dependent upon h (Monaghan 1987, Hernquist and Katz 1989). In the gather interpretation, equation (2.6) may now be written as from which we can shown that equation (2.9) becomes N (p(ri)) = L m 1W(ri1 ; hi), j=l (2.15) (2.16)
PAGE 40
26 where the summation now depends on hi rather than hJ as in the scatter interpretation (e.g., Wood 1981, 1982, Benz 1984, Loewenstein and Mathews 1986, Evrard 1988). Here, the interpolation errors are independent of h However, as pointed out by Monaghan ( 1985), the total mass of the system will no longer be conserved when using the gather approach. The size of the resulting errors can be shown to be ,..., 0 ( h 2 ) and, therefore, are consistent with the other errors in the system. Hernquist and Katz ( 1989) have shown that similar difficulties are present in the scatter interpretation and these are of the same order as those for the gather method. A more important drawback to either scheme is that the kernel is no longer symmetrical with regards to the contributions of one element to another. When this asymmetry occurs, as it will for all systems with a dynamical h it manifests as a violation of Newton's Third Law for the fluid elements of the system. Therefore to conserve momentum in the equations of motion, the formalism must be symmetrized in hi and h 1 There are two approaches used to symmetrize the formalism. The first involves using hi~h1 in place of hi or hJ in the kernel, while leaving the kernel in its former form. Hernquist and Katz ( 1989), R. Drimmel and this author have experimented with this approach and have found it unable to handle systems in which large gradients or shocks are present locally Additionally, Hernquist and Katz have found that such a correction is likely to introduce errors into the integration of the system. Alternatively, the codes used herein use a symmetrization of the smoothing procedure itself. This method, which may be thought of as a hybrid of both the gather and scatter interpretations, is given by (2.17)
PAGE 41
27 in analogy to equation (2.6). This can then be shown to give, in discrete form, N 1 (p(ri)} = L m;2[W(rij, hi)+ W(rij, h;)], j=l (2.18) which will reduce to equation (2.9) if hi = h;. As this equation is merely a linear combination of the two interpretation, the errors introduced by the formalism will again be ,.., O(h2). Steinmetz and Millier (1991) note that in their tests, this method of symmetrization shows a somewhat better damping of post shock oscillations in the fluid in the case of strong shocks. As can be seen, a final advantage to this method is that each particle will have its own well defined size, which simplifies the implementation of both tree construction and force evaluation. When h is allowed to change with respect to time for each individual particle, there are two common ways the algorithmic implementation has been done. Hemquist and Katz (1989) note that the smoothing length should be proportional to the local interparticle I separation, scaling as h oc n:i in three dimensions. Hence, a twostep approach is used to update h such that each particle interacts with a roughly similar and constant number of neighbors. If hf1 is the smoothing length at step n1, and the number of neighbors at that time is Nr1 then hf may be predicted by hf!.= h~l~ [1 + (~) t] I I 2 Nri1 I (2.19) where Na is an input parameter. Then the current number of neighbors Nr is found using the above calculated value of hf. If Nr differs from Ns by more than some predetermined tolerance, then the smoothing length is corrected so that the number of neighbors falls within the tolerance. While this method is stable and compels all of the
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28 smoothed estimates to be of roughly the same accuracy at all points it suffers from the deficiency that it is rather computationally intensive. A second method proposed by Benz (1990), is to write (2.20) where ( is a parameter of order unity By differentiating this equation with respect to time, and then using the equation of continuity to rewrite the righthand side, we arrive at dh 1 = h'\J V. di 3 (2.21) Using this equation, the smoothing length is then evolved dynamically as would be any other hydrodynamical quantity, thereby saving a great deal of computational time. A drawback to both of these approaches is noted by Steinmetz and Mi.ill er (1991 ), who point out that neither scheme guarantees the smoothness of h and stability against large amplitude density fluctuations. As h is dependent on the density, an amplification cycle can then occur wherein the small density fluctuations then amplify the formerly small fluctuations in the smoothing length and so on. They suggest a somewhat different way to evolve h. As the smoothing length can be too strongly dependant on the local density in simulations wherein the distribution is very lumpy, Steinmetz and Millier suggest talcing a ratio between the local density and some average density and coupling h to that quantity. By doing this, they report that density amplification does not occur and the ancillary benefit of greater numerical stability for hydrostatic configurations. This is especially true in those cases when artificial viscosity (see below) is not used. Finally, their scheme reduces the numerical diffusion of SPH and requires no further computing
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29 time than that already required to run with the varying smoothing length algorithms of Hernquist and Katz or Benz The process of estimating gradients in SPH is fairly straightforward By definition, (Vf(r)) = j WVJ(r1)dr1 (2.22) where W l { W [rr', h ( r')] + W [ rr', h(r)] } Integrating equation (222) by parts, we obtain (2.23) if surface terms are ignored. For discrete systems this may be rewritten as (2 24) where W = ![vV ( rij hi)+ W(rii, hi)] by analogy to W. From this point onwards, we will assume that index j denotes a particle label and that the summation is over all SPH particles. Therefore, we will also say that particle j has mass m1, position r1, density PJ and velocity VJ. It is important, at this point, to use what Monaghan (1992) refers to as the second "Golden Rule" of SPH. This rule is that it is better to write formulae with the density placed inside of the operators Thus, we write (p(r)V J(r)) = (V(p(r)J(r))) (f(r)V p(r)) (2.25) which becomes, in the discrete case, N (p(r)Vf(r)) = Lmi[!(r')J(r)]vw. (2.26) j=l
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30 Divergences may be written in discrete formalism, as N (p( r)(V f ( r ))) = L m1 [ r (r' ) f ( r)] VW, j=l (2.27) where the gradient of the kernel is taken with respect of the coordinates of the unprimed particle. Curls may be taken in a similar way. Given this procedure, it is now possible to derive smoothed forms for the hydrody namical conservation laws which may then serve as equations of motions for the particles If equation (2 .18) is used to compute the density then to terms of"' O(h2 ) the conti nuity equation will automatically be satisfied and, as such, will not need to be integrated forward in time. To evolve a particle in phase space, we use the traditional form given by Euler's equation, dri =v dt 1 (2. 28) (2.2 9) where i is the gravitational potential, Pi is the pressure and arsc is an artificial viscosity term, used allow for the simulation of shock waves in the medium. A number of forms may be used for the smoothed estimate of "il:; however it should be noted (Monaghan 1992, Hemquist and Katz 1989) that whichever form is used should be symmetrized to allow for the construction of a consistent energy equation and to conserve linear and angular momentum. Monaghan ( 1992) suggests an arithmetic mean be used, i.e., VP_ (pi P 1 ) +p p; PJ (2.30)
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31 While this solves the difficulties mentioned, Hernquist and Katz (1989) note that this method can lead to instability in the integration of the thermal energy equation when a leapfrog integrator is used. Consequently, they suggest a symmetrization based on the identity, (2.31) which, via the formalism, yields 9 pi .;..., /PYi = L.., m 1 2 9W. Pi j=l Pi Pi (2.32) The introduction of an artificial viscosity is necessary for the code to accurately treat shocks in the fluid medium (Monaghan and Gingold 1983, Monaghan and Pongracic 1985). A form suggested by Monaghan ( 1988) is N arsc = L mjIIij9W, 1=1 (2.33) where IIij is the viscous contribution to the pressure gradient. While a number of different expressions have been proposed for IIij, one that gives an accurate description of the fluid flow near a shock is given by 0'.Cjjij + /3; 1 IIij = Pii Vjj fij < 0 (2.34) (2.35) v r > 0 I] I) ) h (c;+c;) h d f d art I d were v,1 = v v c t e average spee o soun p 1c es z an J, I ]' I) 2 hij = (h;ihj) and Pij = (PiiPi). The first term in equation (2.34) is analogous to a bulk viscosity, whereas the second term, which is intended to suppress particle interpenetration,
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32 is similar to a von NeumannRichtmyer artificial viscosity (Evrard 1988 Monaghan 1988). Typically, o: ,...., 0.5, j3 '"'"' 1.0 and TJ ,...., 0.01 to prevent numerical divergences (Hernquist and Katz 1989) Due to the drawback that this description of the viscosity introduces a large effective s hear viscosity, we have included a switch (Benz 1990 Drimmel 1995b) by multiplying ij, (ij+ i1fi1 ) The factor fi1 = (!;if;), where l(v'v)li Ji = I ( v' v) Ii + I ( v' x v) Ii + 0 .001 f and the curl can be evaluated, as described above, N (v' x v ( r) ) = L m1(vi v 1 ) x v'W. 1=1 (2.36) (2.37) While this factor reduces the shear viscosity, it does so at the expense of not being able to vary the effective shear. Thus, this factor is not applied to the first term in the viscosity equation To close the system of fluid dynamical equations, an equation of state must be added. Three different equations of state may be selected in the codes. The first is the familiar ideal gas law. Written in a convenient form, P = (1 l )pu (2.38) where u is the specific thermal energy and I is the ratio of the specific heats. A second choice is the polytropic equation of state, given by ( 2 39) If 1 + 1 and J{ = cf so then this equation becomes the isothermal equation of state, which is the third option available in the codes. It should be noted here that the polytropic
PAGE 47
33 equation ass ume s that the sy s tem is adiabatic and, therefore require s no de s cription of the evolution of u. To evol v e u we u s e a therm a l energy equation derived from the fir s t law of thermodynamics d u = PdV + Tds. ( 2.40 ) In this equation, V = i i s the s pecific volume and all nonadiabatic effects are included in the change in the s pecific entropy The s moothed version of the thermal energy equation may be written ( Hemquist a nd Katz 1989) as d uj L N ( /P]5; 1 ) r A d = m1 + IliJ Y i J 'vW + , t p p 2 p J=l l J ( 2.41 ) where rA accounts for the heating a nd cooling terms not associated with artificial p vis cosity ( Field 1965 Spitzer 1978) One additional note involving the dynamical evolution of the fluid in SPH. Monaghan ( 1992) has noted that equation (2.28) may be rewritten in a variant of SPH known as XSPH (Monaghan 1989) In doing this equation (2.28) becomes dr N (V )1 = v = v + c m 2 w d t I I J l J=l P i 1 ( 2.42 ) where E i s a constant over the range 0 :S c :S 1. While no dissipation is introduced using this variant XSPH does increase the di s sipation already present. However, it has proven very useful in the simulation of nearly incompressible fluids as it keeps the particles orderly in the absence of viscosity. Finally before we leave the subject of SPH, it should be noted that by introducing a s moothing length that is allowed to vary in both s pace and time the formal form of the
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34 SPH equations change. The equations will now include terms proportional to 'v h and In this case, the correct form of the SPH equations is obtained when the continuum equations are multiplied by W and integrated over all space. The se results must then be integrated by parts. This procedure can be simplified through the use of the following relations In the s catter formulation, (2.43) (2 .44) while in the gather interpretation, I dA) = d(A) dh(r) J A aw dr' \ dt dt dt oh + j [(v'v) 'vr,A]Wdr1 (2.45) (2.46) (Hemquist and Katz 1989) For the mixed formalism used to symmetrize the kernel, the corrections to SPH will be a linear combination of these. From this point, the dynamical equations are obtained in the prescribed manner. Typically, the additional terms acquired due to these extra considerations are smaller that the dominant, physical terms by powers of N t in three dimensions and, therefore, will be unimportant for runs of large N (Gingold and Monaghan 1982, Hemquist and Katz 1989, Monaghan 1992).
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35 Time Integration The codes used in thi s study u s e a standard leapfrog integrator to update the particle positions and velocities Thi s integrator is accurate to second order in the time step flt, a nd has the advantage that in the formal limit as flt + 0, it preserves the Hamiltonian character of the sy s tem For particle i the positions and velocities are updated according to ( 2.47 ) ( 2.48) where the superscripts refer to the time step at which the quantities are computed For the SPH particles in the simulation the acceleration, ayisc, is dependant upon the velocity of the particle. In this case second order accuracy is maintained by updating the velocity in two stages. Initially a predicted estimate of the velocity is calculated using ( 2 49 ) This predicted value is then used to calculate the timecentered acceleration a~+ t This is then used to update the velocity a ccording to equation (248). The use of a higher order integrator is not appropriate here since the long range gravitational forces are computed using a multipole expansion that is only carried out to the first term. A higher order integrator would only amplify the random noise in the acceleration computation. The leapfrog does not suffer from this difficulty as it computes the acceleration only once per time s tep (Hemquist and Katz 1989)
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36 If the thermal energ y equation (2.41) is to be used in conjunction with an ideal gas law equation of s tate (2.38), then it is necessary to advance the thermal energy along with the positions. To avoid complications with the Courant condition (see below), the thermal energy equation is evolved semiimplicitly using the trapezoidal rule n+l. nl 6.t ( n+l. nl) u 2 = u 2 + 'U.. 2 + ti 2 + 0 (6.t3) 1 1 2 I 1 l ( 2.50) similar to the approaches of Lucy ( 1977) and Monaghan and Varnas ( 1988). equation (2.50) will be, in general, nonlinear in u7+t, depending on the form of u.7+( which must be solved iteratively. If the righthand side of the thermal energy equation depended only on the nonadiabatic terms the solution would be relatively straightforward. However since the equation depends on u;+t through Pj, which involves the neighboring particles a rigorous solution of the equation would require the solving of two coupled nonlinear equations for each particle neighbor pair. Instead, an approximate two step approach is used. This method, which is equivalent to a secondorder RungeKutta integrator, first makes a prediction of the thermal energy by solving the thermal energy equation, implicitly assuming that the first term on the righthand side of the equation doesn t change from step n ! to step n + f. The predicted value is then used, along with the predicted velocity, to solve for u.7+t. Finally, the thermal energy equation is again solved implicitly using both the predicted and old adiabatic and viscous terms. The solution of the nonlinear equation in both the predictor and corrector portions of the algorithm is done using a hybrid of the NewtonianRaphson and bisection techniques (Hernquist and Katz 1989, Casulli and Greenspan 1984).
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37 The issue of time step size in an explicit integration scheme, such as the leapfrog integrator, is one of some interest. For the SPH particles, the time step is limited by the Courant condition. If the time step is fixed, the stability of the integration can be assured by varying the smoothing length according to the Courant condition (Evrard 1988). However, as this may limit the algorithm's ability to resolve regions of high density accurately, the codes allow for each particle to have its "own" time step. These are chosen so as to maintain integrational stability, according to the modified criteria derived by Monaghan (1988a) For the artificial viscosity scheme used herein, this may be written as (2.51) where C0.3 is the Courant number. If r or A in the thennal energy equation are nonzero, then additional constraints must be placed on the Courant condition if the time scale associated with these quantities are smaller than the dynamical time. In simulations where the handling of multiple time scales is necessary, the limitation of having a single system time step is very inefficient. As such, the codes used herein allow for each particle to have its own time step This is done in a manner similar to that of Porter (1985) and Ewell (1988). Each individual time step is chosen to be a power oftwo subdivision of the system time step, t:..ts, such that, t:..ts t:..tk = 7 k = 1, 2, ... n; (2.52) where k refers to the time bin level and b..ts and n are input parameters that remain fixed (Hemquist and Katz 1989, Whitehurst 1988, Heller 1991). Each individual time step is
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38 determined via equation ( 2.51 ) wherein the particle i s placed within the large s t time bin (smallest value of k) so that ( 2.53) for particle i. All of the particles of a particular bin are advanced together; the ordering being selected according to Figure 2.1. 0 Bin 1 2 2 3 1 I 8 4 12 6 10 3 5 I 7 9 I 11 Figure 2 l : An ordering diagram to determine the advancement of particles with multiple time s teps 14 13 I 15 Particles are allowed to move to a smaller time step (larger value of k) at the end of their own time step, but are only allowed to move to a larger time step if the two bins are currently time synchronized This insures that the system will be synchronized after each set of 2n+l 1 time steps. The time step used to advance the particle positions and thermal energy equation is ( 2.54) where kmax is the largest time bin (smallest time step) currently occupied by particles in the system. Therefore 6.tpos is onehalf the smallest time step thus assuring that the pressure and gravitational acceleration is computed at the midpoint of each time step as required by the integrator. Additionally the most recently updated value of the velocity
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39 is used to update the positions. If a particle changes the time bin it is in, its position must be updated using an estimated midpoint velocity in order to preserve the accuracy of the integration scheme. This is found using (Hernquist and Katz 1989, Drimmel 1995b) where ( 1 ) ( 1 ) o ld r i ,cor = ri, uncor 1 X 1 + X 8 a i, 6.ti old x D..ti new I (2.55) (2.56) The same procedure may be used to make the leapfrog integrator selfstarting (Hernquist and Katz 1989). If the positions and velocities of the particles are given upon entry into the integrator, then a secondorder estimate of the positions may be obtained from (2.57) These additional terms guarantee secondorder accuracy in the leapfrog integrator without need of higher order integration schemes. Miscellaneous As SPH is formally similar to other simple Nbody methods, it is well suited for the simulation of systems comprised of both gas and collisionless matter. When this is the case, the phase space coordinates of all the particles are evolved using equations (2.48) and (2.49). For collisionless (i.e., stellar) matter, the acceleration is comprised of only the contributions due to gravity and, as would be expected, there is no need for the thermal energy equation. Additionally, since these algorithms solve the hydrodynamic equations
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40 in an ab initio way the correct continuum limit will be approximated and those physical processes that depend on the thermodynamic state of the gas can be rigidly s pecified. An example of this is the inclusion of accret i on of gas onto a large central body (Drimmel l 995a,b, Heller and Shlosman 1994 ). Another example is the inclusion of star formation processes in the simulation. It has been shown by Heller and Shlosman (1994) that such effects are of vital importance in modeling the gas dynamics of the system. However, it is not clear as to how such processes are to be included in the modeling algorithm as the physics of the actual process in the interstellar medium is not ':"ell understood An attempt to model such effects has been made and will be discussed in greater detail in chapter 4 of this dissertation. Optimization of the algorithms used (TNDSPH (Drimmel 1995a) and FfM (Heller 1991)) is achieved on several levels. The greatest improvement in the speed of the code is achieved through vectorizing the tree descent (Makino 1990, Hernquist 1990) and tree construction (Makino 1990, Hernquist 1990). The methods used result in an improvement of speed of ,..., 10 and are not specific to an SPH implementation of the hierarchical tree method. Some minor refinements of the codes involve two differing elements of the algorithms First, the evaluation of the kernel functions ( equations (2.4 ), (2.5) and (2.11)) is done using lookup tables to determine the values of the smoothing and softening kernels from 0 to 2u. Second, the necessary searches for nearest neighbors in the codes are combined into a single search and listing for each particle at the beginning of each time step. Thus, the CPU cost per time step is reduced at the expense of greater memory requirements.
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41 Notwithstanding, unless the number of particles is very large, the codes will be limited in performance by CPU considerations rather than those imposed by memory availability. Tests of the Algorithm Tests in One Dimension A number of one dimensional tests were applied to the codes used in this study. The reason one dimensional testing has been used is that the test problems have analytical answers that may be compared to the output of the codes so as to judge accuracy. The two most commonly run tests are the shock tube test and the colliding gas flows test. It has been found by a number of authors (Monaghan and Gingold 1983, Monaghan 1989, Hernquist and Katz 1989, Heller 1991, Drimmel 1995b) that SPH does a fairly nice job of reproducing the analytical results over a large range of Mach numbers. One cautionary remark should be made here It has been seen (Monaghan and Latanzio 1986) that SPH can have difficulty discerning structure in small regions with high densities (Hunter et al. 1986). This is due to a lack of resolution brought about by an insufficient number of particles needed to define a shock region. While some improvement can be made in the results of a specific system be modifying a and /3 in equation (2.35) (Heller 1991), generally optimal values for these parameters do not exist (Hernquist and Katz 1989). It should also be noted that when using the SPH algorithm, there will always be some postshock oscillation in the physical quantities, though these are reduced when using the version of dynamical softening put forth by Steinmetz and Millier (1991 ). Fin~lly, Drimmel ( 1995b) has shown that to prevent interpenetration of particles in the colliding gas flows problem, it is necessary to employ an artificial viscosity having both linear
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42 ( a =I= 0 ) and nonlinear (/3 =I= 0 ) terms When using an artificial viscosity in conjunction with a polytropic equation of state Drimmel ( 1995a) has also shown that an effective cooling is introduced into the system. This effect can be lessened by using the viscosity switch given in Equation 37 Tests in Two Dimensions We have done a two dimensional test of the algorithms by analyzing the small oscillation period of a Maclaurin disk. The oscillation properties of a two dimensional nonrotating disk may be found from analyzing the system using the two dimensional virial theorem, 1 d2 I '> I
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43 and, assuming the surface terms to be negligible Ps V + 0. (2. 63 ) If we now introduce a small perturbation, c, into the system s o that R = R0( l + t:), (2. 64) then the virial theorem may be rewritten as f M R2 2 P o ( l )2(yt)M f Gl'vln 2 D 0l + c D 1c., Vo Ro (2.65) when terms of second order or higher are dropped. Expanding the first term on the right side using a binomial expansion and noting that, in equilibrium, 2&Mn = Ji GRMo, V o o equation (2.65) may be simplified to read .. 2Ji GMn [ 3 ] c + 1 c = 0 !2 R g 2 (2.66) Finally, if we assume oscillatory solutions of the form, c = c0eiwt, then the solution is easily arrived at and given by w= (2. 67) As can be seen, the solution becomes physically meaningless for < !For values l ess than this critical the system is unable to support its own weight and will collapse beyond the linear regime. Note, this derivation is true for all disks if the factors Ji and f2 are known. Thus in two dimensional calculations = has the same significance as d oes = in three dimensional problems.
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44 For a Maclaurin disk it can be s hown that Ji = /0 and h = f a nd, therefore the period of o s cillation will be given by 271" T=1 ( 2 .68) [ G MD (, ~) r Simulations show the s y s tem undergoing regular period oscillation. The analytical values for several values of I are given in Table 2 1 along with the values produced by the codes. As can be seen the SPH codes reproduce the analytical result to better than Table 2.1: Period of o s cillation for two dimensional Maclaurin disks / T anal (yrs ) T code ( yrs) 2 2 5 3 X 108 2 .5 5 X 108 l.75 3 .58 X 108 3.85 X 108 1.6 5 667 X 1 08 5 .6 8 X 108 1 % Given that these simulations were done with only 5000 particles we see that with even a relatively small number of particles, SPH is able to very accurately reproduce the behavior of mildly unstable systems. It should be noted that due to the effective di s sipation introduced by the artificial viscosity this oscillation is damped out over time For a system that is rotating, with a the virial theorem can be written as ( 2.69 ) /3M R 2 2 where kRo = n2 w0 Again assuming oscillatory solutions, the solution is ( 2.70)
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45 or WR= WN+2/ i~ = ;;t (2 71) where w N R is the oscillation frequency of the nonrotating case and t is the OstrikerPeebles (1973) ratio of rotational kinetic energy to gravitational potential energy It should be noted that the system will only be stable over a range of and t such that ( 2.72) Simulations have been done to test the validity of the code in this case and the analytical results were found to be reproduced as long as the system stayed within the linear regime. As is discussed below, if t ;s 0.14 the system will be stable against the formation of nonlinear modes Tests in Three Dimensions Tests of the code have been performed in three dimensions by investigating the radial collapse of initially isothermal gas spheres (Hemquist and Katz 1989) The results were found to be in excellent agreement with those obtained using finitedifferencing (Thomas 1987) and particlemesh (Evrard 1988) schemes. This is an excellent test of the code due to the large range of smoothing lengths and time steps needed in the problem Another test that has been conducted (Heller 1991) is very similar to the two dimensional tests conducted above Here, Heller simulated the radial oscillations of a n= 1 polytrope (Chandrasekhar 1967) and found that the code was able to accurately reproduce the expected pulsation frequency. Hence, we are confident that the algorithm presented here can accurately model physical phenomena in fully three dimensional settings.
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46 Specifics of the Codes Used TNDSPH This code was developed from the TREECOD hierarchical tree algorithm provided by L. Hemquist ( 1987) to our group. The SPH implementation was done by R. Drimmel ( 1995a) with assistance from N Hiotelis. Written to run on an IBM VMS platform the code is fully vectorized and has separate two and three dimensional versions. Much of the initial work in this modeling program was done with this version of the code and it is mainly used, at this time, to confirm results produced by the FTM code. The main properties of this code are listed in Table 2.2. FTM Provided to our group by C. Heller (1991), the FTM code was originally written as a C implementation of the hierarchical tree/SPH algorithm. The code is now configured to run on a number of UNIX platforms and is written in FORTRAN. This code is used for the bulk of the modeling program at this time due to its more flexible output and inclusion of star formation routines. The code allows for a number of switches to be set at run t ime, further enhancing its flexibility. The code is further detailed below in Table 2.2.
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47 Table 2.2: Propertie s of the Codes u se d in the Present Modeling Effort Property TNDSPH FfM SelfGravity Always Switch 2D/3D Kernel Switch Switch Multistep Always Switch Quadrapole Switch Switch Dynamical Softening None Switch XSPH None Switch Artificial Viscosity Switch Yes No External Potential Hard Coded Switch Accretion Yes Yes Star Formation No Switch Boundaries No Yes Output ASCII IEEE/IBM Binary Black Hole No Yes
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CHAPTER 3 INITIAL CONDITIONS Introduction While it is vitally important to u se computer algorithms that are both accurate and computationally efficient, it is just as important to begin a simulation with initial conditions tha t are physically meaningful and analytically tractable In practice this requires tha t the initial disk must be con s tructed from a set of meaningful assumptions that will gi v e insight into ob served phy s ical systems. Additionally it is important to be able to v ary certain physical parameters of the stellar system s uch as the first two integrals of motion ( energy and angular momentum) in an easy, systematic way in order to be able to explore a parameter space of interest and relevance. Lastly the set of initial conditions used should resemble the properties of real, observed disk s ystems or, if they do not as in the case of the Kalnajs/Hohl ( KH) disk (Kalnajs 1970 1971, 1972 1976a b 1977, Hohl 1970a,b 1971, 1972 Hohl and Hockney 1969 Hockney and Hohl 1969) they should have other physically important properties that warrant investigation The construction of such sets of initial conditions has been referred to, by s ome, as something akin to an art. Certainly, it is true that a number of subtle techniques must be used to construct initial disks that are in radial and azimuthal equilibrium, easy to generate an d that mimic the possible evolution of observed disk systems. As this topic is of some interest and importance we choose to spend some time here on the details of disk building for numerical integration. A good review of the subject has been written 48
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49 by Sell wood ( 1987 ) who, in addition to the fundamental considerations of the s ubject discu ss es the techniques u sed to obtain quiet starts ( i e. those initial conditions that are sufficiently close to equilibrium that they do not produce undesirable dynamical effects during adjustment to equilibrium "). Observation of Isolated Galactic Disks When observing disk systems, a number of useful properties can be measured. Of greatest interest to this present s tudy are a measurement of where the mass in the sys tem lies, i e. the density (s tellar and gas) and how the mass moves, i.e., the rotational velocity curve. Additional useful information is the stellar velocity dispersions around the mean rotational velocity. As the measurement of these quantities is a complicated bu s iness we will state only the general results with a few important notes to be made when necessary In measuring the density of the system, the issue is first complicated by the fact that the observations are usually made of disks that are inclined to the plane of observation, rather than face on. Even when this effect is accounted for the observations will still only produce a surface density, I;(r, ), where is the azimuthal angle. (I;(r, ) has units of mass/area.) To produce a volume density an assumption must be made as to the distribution of matter in the z direction. Oftentimes it is assumed that p(r ,,z) = I;(r,)g(z), where g(z) is some z distribution with units of length1 A final complication is that there is an assumption of a proportional relationship, that is constant in all regions of a disk between the density and the measured surf ace brightness of the system. While there is observational evidence in support of this assumption in the outer portion of the disk there is also much that calls it into question when considering
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50 the innermost region of disk or barred disk sys tems A great deal more work need s to be done to s ort out thi s problem in its entirety. Nevertheless the results reported here are based on the claim that this assumption is true Observations in the visible s pectrum, as well as ,\ 21 cm data suggest that the surface density in many dis k like sys tems may be fitted by an exponential curve in the r direction (de Vaucouleur s 1959 Freeman 1970). While it is entirely possible for the data to be fitted using other function ( as will be discussed below), this seems to be the simplest function to use While data for the determination of the z distribution of the mass is sketchier, a common assumption is that it may be described using isothermal sheets distributed perpendicular to the disk plane (Spitzer 1942 Bahcall and Soneira 1980). Another possible assumption is that this distribution may also be modeled using an exponential function in the z direction. La s tly it should be noted that it is usually found that the gas component of a disk system has a greater physical extent in the r direction and a lesser extent in the z direction than do the stellar counterparts. Consequently, it is usually assumed that the two components of the system will have differing scale lengths. To measure the rotation curve of a disk system, there are two main techniques. The first is to infer the rotational velocity of the stellar component by using the rotationally induced Doppler shift of particular spectral lines of the system as a function of r A determination of the velocity dispersion may also be made in a similar way through measuring the broadening of the spectral line and assuming a distribution function (usually a Gaussian) to describe the stellar motions that give rise to this feature. This technique is quite difficult and the procedures used to separate the rotational velocity from the velocity
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51 dispersions are s ubject to a number of assumptions. The second, somewhat more reliable, method is to make a similar measurement of the ,\ 21 cm lines of neutral hydrogen gas. By measuring the gas one also gains a clearer picture of the underlying potential of the system. Both effects, the greater reliability and the ability to probe the spatial variability of the potential, are due to the fact that the gas component is constrained to move along hydrodynamic streamlines Therefore, the velocity dispersions of the gas will be much lower than those of the s tellar component of the disk which means that the measurements of the rotational velocity of the gas will be more accurate. The observations of galactic disk systems reveal the following general information regarding the rotation curve. Near the center of the disk, the velocities rise linearly at a fairly rapid rate This is seen mainly in the stellar component of the disk, though ,\ 21 cm and CO observations confirm this. At some radius, the rotation curve flattens out or in some cases turns over and begins a slight decrease. These data come mainly from ,\ 21 cm observations because usually the gas disk extends well beyond the visible stellar disk. As will be discussed below, several different analytical fits can be made to the rotation curve data. Finally, it should be noted that the rotation curve data again for reasons to be discussed below is perhaps the most reliable and useful tool in studying disk systems. Analytical Disks General Discussion To establish a set of selfconsistent initial conditions from which we may proceed, it is important to have a theoretical background from which to start. As these analytical
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52 solutions must eventually correspond in some sense, to observable systems, it is germane to the topic at hand to review the basic observational evidence in light of the theoretical framework. From this, we may extend what is available analytically by using justifiable assumptions to produce a set of initial conditions that meet the criteria set forth at the beginning of this chapter. When considering selfgraviting systems, we begin from three fundamental suppo sitions. First we assume that Newtonian mechanics, namely the second law is the dominant consideration in understanding the dynamics of the system and its individual components (i.e., relativistic effects may be ignored). If we assume that the particles obey the second law and are confined to circular orbits (this constraint shall be relaxed considerably), then through the use of potential theory we can write diI> dr r (3.1) where iI> is the gravitational potential of the system and r the radius from the center of the system to the point of interest. Second, we assume that we may represent the gravitational interaction via a smoothed potential function and therefore, Poisson's equation may be written as 'v2iI>(r,,z) = 41rGp(r,,z), (3.2) where p(r,,z) is the volume mass density of the system. Third, we assume that there is a distribution function, f (r, , z, vr, vip, Vz, t), that will describe the individual particle motions of the system such that equations (3.1) and (3.2) are satisfied. It should be noted that while knowledge of J(r, , z, Vr, vip, Vz, t) will uniquely determine p (r, , z) and, therefore, v(r, , z), the converse is not true.
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53 Observation a lly as previously discussed, we are a ble to determine reasonably good estimates of p ( r, and in doing so relate p ( r
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54 one may assume a density distribution for the initial disk and from this determine the rotat i on law that will determine the motion of the particles in the system. Here again, one must pick a form for the distribution function and, almost invariably the disk will begin somewhat out of equilibrium. This is the method of Fall and Efstathiou ( 1980) and Hemquist (1993b). Kalnajs/Hohl Disks In two separate series of papers Kalnajs (l 970 1971, 1972 l 976a,b 1977) and Hohl (l970a,b, 1971, 1972 Hohl and Hockney 1969 Hockney and Hohl 1969) investigated a disk characterized as an infinitely flattened Maclaurin spheroid. Given a density profile of i :S 1 i > 1 we can derive, via the Poisson equation, the potential, or, relating this to the velocity we can write V, = n, r (3.3) (3.4) ( 3.5) showing that these disks will rotate as rigid bodies. Additionally, a distribution function for this system can be derived analytically. This distribution function can be written as J(E, L,) = { :[(n] !l2)r2 + 2( f!L,)rl [ ... ] > 0 [ ... ] :S 0 ( 3 6) where (3.7)
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55 In analytical studies of these systems, Kalnajs ( l 972) found that they were axisymmet rically stable for random to rotational kinetic energy ratios greater than threehalves Numerical studies (Moore 1991) have verified this conclusion. In addition, Kalnajs was able to show that the dominant mode in the system was an m=2 (bar) mode. This phe nomenon had already been observed in the work of Hohl ( 1971) who had shown that disks with insufficient random kinetic energy, as parameterized by the local Toomre Q parameter (1964) (see below), developed strong bar responses. While this type of disk only approximately models the inner region of real disk systems, it points to the fact that the integrals of motion play an important part in determining the global evolution of the system. Toomre Disks In their 1959 communication, Burbidge, Burbidge and Prendergast detail procedure by w h ich a mass determination of an observed, three dimensional disk can be made using the rotation curve. By using the equation of motion for the gas in the system, (3. 8) and assuming that a similar equation can be derived to describe the stellar motion in the d isk (the socalled "stellar hydrodynamic" method) where (generally anisotropic) velocity dispersions will appear instead of the pressure term above, the authors are able to determine the force at any point in the disk. This is done by assuming that the equi d ensity surfaces in the disk will be selfsimilar ellipsoids (homoeoids). Also, if circular motion is assumed, then the pressure term in equation (3.8) will be negligible.
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56 Thus the gravitational force due to a uniform shell of equatorial thickness da will be dF = p (a) ox(~aa, k ) da, (3.9) where p(a)x( r,a,k) will be the force per uni t mas s in the equatorial plane of an oblate 1 (a2 c2)! spheroid of density p( a), semimajor axis a, and eccentricity k = a The total force due to all such spheroids will then be given by F(r) = v2 = fr p(a)oxda, r oa ( 3 10) 0 smce, by Newton's theorem shells for a > r do not contribute "to the integral. By substituting the form of the exterior potential of a uniform spheroid into equation (3.10) and taking the derivative we arrive at T 2() C( k2)t f p (a)a2da V r 47r 1 1 o (r2 k2a2)'i ( 3 .11) which must be solved for p (a) given v(r). From this point we follow the work of Brandt (Brandt 1960, Brandt and Belton 1962) who notes that the mass of the uniform spherical shell being considered is 1 dm(r) = 41ra2 (1 k 2 ) 2 p ( a k )da, and, therefore equation (3.11) may be rewritten as T 2( )cf dm(a k) V r 1, or in a more convenient form, T (r2 k2a2)'i 0 v2(r) = cf [dm(a,k)] ad a 1 0 ada (r2 k2a2)'i (3 12) (3.13) (3.14)
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57 By noting that the term in the brackets ha s the unit s of s urface den s ity as the homoeoids are infinitely flattened i.e., as k ......., l we may finally write T 2 J ada v ( r )=G O"(a) 1 o (r2 a2)2 (3.15) where 17( a) is the surface density of the now infinitely thin disk. By use of a simple substitution, this integral transforms into an Abelian form and thus may be inverted (Arfken 1985) thus giving T O"(r) = 2_.:!_ J v 2 (a)ada1 G1rr dr O ( r2 a2)2 (3.16) To calculate the mass that is contained by those homoeoids interior to some radius r we use T T M(r) = J O"(a)ada = 2._ J v2(a)ada1. o trG o (r2 a2)2 (3.17) If the substitution a = r s in() is then used, equation (3.16) may used to analyze numerical data To further examine the process of going towards the limiting case of a flat disk, we consider a homoeoid with an axial ratio, ~. When this quantity becomes very small, the attraction of the spheroid of interest on an exterior point will be very close to that of a nonuniform disk in which the surface density is allowed to vary, i.e., I { (r)2}t O" = 2cp 1 (3.18) As we pass to the limit, ........ 0, we require that the mass remain constant in each cylinder of unit area or cp = 170 Using equation (3.18) along with this restriction we may then
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58 calculate the mass contained in the disk of radius r to be 47rO"o { 3 ( 2 2)t} maar 3a (3.19) By differentiation, we may also conclude the relative mass distribution within the flattened homoeoid with a circle of radius r to be 47rO"o { ( 2 2)t (a2 r2)t} dm = 3 2a 3 a r + r 2 da. (3.20) Using this formalism, Mestel (1963) was able to construct the surface density distribution of an infinitely thin disk of radius Ro, maintained in equilibrium solely by rotational motion. By calculating separately the contributions from those shells that lie wholly within the radius of interest and the contributing portions of those that do not, Mestel is able to arrive at the expression for the surface density This is done by defining, Mr(r) = M1(r) + M2(r), where M1(r) is the mass contributed by infinitely flattening homoeoids that lie entirely within radius rand M2(r) is the mass contribution from infinitely flattening those homoeoids that have only portions of their mass within r This is illustrated in Figure 3.1, where the homoeoids labeled 1 and 2 lie entirely within r and thereby have all of their mass included in M1 ( r) and those labeled 3 and 4, which contribute only a portion of their mass when flattened to M1 ( r ), and thus have the remainder included via M2 ( r). To calculate M1 ( r) and M2 ( r), we use (3.21) and (3.22) r
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z 4 : 3 2 59 I a ~ Figure 3 1 : A schematic diagram illustrating the mass calculation u sing homoeoids Using this formalism, Mes tel then calculates the surf ace density required to produce a disk with a uniform rotational velocity (i.e., a flat rotation curve). This surface density is found to be ~(r ) = 2~;r { 1 ~sin1 ({D) } (3.23) where V is the value of the constant velocity. We will refer to this disk as the finite Mestel disk. At this point it is useful to note the work of A. Toornre ( 1963). Toornre noted that while the method of Burbidge, et. al. and Brandt was quite useful it involved the evaluation of a double integral equation. As this can be cumbersome, he developed an alternative solution to the same problem considered above using FourierBessel t ransforms. By assuming cylindrical geometry Toornre writes the potential of the system as (3.24)
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60 Noting that this potential satisfies Poisson's equation in three dimensions if the mass s urface density is written as 1 ( d ) k d<7(r) = 2 G' d d = ? 0 J0(kr)dk, 7r Z z =O+ 'lr (3.25 ) and requiring that the surface density be expressed as a Bessel integral 00 <7(r) = J J0(kr)kS(k)dk ( 3.26 ) 0 where 00 S(k) = J J0(ku)u<7(u)du, (3.2 7) 0 from the FourierBessel Theorem (Morse and Feshbach 1953), then the actual potential can be written as 00 (r,z) = 21rG J J1(kr) S(k)e lklzdk. 0 Therefore we can now write v2 (8) Joo ; = or z=O = 21rG J1(kr)kS( k)dk 0 Following similar logic, we can also write 00 00 : 2 = J Ji(kr)k J v2(u)J1(ku)dudk. k=O u=O (3.2 8) (3.29) (3.30) Equating equations (3.29) and (3.30) and noting that the transforms must be the same, we are able to show that 00 S(k) = 2:G j v2(u)J1 (ku )du, (3.31) k=O
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61 and, therefore, 00 00 o(r) = 2:c J l o(kr)k J v2(u)J1( ku)dudk. ( 3.32) k=O u=O As this is a bit unwieldy to be of direct use we integrate the inner portion of the expression by parts (assuming that v(O ) = v( oo) = 0) thereby reversing the order of integration and obtaining 1 Joo dv2 Joo o(r) = 2 r:G du J0(kr) J0(ku)dk du. (3.33) u=O k=O Given equation (3.33), we now have a second, somewhat simpler, method to take an analytical rotation curve and produce a density profile. Toomre notes that it is possible to explicitly integrate equation (3.32) if one chooses as a rotation curve 2 2 c2 v (r)=V ( b ,r)= 1 (1+~)2 (3.34) It is then possible to obtain the corresponding surface density, o( b r) = c 2 {!__ 1 __ } 2r:G r (r2 + b2) t (3.35) where b is a scale length parameter that may be thought of as determining how dense the central region of the disk is and how rapidly that density falls off. C is a constant that may be evaluated by taking the maximum of V at r~ = 2b2 If this is taken to be true, then C2 = 3'f V2 While this surface density has the difficulty of possessing a singularity at its center, its velocity distribution initially rises steeply as radius increases, flattening out at higher
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62 radii as is seen in many of the galactic disk observations, and Toomre reports its mass to be finite. It may be noted that from the above pair of velocity and density distributions, it is possible, due to the linearity of equation (3.32) in v2(r), to create pairs of functions that more accurately model observed disk systems (i.e., those without singularities in their centers). By taking the derivative of equations (3.34) and (3.35) with respect to ) 6, we arrive at (3.36) and c2 1 0"1 ( 6, r) = 1 3 21rG (62 + r2)2 (3.37) Toomre refers to this pair of functions as "Model 1 ". Further models may be obtained by using (3.38) and (3.39) (Toomre 1963, Hunter, Ball and Gottesman 1984) It is also possible to generate the nJ'h model in the family by integrating the n'h model with respect to b. Using this relationship, it is possible to produce the lowest model in the family, the n=O disk, given by c2 0"0( 6 r) = ! 0"1(6, r)6d6 = 0G 1 1 21r (r2 + 62)2 (3.40)
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63 and V0 2( b r) = ! V/(b r)bdb = C5 [1 b 1 l ( b 2 + r2)2 (3.41) A final detail in this portion of the modeling process is that for all of the Toomre disks, the fraction of the mass outside of any given radius falls off only as rt. Hunter Ball and Gottesman (1984) have used Brandt and Mes tel' s method of infinitely flattened homoeoids to truncate the Toomre series at radius, RD, While the rotations laws remain the same, the density distributions have additional factors that modify the surface density. The consistently truncated Toomre n=O model can be rewritten as Va2(r, b) = C5 { 1 (r2:b2),:} r ::; RD, (3.42) 2c5 {sin1 (1~ .L2.) b tan1 (Rv(r2+b 2)t ) } r > RD, 7r r (r2+b2)f b (r2R1)f and { C2 ( r ) tan1 ((R1r2l) r::; RD, <7o(r, b) = (r2+b2)f (r2+b2)f 0 r > RD, (3.43) whereas the consistently truncated Toomre n=l model is given by and r S Rv, r > Rv, (3.44) { C1 { (( 2 2)!] 1 ((Rbr')t) Rbr2 t } o1(b, r) = r + b tan (r'+P) f + Rb+b' ( r ,+b') r S Rv, r > Rv. (3.45) 0 The successive models may be found using the procedure described in equations (3.38) and (3.39). Representative graphs for the rotation curves and density profiles of Toomre n=O and n= l disks are shown in Figures 3.2 and 3.2.
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64 Cold Rotat ion Curv e:Toomre n 0 dis k .... 0 ._._~~ .......... ........ ~~~'~~~'~~~'~~~'~~~..L.....~~_._, 0 0 5 10 15 20 25 J O 3 5 Ra d ius ( kpc) S urface Den.si t y Plo t :Toomre n0 d i ~!,. 5 10 1 5 20 Rodi ua (kpc) Figure 3 .2: The cold rotation curve and surface density for a consi stently truncated Toomre n=O disk 4 0 25
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65 Cold Rotati on Curve:Toomre n 1 di:,k 0 0 10 1 5 20 Radius (kpc) JO 40 Surface DenJiity Plo t:Toomre n1 di,1,, 10 15 20 Rodi UI (kpc) Figure 3.3: The cold rotation curve and surface density for a consistently truncated Toornre n=l disk 25
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66 Other Initial Condition Formalisms Hemquist Hemquist (1993b) has devised a multicomponent method used by him and his collaborators to construct warm, compound galactic models based on taking the moments of the collisionless Boltzmann equation (CBE) A generalization of earlier methods (Hemquist and Quinn 1989, Hemquist 1989 Quinn, Hemquist and Fullager 1993, Barnes 1988, 1992) involving bulges, disks and haloes, he assumes a spatial density distribution for all three components, the disk being exponential in the r direction and having a sech2 distribution in the z direction (he assumes the disk to be isothermal). Using moments of the CBE, he is then able to specify the dispersions for each component. In tests, Hemquist found that the disks constructed using his approach were fairly close to equilibrium, requiring a no greater than 10% adjustment in any component to reach a final equilibrium state for models constructed to be stable The strength of this approach is its fully three dimensional, multicomponent nature. This allows a great deal of variability in the systems to be studied. However, the shape of the rotation curve can not be affected directly (it must be changed by changing the density distributions of one or more of the components) and, as such, it seems that a systematic study of parameter space in relation to the integrals of motion would be somewhat difficult. Given this shortcoming, control over the system's global stability (local stability is taken care of by the use of the Toornre local stability criterion (Toornre 1964)) is indirect at best since there is no easy way to adjust the initial disk's "coldness"
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67 Fall and Efstathiou By assuming an exponential density profile, Fall and Efstathiou ( 1980) laid down the foundation for constructing a selfconsistent disk/halo system based upon observations. Building on the work of Mestel ( 1963) and Freeman ( 1970), the authors attempt to construct a disk that in profile looks much like a lenticular (SO) galaxy embedded within a dark halo. Beginning with a surface density and z distribution, ( ) (a2MD) (o r ) CJD r = e 21r (3.46) f (z) = (2 ~9) sech2(;9 ) (3.47) and a rotation curve, ( r 2 ) t [ ( r 2 ) ] t VD = V M 2 2 1 ln 2 2 rm+r rm+r (3.48) 2 where a is the radial scale length, Hg = 3.,/J9 rm is the radius of maximum velocity and rt is the disk cutoff radius, the authors are able to derive a halo density function that is selfconsistent with the disk. Hence, by assuming both a density distribution and a rotation curve, Fall and Efstathiou are able to deduce what the form of the halo must be. Here again, we have the strength of the model being closely matched to the observations, but difficult to vary in terms of the integrals of motion. Additionally, it should be noted that the model assumes isothermal behavior in the disk. Calculation of Velocity Dispersion The Toomre formalism described above assumes what is known as a "cold rotation curve" In other words, all of the energy of motion for the individual disk particles is
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68 restricted to the circular motion necessary to offset the centripetal force of gravity. When observing stellar motions in disk systems, one usually finds that this is not the case, but rather that large velocity dispersions act in a manner analogous to the pressure in a fluid sys tem. Thus the real disks are supported through a combination of circular motion and "pressure" gradients in the radial velocity dispersions. As this is the case, it would be good to be able to easily control the how the kinetic energy is partitioned between the ordered motion of the system (i.e.circular motions) and the random motions of the system (i.e.dispersional motions). Using a method known as "stellar hydrodynamics" we are able to by taking moments of the collisionless Boltzmann equation (CBE) quantify this process in an easy straightforward fashion. The treatment discussed here follows the method developed by Hunter and Moore ( 1989) and reported by Moore (1991 ). Beginning with the CBE in two dimensions, we have in polar coordinates a f a f vlP a f a f a f 8t + Vr or + :;:&> + v;. av;. + VIP &VIP = 0 where the distribution function f = f ( r, >, Vr V,p, t). Noting that Vr = r = a r (3.49) (3.50) (3.51) and assuming azimuthal symmetry, (i.e.ff = O ; ij = 0) we can rewrite the CBE as af + v;. of+ [vj a
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69 where Q = Q( Vi) and d, v i s defined to be the volume element in velocity space. For the zeroth moment of the CBE, we take Q = mv? = m and upon integration, we obtain a continuity equation for the system, av(r, t) a(vVr) vVr at + or + r= o ( 3 54) where v = J mfd,v, Taking the first moment, Q = mVr, we arrive at a(v"Vr') 8 [ ( 2 2)] vV V ( 2 2 ) r+ V o r + :;: (7 r + V r = 0 (3.55) where we have used the identity, v;,2 = <7; + Vr 2 <7 r being the random motion of a particle and Vr being its ordered motion. Using the continuity equation derived above, we may rewrite this as 8Vr T 8Vr Vat+ Vvr Or V ( 2 2 ) a(v<7;) V<7~ +<7r <7,;. + = 0. r 'I' or r (3.56) If we seek stationary solutions to this equation, Vr = O; = 0 and that the azimuthal velocity may be parameterized by V V 2 2 V(7 + (7 (7
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70 By a gain res tri c ting our s elve s to time independent model s with no bulk radial motion allowed the above relation becomes 2_a ; ( dlnV,p) a l+ d l 2 n r ( 3 59 ) U s ing this e x pre ss ion we may now s implify equation ( 3 57), the dis persion rel a tion to d ( 2 ) v 2 ( dln Vef>) = vd4> + ~ k2V2 d vcrr + ar 1 dl d o' r 2r n r r r or !}__( 2 ) 2 ( dln V0 dlnk) __ d4> ~k2 2 d var + a r 1 d 1 d 1 // d + Vo r 2 r n r n r r r At the present time we assume that d4> dr v 2 0 r ( 3.60) ( 3 61) ( 3 62 ) and that k is spatially invariant throughout the initial disk which reduces the dispersion relation to its final form !!__(vcr2 ) + ~a2 (1 dln Vo) = ~( 1 k2)v2. dr r 2 r r d ln r r 0 ( 3 63) Due to the complexity of most analytical velocity curves, it is usually necessary to solve the dispersion relation using numerical methods. As previously mentioned, knowledge of the velocity curve and through it the density distribution and velocity dispersions do not uniquely determine the distribution function, however they do provide useful constraints as to its form. After experimentation, it has been determined that the macroscopic quantities of the system (i. e., the energy and angular momentum) are fairly
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71 insensitive to the choice of initial distribution function ( Moore 1991) provided that the choice is well bounded in velocity space. The simple choice that we find works well and may be close the actual distribution function is a local Gaussian form, with unequal velocity dispersions in the r and directions. In practice, the high velocity tails of this function are truncated as the particles in that region of the distribution would have escaped the system. Tests have shown that the inclusion of such a truncation reduces the number of escaping particles in an axisymmetrically stable model by 90% without significantly altering the global parameters of the system. Stability Criteria Toomre's Local Stability Criterion In his 1964 paper, A. Toomre discussed the problem of local stability in disk systems using hydrodynamic considerations In presenting the derivation here, we follow the outline presented in Binney and Tremaine ( 1987). If the thickness of the disk is assumed to be negligible and motions of the gas are restricted to the z=O plane, Euler's equations may be written as 2 OVr OVr V, OVr v a~ 1 Op Ot + Vr Or + :;:OqJ :;:= Or I; Or' (3.64) 0V, OV, V, 0V, VrVd, l 8~ 1 op 8t + Vr Or +:;:OqJ + r= :;: OqJ rI: {)rp' (3.65) where I: now represents the surface density of the disk so as to reduce notational confusion. Choosing a polytropic equation of state, p = J{I;"f' (3.66)
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72 we may then write that the s ound s peed is given by (3. 67 ) If we then define a s pecific enthalpy as ( 3 68 ) Euler's equations are s implified. Assuming we may linearize these equations we s ay, v r = v ro+ vr1, v = V,o+v~n, h = ho+h1 = o+1 where the quantities s ubscripted with a one are assumed to be s mall perturbation s of the background zero subscripted quantities. First, we note that for the unperturbed disk, Vro = 0 and 88 ~0 = ?/f = 0. This reduces the Euler s equations for the unperturbed disk to 2 vO = i_(o + ho), r dr ( 3 69) as would be expected. Additionally as the second term on the right hand side of the equation is smaller than the first term by a factor of (/!:;) 2 it will be neglected and therefore we may write ( do) t V,o :::'. r dr = rO( r ), (3.70) where O(r) is the circular frequency. Using these and keeping only terms of first order in the perturbed quantities, we write the Euler equations as ( 3.71) and 8vq,1 [d(rO) n] n8V,1 n8vip1 ~i_( h ) 8t + dr + Vr1 + 8 + 8 r 8 1 + 1 ( 3.72)
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73 At thi s point we find it con v enient to introduce two definitions to s implify the equations. Fir s t we define B(r) = + n = n r 1 [d( rrl) ] 1 d n 2 dr 2 dr ( 3 73) and then u s ing the epicyclic frequency we have (3. 74) Assuming solutions of the form, 1P1 = 1Pa(r)ei(m>wt), we may then solve Euler s equations for Vra and V>a These are given by ( 3.75) and ( 3.76) where = r,,2 ( mn w2 ) and, by using the same linearization procedure on the equation of state, h a = I
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74 To close our system of equations we need to determine the relationship between ~ a and a using Poisson's equation. To do this, we invoke the WKB approximation to obtain local solution to the density perturbations. The perturbed potential may be written a(r) = F(r)eif( r ) = F ( r)ei f kdr, ( 3.79) where k = d~~r) and lkrl 1. Using this, we note that Poisson's equation is approximated within O ( i krl1 ) and therefore, Ea and ha also contains the factor, eif ( r ) As this term varies rapidly with radius, we may drop those terms that are proportional to (a + ha) 'k(A'. h ) dr z 'a + a ( 3.80) we may simplify the velocity solutions, (mn w)k(a + ha) Vra = .6. (3.81) and 2Bik(a + ha) Va = .6. (3.82) Similarly, we may write d(rEovra) .k _, dr = z rwo, (3.83) which is of the same order as equations (3.81) and (3.82). Thus, this term dominates over the third term in equation (3.78) by 0( lkrl) and therefore we drop the third term. Given this we may write the continuity equation as (mn w)Ea + kEovra = 0 (3.84)
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75 which allows us now to eliminate Vra, ha and a, leaving ( k2v; 27rGEolkl) ''a 1 + .6. .6. 0. (3.8 5 ) As we require that the quantity in parenthesis vanish, we may write the dispersion relation for a gaseous disk in the tight winding limit as (3. 86) Using a similar derivation, one may derive the equivalent dispersion for a s tellar disk. Assuming axisymmetry, we require m=O and so equation (3.86) becomes (3.87 ) If w2 < 0 then an exponentially growing solution is allowed and the local region is unstable. Therefore, if (3.88) we are at the line of neutral stability and thus, by solving the quadratic we see that local stability requires Q VsK, l => 7rGEo for a gas disk. For a stellar disk, this becomes Q = VsK, > 1. 3.36GEo (Julian and Toornre 1966). (3.89) (3.90)
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76 OstrikerPeebles Parameter In their paper discussing the stability in numerical modeling of disk systems ( 1973), Ostriker and Peebles introduce a dimensionless parameter, t which is characterized by T t = lvVI' ( 3.91) where T is the rotational kinetic energy of the system and W is the system's potential energy. If the random kinetic energy of the system is represented by IT, then the vi rial theorem may be written as or 1 1 T+ IT= W 2 2 IT cl 2 T . (3. 92) (3.93) Given that > 0, we may say that O < t < l Ostriker and Peebles found that for models in which t 0.14 c :5 5 ) the system was globally unstable to the formation of an m=2 (bar) mode. While this conclusion was merely based on a number of numerical simulations run by the authors, the conclusions reached are supported by the work of a number of other investigators (Sellwood 1981, Zang and Hohl 1978, Efstathiou et. al. 1982, Frank and Shlosman 1989). The rotational parameterization parameter, k defined in the section describing the derivation of the dispersional velocities of our disks, can be s hown to be related to the OstrikerPeebles parameter by k2 t= 2 (3.94) This being the case, we find that we may easily vary the global stability of a disk system to the formation of a bar mode by varying k
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77 Since the publication of the stability parameter a number of examples have been found to violate the parameter. Vandervoort ( 1983) found that the point of marginal stability varied between 0.1 28 6 < t < 0 1 88 2 for uniformly rotating stellar spheroids. While technically in agreement for the e = 1 case the models run are in violation of the OstrikerPeebles (OP) parameter otherwise Another exception to the empirical parameter is the case of BodenheimerOstriker gaseous spheroids ( 1973) where the system is stable for values oft 0.24 a clear violation of the OP parameter. Other examples of this type of violation include TohlineDurisenMcCollough gaseous spheroids ( 1985) and Woodward, Tohline and Hachisu gaseous tori (Tohline and Hachisu 1990 Woodward et al. 1994). Finally, Miller (1978) showed that models that are represented by a ToomreZang (Zang 1976) stellar disk (these are Mestel disks in which only a fraction of the particles are allowed to respond to perturbations in the system, while the rest are kept 'frozen in their circular orbits) are able to violate the OP parameter by remaining dynamically stable at t=0.248 As will be shown in Chapter 6 of this dissertation, we have found further exceptions to this rule by including counterrotating angular momentum (Davies and Hunter l 995a,b) While a number of violations of the OP parameter do exist, the criterion works as a very good empirical guide for determining the stability of a simple disk system to the development of a bar mode. The failure of this criterion to accurately predict stability in a wider range of models stems from that fact that it is only an approximate relation deduced from numerical experiment, not analytical consideration. Additionally, it takes into account only one of the integrals of motion of the system, namely the energy. A
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78 more accurate description of the stability criterion for such systems may have to take into account other integrals of motion. Christodoulou' s Parameter In two papers recently submitted to the Astrophysical Journal D Chri s todoulou, I. Shlosman and J. Tohline (I 995a, b) propose a new criterion to gauge the global stability of a rotating system. This stability parameter can be written as (3.9 5 ) where (3.96) L is the total angular momentum of the system, fl1 is the Jeans frequency introduced via selfgravity and W, as before, is the total potential energy of the system. For stellar systems, o: 0.254 0.258 will provide stability to the development of nonaxisymmetric modes. Note that this new criterion is dependant on two integrals of motion and is therefore able to more accurately predict the behavior of those systems whose angular momentum is not included in a simple way. In can be shown that equation (3.95) may be rewritten as, 5 f L/M o:4 fl1af (3.97) where f=l for disks, M is the mass of the system and a1 is the equatorial radius. The term fl1af therefore represents the maximum angular momentum of a circular orbit in the equatorial plane of the system. When tested over a number of differentially rotating
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79 sys tems this parameter accurately predicts the formation of bar modes in the systems investigated, including those where the OP parameter does not. GENSTD GENSTD stands for GENerate Standard Toomre Disk. This is the program used to develop the initial conditions used in this modeling effort. The code is written in FORTRAN and presently there are versions that will build consistently truncated Toomre n=O and l disks using the dispersion relation given in equation (3.63). The version of GENSTD that builds a two dimensional n= 1 disk is listed in the appendix. The generation code allows the user to input as runtime parameters, the radius of the disk, RD, the mass within a unit radius, M, the total number of particle m the simulation, N, the shape parameter, b, and the heating parameter, k, as well as certain other algorithmic parameters to be described shortly. After initializing several constants, the code preliminarily calculates the mass distribution of the disk and the actual particle number and individual particle mass. As the code does not calculate the particle velocities from the analytical expression for the velocity curve (for reasons to be discussed below), but rather uses the startup routine from the hierarchical tree algorithm's leapfrog integrator, the necessary lookup tables are constructed. At this point the mass distribution of the disk is built. This is done by starting at the outer radius of the disk and calculating the mass within a thin ring. This is calculated using the analytical forms of the truncated disk. From this the number of particles in the ring is calculated and those particles are placed randomly within the ring using a MonteCarlo method This process is done for rings inward of each other until the entire disk is laid down.
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80 At this point the average acceleration on each particle in a ring is calculated using a simple Nbody loop. This is done so that the effect s of gravitational s oftening will be taken into account. If the velocities are calculated from analytical formula (which do not take softening into account) the disk will be unstable to the formation of m= 1 ring modes and will spend the first several dynamical times readjusting. Once the accelerations are computed, the average circular velocities for each ring can be calculated. If the velocity of a ring is less than zero which may occur near the center of the disk due to numerical noise, then analytical value is then substituted. Once this is completed, the size of the dispersions for each ring are computed using the dispersion relation (equation (3.63)) and equation (3.59). The dispersion relation is solved numerically using a fourthorder RungeKutta method The velocity is then assigned to each of the individual particles with the size of the individual dispersion s selected using the truncated Gaussian described above as calculated using the GASDEV subroutine. Once done, the program writes a file containing the position, velocity and mass information in a from usable by the TNDSPH code. If the data is to be run using FfM, the file is then run through a conversion program that produces output suitable for input into the FfM code.
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CHAPTER 4 NUMERICAL MODELING OF DISK SYSTEMS Introduction The basic formalisms used in this modeling effort, namely the codes and the initial conditions, having been put forth, we shall proceed to the actual modeling of disk systems and the issues relevant therein. Given that the systems modeled are only approximations of real physical systems due to the very large difference in particle number between our models and real galactic disk systems, we shall discuss what effects this will have on the validity of the simulations reported in this dissertation. Additionally, methods of representing of star fomiation effects will be considered as will be the difficulties inherent in describing processes in the interstellar medium (ISM). Finally, a discussion of runs for simple Toomre n=O and n= l stellar disks will be undertaken to provide a background for the work to be presented in chapters 5 and 6. It should be stressed here that the models presented herein are global in nature and focus on the formation and dynamical importance of the m=2 (bar) mode of the system. Relaxation Effects One of the most important questions concerning simulations of the type described here is whether these models, with a large but limited number of particles, will accurately model a physical system made up of a much larger number of particles. As is well understood (Miller 1964 ), such simulations will not follow the exact representation and 81
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82 evolution of real phy s ical systems in pha s e space due to roundoff error and discreteness effects. Nevertheless, i t can be shown that in, a s tati s tical s ense s uch s imulation s will model the global behavior a phy s ical s ystem A vital consideration, however is the effect that model particles of much greater masses than those in observed disk systems will have on the time scales for the global evolution of the system. This will be discussed for the three dimensional case and then con s idered in a two dimensional context. For the derivations in chapter 3 we have relied on what is sometimes called the "s moothed potential approximation Namely in our use of the collisionless Boltzmann equation we have assumed that the potential generated by the stellar component of the disk varies slowly and uniformly in both space and time Given that this potential is created by a finite number of particles, obviously this is not exactly true. As such, an estimation of the divergence from this assumption is essential to understanding how much confidence may be placed in the results obtained by such "nonsmoothed" methods If we consider that a model disk will be constructed of N particles, each with mass mi, we can focus on the motion of a single star across the model. To obtain an estimate of the difference between the motion of this individual particle in the model system, built from a finite number of particles and its motion in a smoothed potential we u se the derivation found in Binney and Tremaine ( 1987). From this derivation we find that the velocity of the particle over a crossing time of the system will differ from its smoothed potential path by 6.vi ln A ?=8 N' v~ (4.1)
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83 where ln A = ln ( 6~J, bmin = G~T and ~vi is the deflection from the expected path due to disceteness effects. If the star is able to cross the system several times during the simulation, the velocity will change by this amount each time. Therefore the number of crossings required for the star's velocity to change on the order of itself is (4 .2 ) The amount of time this takes is called the relaxation time and is given by frelax=nrXfcrossing, where fcrossing is the crossing time defined as f cross= f. If we take a reasonable value for A, such as A N, then we may say that the individual encounters will perturb the star from its smooth potential course on the order of O l N trelax ln N icrossing (4.3) While the derivation used here is of an approximate, order of magnitude nature, a more detailed derivation (Spitzer and Hart 1971) finds little difference except to take a system's density into account. More accurately, the relaxation time relation is given by 0.14N r;f trelax = ln (0.4N) V GM' (4.4) where rh is defined to be the system's median radius and M is the total mass of the system. As can be seen, while relaxation effects will have little impact on the evolution of a physical galaxy containing,....., 1011 1012 particles, owing to the system's relaxation time being longer that the Hubble time; such collisional processes will have a much greater effect on systems with ,....., 105 106 particles
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84 For a strictly disk (i.e., two dimensional) system, Rybicki ( 1972) has considered the same problem. For such a two dimensional system, he finds that k trelax = 2tcrossing ( 4.5) where k is the ratio of the ordered kinetic energy to total kinetic energy given in chapters 1 and 3. As can be seen, for a strictly two dimensional system, the relaxation time is always on the order of the crossing time and consequently, systems simulated in this way will never be able to adequately describe systems using the smoothed potential approximation. Fortunately there is a consideration that still needs to be factored into the two dimensional derivation. The calculations that lead to the above conclusions in both two and three dimensions assume an inverse square law force dependence. In our numerical simulations, we soften this force so as to avoid near singularities. If the calculation is redone, taking softening into account we arrive at (1 k/tN trelax = 2 R D tcrossing, ( 4 6) where E is the softening length. For the systems considered in this study, assuming N ,....., 3 x 105, trelax ,....., 4 otcrossing (4.7) Therefore we may conclude that, while collisional effects may play a role in the evolution of the systems considered, errors introduced by the secondorder integrator and numerical roundoff will be of comparable size. The validity of the above arguments must be tested by numerical experiment. From numerical experiments performed at the N ,....., l 05 particle
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85 number size, the time scales as roughly N. Given this, the time scales over which the physical processes proceed will be much shorter in models with low relative particle number than those same processes in physical systems. While collisional relaxation may not be of great importance, we can not make similar claims regarding collective effects. Since the inverse square nature of the gravitational interaction means that such effects are long range in nature, it is likely that adjustments in the density distribution may be due to collective effects. Star Formation Given that the process of star formation is not well understood in the interstellar medium, any approach to model these processes numerically in the context of a much larger system will, of necessity, be very schematic Still however, one may gain insight into the effects of including such processes on the global evolution of the system if the implementation of these considerations are modeled in such a way as to be physically reasonable and so that they produce results in rough accord with observations. For this to be the case, star formation processes must take into account the local characteristics of the system at the point of interest. Additionally, as time the time step of the system can be on the order of "' 105 106 years, stellar winds and supernovae of OB stars also must be accounted for, at least schematically The simplest approach to this problem is to use a star formation criterion based on the Jeans' length and the mass of the local region. This may be done by comparing the density of each SPH particle with some critical density, Pc If the SPH particle's density exceeds
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86 this amount and the particle's velocity divergence is negative, star formation is assumed to take place. Besides taking into account the Jeans' instability, P c may also represent those effects due to relative motions, magnetic fields and temperature gradients. This is the approach used by Heller and Shlosman (l 994) in their investigations. While simple and easy to implement, this criterion has the shortcoming that stars form almost exclusively in the bar region of the disk. This conclusion is not supported by the observations, which show star formation occurs in both the bar and spiral arms. Friedli and Benz ( 1993, 1995) have investigated three differing star forming criteria. These three criteria are the criterion used by Heller and Shlosman (1994), Toomre's local instability criterion (1964) and a negative change in the local entropy. They have found that the local Toomre stability criterion for gas, with a suitable choice for the cutoff value, reproduces the observations in the most accurate way. Their criterion is to chose a value for Q below which star formation will take place in a particle. The choice of this number is somewhat arbitrary and likely depends sensitively on the exact physics of the ISM. Nevertheless, Friedli and Benz show that choosing 1 .:S Q .:S 1.5 leads to star formation occurring in both the bar and spiral arms. In estimating the local value for Q, the authors follow the suggestion of Elmegreen ( 1993) and replace "' with cA, where c=2.83 and A = r9~ is the Oort constant. Additionally, they invoke a temperature condition that requires a gas particle to have a temperature less than some critical temperature so as to have their method simulate star formation in cool regions of molecular clouds. Hence, this second selection criterion forces star formation to occur in those regions where the cooling timescale is shorter than the dynamical timescale.
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87 Mihos and Hemquist ( 1994c) have included star formation in their TREESPH models m a somewhat different way. Noting the above stated difficulties in modeling the exact physical processes in the ISM, they instead rely on observational arguments to parameterize the rates of star formation in their systems This is done by assuming that the physics governing the formation of stars in the ISM is well described by a Schmidt law ( 1959) given by (4 8) where n ,...., 1.5 This value is based on several observational constraints (Kennicutt 1983, 1989, Lord and Young 1990) as well as past numerical modeling (Larson 1969, 1987, 1988) The authors, do not, however, adopt modifications to the Schmidt law to include the effects of radial cutoff, gas density thresholds or galactic shear. They argue that such effects are based on a model derived from gravitational instability arguments which are rooted in linear stability analysis. Once star formation criteria have been selected, an algorithm is developed to deter mine the effects of the pursuant processes on the surrounding ISM. In the Heller and Shlosman scheme, energy is deposited back into the surrounding medium over two time steps. Each amount of energy is equivalent to the ,...., 1051 ergs released during a super nova event. The first time step's deposition of energy is to model OB winds while the second impulse is used to model the supernovae themselves. In their algorithm, there is no conversion of gas mass to stellar mass. Katz ( 1992) proposed, in a TREESPH algorithm, converting some of the gas mass into collisionless stellar particles assuming a MillerScalo ( 1979) initial mass function
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88 to determine the percentage of the original gas mass that contributed to supernovae and the amount that has been converted into stellar mass. Summers (1993) and Navarro and White (1993) follow a similar approach, but smooth the input of supernovae energies. Navarro and White also split this energy into a thennalizing component and a kinetic component. The approach of creating new stellar particles has been used by a number of authors (Elmegreen and Thomasson 1993, Steinmetz and Muller 1994, Friedli and Benz 1995), but as Mihos and Hemquist point out (1994c), this approach is not without its difficulties. They note that there will be a marked increase in the total number of particles in the system of low mass which will lead to a significant degradation of computational efficiency. Moreover, as these particles will be of much lower mass than the original stellar particles in the system, they will be rapidly heated through two body interactions. Mihos and Hemquist (1994c) solve both of these problems by introducing a hybrid SPH particle. In their algorithm, each SPH particle has two masses, Mtot and Mgas The first mass is used when computing the gravitational forces on the particle, the second in SPH calculations. At the beginning of the simulation, these two masses are equal for all SPH particles. When an SPH particle undergoes star formation, a percentage of the gas mass is lost to represent conversion into stellar mass. When the gas mass of a particle reaches 5% of the total mass, the particle is converted into a collisionless particle with the remaining gas spread among the nearest neighbors. While this method alleviates the difficulties mentioned above, it assumes that the collisionless and gas components of the hybrid particle are dynamically coupled This
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89 Hkely compromises both the hydrodynamical and collisionless calculations to some extent, most strongly at the time when roughly half of the hybrid particle is in each component. Summers (1993) resolves this problem by converting each SPH particle totally into a collisionless particle, but his approach suffers from the disadvantage of strongly limiting the star formation resolution of the algorithm. Finally, Friedli and Benz (1995) limit unchecked particle growth by limiting the maximum number of times each SPH particle can undergo star formation In our simulations we will adopt the method of Heller and Shlosman as a simple first step towards modeling these processes. Numerical Models Units A brief word should be said here about units. Due to the fact that we are free to chose what a machine unit will be in physical tenns, we have found it convenient to cast our units in such a way that a machine time unit is approximately one crossing or dynamical time The following are the units chosen and their conversions into physical quantities. lLut = 10 kpc, 1 Mut = 1011 M0 = 2.998 X 1041 kg, l Tut = 4.71 X 107 yrs ltdyn, 1 Vut = 207 !:, G 1 6 67 1011 Nm~  X Tg'l Henceforth, all model quantities will be reported in these units.
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90 Toomre n=O Disks A typica l stable run for a consistently truncated Toomre n=O disk is shown in Figure 4 .1. The parameters for this run are : RD = 2.5L ur, b = 0.5 L ur, M = 2.355Mur, k = 0.4 and N = 32, 000. The OP stability parameter is given by t = 0.08, thereby predicting stability to the development of a global bar mode. For Figure 1, only half of the particles are plotted to enhance clarity. The system shows a weak m=2 mode after 30 t dyn, but is otherwise longterm stable. However, during this time, the system shows significant axisymmetric evolution from its initial equilibrium configuration. The time evolution of t he surface density in this system is towards that of an exponential disk with a scale length of 0.35Lut Additionally, this evolution can be seen in the change of the rotation curve of the system. The initial constantly rising rotation curve evolves to a configuration very similar to that of an exponential disk given by (4.9) and (4.10) as proposed by Freeman ( 1970) This result is similar to work done by Hohl (Hohl 1971, Zang and Hohl 1978) on KH disks. This behavior is shown in Figure 4.2
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crhtl_OOOOOOO dmp >0 N I .. I 2 crtst1_0250000. dmp >0 N I .. I 2 9 1 Stellar Disk St ellar Disk t = 0.000 crtst 1_01 00000.dmp >0 N I .. I 0 2 0 St ellar Oiik St e lla r Disk t 25 000 c rt s t 1_0400000. dmp >0 N I .. I 0 2 2 Figure 4.1: The t i me evolution of a Toomre n=O di s k c on s truct e d to be s table to nonax i symmetric modes. t = 10 .000 t= 40 000
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"' ..: 0 u > ;;; "' O'.) 0 "' 0 ... 0 N 0 0 N I 0 .... _, I "' I 0 D 92 D isk Velocit i es crts t 1 0 400000.dmp t +0.00() ... 0 5 1.5 2 2 5 Radius Disk Surface Density 0 5 1 5 2 2.5 Radius Figure 4.2: Plots of the rotation curve and surface density versus radius for a Toomre n=O disk at t = 40 tdyn
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93 For an n=O disk run with k = 0 6 and OP stability parameter; t = 0.1 8 we predict that the system will be unstable to the formation of an m=2 (bar) mode As is shown in Figure 4.3, the system forms a very strong slowly rotating bar. The bar potential is found, at its maximum, to be 55% of the potential of the initial background disk. In fact most of the disk matter ends up supporting the bar pattern. The global evolution of the system is that of the disk forming a bar that, through its growth, consumes most of the disk. Figure 4.4 shows that strength of the m=2 mode at a function of time. Note that the bar s strength oscillates over a fairly large range and has not reached an equilibrium after t = 28t dyn. This is due to the ability of the bar to efficiently transport angular momentum out of the inner regions of the system to the remaining disk component, thereby causing those particles to orbit at increasingly large radii. The cause of the large bar formation is due to the relative weak shear of the system. As the rotation curve has no turnover point, the system does not possess the shear necessary to truncate the disk inside the system. Consequently the bar grows to the extent the material will allow adjusting its pattern speed accordingly Finally, as can be seen in Figure 4.3, the system initially develops spiral features that are fairly short lived. These features are driven by the bar mode initially and are destroyed shortly thereafter by dispersional heating due to twobody effects. As discussed above, while larger particle numbers will slow this process, it is unknown if spiral arms can persist for a Hubble time if N ,..._, 1011 101 2
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tOte1tJ_OOOOOOO.dmp tOt .. tL0120000.dmp I t0teet3_0240000.dmp I 5 Ste llar D i sk t 0 .000 SteNor Oi1k t 1 2.000 SteRor Oi1k t 24.000 94 t0tut3_0060000.dmp t0t11tJ_O 180000.dmp I tOtatl_.0300000.dmp I Stellar Dis k l 6 .000 S taller Oik l 1!1.000 t .l0 .000 Figure 4.3: The time evolution of a dynamically unstable Toomre n=O disk
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:5 O' C vi 0 m CX) 0
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96 density and rotation curve of the disk at t = 30tdyn are shown in Figure 4 .9. As in the n=O stable case discussed above, the initial configuration seems to evolve towards an exponential density distribution with a scale length of 0.26Lut, roughly half of b. Shown in Figure 4.10 is the evolution of a bar unstable Toomre n= I disk with k = 0. 7 (t = 0.245). In this simulation, the bar mode does not grow to dominate the disk. This is due to the fact that the rotation curve does reach a maximum and tum over, thereby providing the shear necessary to truncate the bar well within the disk Accordingly, the pattern speed of the bar is nearly twice that of the similar n=O case. Stable models including gas and star formation are shown in Figure 4.6 and Figures 4 .74.8 respectively The gas is initially distributed as an exponential disk with a scale length equivalent to that of the Toomre stellar disk. For the model displayed in Figure 4.6, 1 =2 as does the Toomre local stability parameter, Q. In Figures 4 7 and 4.8, where star formation is allowed, = 1.5 and Q=2. As can be seen upon examination of both Figures 4 6 and 4.7, flocculent spiral structure develops in both disks throughout the simulations. However, in the model where star formation is allowed, energy deposition from supernovae prevent the gas from forming filamentary structure as is seen in the model lacking star formation processes. Figure 4.8 shows the star formation maps for the model in Figure 4.7. Here, the main limitation of the critical density criterion shows itself as star formation is confined exclusively to the central regions of the disk, with no activity in the spirallike structure. .....
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Stellar Disk t0test 6_0000000.dmp t = 0 .000 St ellar Dis k tot est6_D200000. dmp t 20.000 .. ~~~~~~~~ N I ,_ ... 4~_'2~'o~~' X 97 Stella r Disk t0test6 _0100000.dmp t= 10.000 St ellar Disk tOte1t6_0JOOOOO.dmp t 30.000 ,_ 0 N I ,_ ... 4~_ ... 2~~o~'2'4 X Figure 4.5: The time evolution of a dynamically stable Toomre n=l disk
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98 Star Formation Map QStst 1 _0000000.dmp t0.000 g stst1_0100000.dmp N I Sto r Formation Map t 10.000 'T _'.'2~o.,__ ___ __,'___ _, S t a r Formation Map QStst 1_0200000.dmp t 20 .000 oststl_0.300000.dmp Star Formation Map ~,, N I ~~~~~~ ' 2 Figure 4 .6: The time evolution of the gas component in the model displayed in Figure 4.5 without star formation t J0.000
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gst s t.2...0000000. dmp g1tst:2.....0200000.dmp 99 Star Formatio n '"4op t = 0 .000 S ta r Formation Map t 20 .000 gstst201 00000.dmp N I S tar Formation Map t= 10 .000 r _L..~_'2~'o~'' X S to r F o r m a ti on '"4ap g,tlt.2_0300000.dmp t 30.000 Figure 4 .7: The time evolution of the gas component in the model displayed in Figure 4.5 with star formation
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gstst20000000. dmp >0 "' 0 I I_, 0 5 g s tst20 200000. dmp "' 0 "' 0 I 0.5 S ter F o rmatio n Ucp 0 5 Star Formation Ucp 0 5 t 0.000 t 2 0.000 100 g s t st2._0 t 00000. dmp g s tat20300000. dmp "1 0 >0 "' 0 I Ster F o rmation U cp t :s 1 0 .000 Star Formation Uap t 30.000 Figure 4 .8: The star formation maps for the model displayed in Figure 4.7
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. f 0 .; > 00 0 .., 0 .... 0 N 0 0 N 0 .... I .., I 0 0 101 Oisl< V e loc itie, crtst3_0400000.dmp t 40.000 0.5 1.5 2 2 5 Ra d ius D isk Sur f oc.e Den sity crtst3 t = 40.000 0 5 1 5 2 Rodiu Figure 4 9 : Plots of the rotation curve and s urface den s ity ver s u s radiu s for a s table Toomre n=l disk at t = 4 0 tdyn
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t0test5_D000000.dmp tOt.nt.S_O 120000.dmp 'i I t0t.H t ~_0240000.dmp ., I 5 102 Stellar Disk I 0 .000 Sl llor Dia k t 1 2 .000 2 t 24.000 t0teat50060000.dmp N I St ellar Diak t 6 .000 ;_~.~,~~~ t0tH t.S_011!10000 .dmp N I I tOtfft!LOJOOOOO. dmp ., I Stellar Diak t 1 8 .000 Staller Diak t J0. 000 Figure 4 .10: The time evolution of a dynamically unstable Toomre n=l disk
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CHAPTER 5 ENCOUNTER SIMULATIONS Introduction Having established the evolution of the model program disks in isolation, we tum to the effects on the i r behavior of an encounter with another large, but lesser mass object. In the first section of this chapter we will examine the behavior of systems comprised of only stellar particles to determine what affect, if any, an encounter will have on the global, stellar dynamics of the system. Following that, we will examine the differences in the behavior if systems with both stellar and gas particles are used. In these simulations, the gas particles will not be allowed to undergo star formation but rates of gas inflow and exchange will be calculated as diagnostic indicators The purpose of these simulations will be to identify the behavior of the gas during the encounter. In the final section we will examine the effects of including star formation on the evolution of the gas. Results from these simulations will be reviewed in chapter 7. The models run will be divided into three types, each with two cases The types will be 1. a single massive point particle in a nonmerging, positive energy interaction with a marginally stable consistently truncated Toornre n=l disk, 2. a nonrotating KH disk in a nonmerging, positive energy interaction with a marginally stable Toomre n= I disk, 103
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104 3 a marginally stable Toomre n= I disk in a nonmerging interaction, positive energy interaction with a second disk of the same type. Additionally, each type of encounter scenario will have two cases. The first case will be when the angular momentum of the interaction between the two systems is positive while the second case will be when this quantity is negative. Note that in considering the sign of the interaction angular momentum we do not include the angular momentum of the individual systems. In all cases, the angular momentum vector for the individual systems is positive. Here, the interaction angular momentum will refer only to the quantity associated with the encounter itself For the masses for the massive particle and the nonrotating disk, we adopt a value of 0.6Mut, which is approximately 44% of the large disk mass. The radius of the nonrotating disk is chosen to be 0.5Lut These values have been selected to model the encounter of a Magellanic sized dwarf galaxy with an average sized spiral system. Finally, the energy of all of the modeled encounters has been chosen to be positive so as to model nonmerging scenarios. For the Toornre n=l disk, the term marginally stable in the type descriptions refers to the way initial conditions are established. For this disk k=0 45, which is just below the OP parameter of k=0.53 (t=0.14). Those models containing gas are constructed so that 5% of their mass is in SPH particles. The gas distribution is constructed with an exponential surf ace density with a scale length of 20% of the disk radius The models shown have all been run using a polytropic equation of state unless otherwise noted. Additionally, none of the systems have been constructed with massive haloes. The models run are shown in Table 5 .1.
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105 Table 5.1: Encounter models run Model Primary Secondary Type Case Gas SF Mass Mass enctstlng l.37 0.6 l p enctst2ng l.37 0.6 I n enctst3ng l.37 0.6 2 p enctst4ng l.37 0.6 2 n enctst5ng 1.37 1.37 3 p enctst6ng l.37 l.37 3 n enctstlgs l.37 0.6 l p v enctst3gs l.37 0 6 2 p v enctst5gs l.37 1.37 3 p v enctstlsf l.37 0.6 l p v v enctst3sf l.37 1.37 2 p v v Stellar Models Massive Particle Encounters The evolution of a purely stellar, marginally stable disk system with a single massive particle is shown in Figures 5.15 .2. Figure 5 1 shows the model with the massive particle approaching with a positive interaction angular momentum while Figure 5.2 shows an interaction with negative angular momentum. As can be clearly seen, the global development of the disk is quite different depending on the sense of interaction angular momentum. In the positive case, the massive particle is able to gravitationally acquire a number of the disk particles. Additionally the encounter excites a weak bar and two armed spiral feature in the disk While the spiral arms are quickly destroyed due to dispersional heating, the bar is long lasting and apparently stable. While the negative
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106 energy encounter is able to scatter a number of particles into escape trajectories the massive particle is not able to capture any mass from the disk, nor it is unable to excite any long term nonaxisymmetric structure in the disk. Neither interaction is particularly successful in developing the bridge/tail structure reported by Toomre and Toomre ( 1972). This is likely due to the more passing nature of the encounter in this simulation, the lower mass of the second system and the lack of gas in the models to more accurately trace the potential. The differing behavior in these two cases can be explained by thinking of the impulse imparted to a single disk particle by the massive particle during the encounter. In the positive case, as the massive particle approaches, the smaller particle is slowed in its orbit and thus its orbital radius increases. As the massive particle passes by the disk particle may then be easily captured by falling into the potential well of the massive particle or scattered from the system if the passing impulse of the massive particle imparts new energy. In the negative case, the particle is sped up as the massive particle approaches causing it to spiral in closer to the center of the disk. As such there are fewer particles near the edge of the disk and none whose kinetic energy is small enough to allow capture by the massive particle Even though these particles are slowed and scattered by the passing of the massive particle, the time scale of the encounter is too short to allow for any of the particles to be captured. The explanation of the long term stable, weak bar mode is more difficult, but as the disk is only marginally stable there are two possibilities. The first is that the initially one armed (m= 1) disturbance is swing amplified (Toomre 1981) as it passes through the
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107 center of the disk This may well be the cause of the development of the m=2 s piral structure from the original m= I s trong perturbation introduced by the encounter but it does not account for the apparently s table bar left behind after the spiral s tructure has dissipated. The second possibility is based on a more global argument. As the massive particle removes a portion of the mass of the disk, the disk system's potential energy and Jeans' frequency, n 31rGA1 1 4R3 0 (5.1) (Christodoulou 1994a) are lowered. As such, both the Christodoulou and OP stability parameters are raised As such an increase would only need to be small in this case, it would relatively easy to drive the system into marginal instability in this fashion The question here is what is the net increase of the disk system's rotational kinetic energy and angular momentum. For the positive angular momentum case, approximately l 0% of the original disk mass is lost to the disk with onethird of that being captured by the massive particle and the remainder being scattered. For the negative angular momentum encounter, the disk loses only 4.2% of its mass, none of which is captured. While the disk loses over twice the mass in the positive encounter as it does in the negative encounter, it is unlikely that this is the only mechanism responsible for the bar formation. It seems that a possible net increase of the rotational kinetic energy in the disk caused by the negative encounter may further stabilize the disk against the formation of a bar.
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ancts\1 ,._0040000.dmp .. >0 .. I 2 enctsttnG,..()120000.dmp stellar D11k l 4.000 Stellar Diak t 12 000 !:! ...... ~............ ~~~..~~............. ....... >a .. I 108 enctlt1n9J)OBOOOO.dmp .. >O .. I 2 enct.stln;...0160000 .dmp St1Hor 01111 0 X Stello, D iek I 8.000 t 1S.OOO !:! ....... ~~~...~............. .. >a 0 .. ~,o~so"~...... s~...__,,o X Figure 5.1: Plot of particle positions for a positive interaction angular momentum, purely stellar massive particle encounter.
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en ctst2ng_0040000.d mp >0 N I .. I 2 e nctst2n ~ O 1 20000. dmp Stellar Disk t 4 .000 St ellar Disk ta 12 .000 109 enctst2ng_0080000. dmp N I .. I 2 enctst2n~D160000. dmp Stetler Disk t= 8 .000 St e llar Disk ta 16 .000 o~~~~~~~~~~~ >0 "' I >0 "' I ~._ ____ ..._ ___ ~..,__~~~~'' 'to 5 10 Figure 5.2: Plot of particle positions for a negative interaction angular momentum, purely stellar massive particle encounter
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110 Dwarf /Disk Encounters Figures 5.35.6 show the evolution of the two cases of the dwarf/disk type of encounter described above. Figure 5.3 and 5.5 show the time evolution of the positive and negative interaction angular momentum encounters respectively As before, only the positive case is able to produce structure in the primary disk. Also, as before, the less massive KH disk is able to capture some of the mass of the primary disk. However, the ability of the dispersed mass to capture particles is somewhat reduced from the massive particle type of encounter. In the present positive case simulation, 8.6% of the Toomre disk's mass is lost (compared with 10% above) and 2.6% of the Toomre disk's mass is captured by the less massive KH disk. The remainder of the lost mass is scattered into escaping orbits. The lower amount of perturbation is most likely due to the more dispersed distribution of the mass in the secondary system. In the negative case, the mass capture by the small disk is negligible. Here the mass loss by the Toomre disk is only 2.4% of the original disk mass. All of this is lost to escaping orbits. Interestingly, in this case, the KH disk also loses 1 % of its mass. This shows the effect of dynamical friction on these tightly bound systems. Mass loss during the encounter in the secondary, dwarflike system can be neglected in these cases. Figures 5.4 and 5.6 show particle positions for each disk individually. As can be clearly seen, in the positive case nearly all of the particle capture is done by the KH disk, while in the negative case the opposite is true. As before, the positive case induces bar structure in the Toomre disk. Similar development is inhibited in the KH disk as it is constructed with no net rotational motion.
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enctst.Jng.._0040000. dmp ,n I S tellar Di!k l 4 .000 I 11 enc t !lt3ng.._0080000 .dmp "' I S tellar Dis k l 8 .000 1 5 r_~,~2~~o~~~~' Ste llar D isk S t ellar D i sk enctst3ng.._O 120000.dmp t 1 2 .000 enctstJng_O 1 60000.dmp ,n I s >0 ,n I 5 Figure 5 .3: A plot of particle positions for a purely stellar dwarf/disk encounter. Interaction angular momentum is positive l 1 6 .000
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e ncbt.Jng_ OOBOOOO. dmp "' I 5 e nct stJng_.0 1 60000.dmp St eller Disk t 8 000 St ellar Disk t 16. 000 "' I 112 enctstJng_.0080000 dmp >N I .. I 2 en ctst Jng_ O 160000. dmp >S t eller Disk Slello r Disk Figure 5.4 : A detailed plot of stellar particle positions for each disk individually for the simulation shown in Figure 5 3 t 8 000 t 1 6 .000 1 0
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enctst4ng_0040000.dmp "' I St ellar Disk t= 4 .000 113 enctst4ng_OOBOOOO.dmp "' I Stella r Disk t = 8 .000 ~~~~~~~~ ' 5 2 St ellar D i sk 0 X Stellar Disk enctst4ng_0120000. dmp t 12 .000 enctst4nQ...0160000.dmp >0 "' I 5 >0 "' I 5 0 X Figure 5 .5: A plot of particle positions for a purely stellar dwarf/disk encounter. Interaction angular momentum is negative t 1 6 .000
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encbt4ng._0080000.dm p >0 "' I 5 e n ct st4ng_ O 160000. dmp Stella r Disk t 8 .000 Stellar D isk t 1 6 .000 e ,,~,.~~~~....,~~.~~~~..........., >0 "' I 0 ... ,..\.,~. .;,;'; ..,'. .. ) _... 'Ali;? ~ 1 ,;..,.: 0 1 ...... ... ~. ...__. ___ .... 5~'o~'5'10 _, 114 enctst4ng._0080000 d mp "' I 5 enc t st4ng_ O 160000. d mp Stella r Dis k t = 8. 0 00 ' .. ( .. 0 Stellar D isk t 1 6 .000 0 ,,,.,.~~~r,~~~,.~~~~......., "' I 0 .... ... .,,. I ...__. ___ .._5 ____ .._0 ____ ..,_~'10' Figure 5 .6: A detailed plot of stellar particle positions for each disk individually for the simulation shown in Figure 5 5
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115 Disk/Disk Encounters Figures 5 .75.10 show the evolution of the encounter between two equal mass Toomre n=l disks constructed, as before, to be marginally stable. As can be seen in Figures 5 7 and 5.8 the positive angular momentum interaction causes a great deal of scattering from both disks and produces some mutual capture and mixing of material. However, this mixing is very limited and would be difficult to trace observationally It is likely that a much closer encounter is necessary to produce significant exchange of mass between the disks. However an interesting note is that the mass that is captured quickly mixes into all regions of the capturing disk. The total amount of scattered mass is 5 5% of the sum of the two disk masses, with roughly equal amounts from each disk. Both disks in the positive encounter develop weak bar modes upon closer examination and a tenuous bridge of material is left between the two separating s ystems after the encounter. The negative angular momentum case shows little of the structure found in the positive case. This case is shown in Figures 5.9 and 10. The amount of material scattered by either disk is very small comprising only 0 8% of the total mass of both systems. No mass is captured by either disk from the other. While there are some tidal remnants of the encounter after 20 dynamical times, these features are quite weak and may be reabsorbed by the disk at later times. Observationally, it would be very difficult to determine if an encounter had occurred by looking at either the stellar kinematics or either disk's trajectory as neither shows a strong deflection from its original path.
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116 Stellar Disk St ellar Disk enctstSng._0040000.dm p t=4 .000 enctstSng._0060000.dmp ,~ 6.000 0 0 >N I "' .. I I 2 0 0 St ellar Disk Stellar Disk enctstSng_..0080000.dmp t 8.000 enc tst5ng_ O 1 00000.dmp t 10.000 >"' I '~ : .~ 0 1 0 "' I 0 I ~ .. ~ .: t f f;i 10 Figure 5 .7: A plot of stellar particle po si tions for a purely stellar disk/disk encounter. Interaction angular momentum is positive 15
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encbtSnq....0060000 dmp "' I 5 encbt5nq.... 0 1 36000. dmp "' I 0 I S t ellar Disk t = 6 .000 St ellar Oislc t 13 .600 117 encbt5nq....00 6 0000.dmp "' I 5 e nc ts t 5ng_0 1.36000 .dmp 0 >"' I I "' I 5 St e lla r D i slc St ella r Disk 10 Figure 5.8 : A detailed plot of stellar particles for each di s k individually for the s imulation shown in Figure 5 7 t = 6 .000 t 1 3 .600 1 5 20
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e ncbt6ng_0040000.dmp 5 encts t 6 n9012DOOO. d m p >0 "' I 0 Stellar Disk t= 4 .000 St ella r D isk ,_ 1 2 000 I L.... __ ......_ ___ ......_ ___ ......_ ___ ......_ __ ..._._, 0 1 0 15 118 e nctst6n g_0080000.d mp 0 "' I encbt6ng_O 1 60000.dmp 0 ,? I St ella r Disk t = B .000 1 0 Stellar Dis k t 16.000 10 20 Figure 5 9 : A plot of stellar particle positions for a purely s tellar disk/disk encounter Interaction angular momentum is negative
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e nct!!lt6nq._00 80000.dmp "' I enctst6n~O 1 60000.dmp 0 0 I '. ... .... ~;'.!''.: St e lla r Disk t::t: 8.000 10 St e ll a r Disk t1 6 .000 10 20 119 e nctst6ng._00 80000.dmp "' I enctst6ng_O 160000.dmp >0 I Stella r Disk Stella r D i sk 10 Figure 5 .10: A detailed plot of stellar particles for each disk individually for the simulation shown in Figure 5 9 t::z 8 .000 10 t16 .000 20
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Massive Particle Encounter 120 Stellar/Gas Models As shown above the retrograde encounters produce little development of the s tellar component in either system in a nonmerging encounter. Therefore, we will concentrate on positive angular momentum encounters to understand the effects such scenarios have on the gas in a disk system. This simulation is identical to simulation enctst Ing, except that the disk is now is possessing a gas component that is 5% of the total disk mass. The density distribution of this component is that of an exponential disk with a scale length that is the same as the shape parameter b of the Toornre disk. Initially the gas is supported mainly by rotation and has 1 = 2 and, initially, the minimum value of Toomre's local stability criterion, Q, is also 2, guaranteeing stability against axisymmetric disturbances, but not nonaxisymmetric perturbations. In this model, no star formation is allowed to take place. Also, Q for each SPH particle is allowed to evolve freely as there are no restrictions placed on the SPH particles' motions. Figure 5 .11 shows the evolution of the stellar component over the course of the simulation. One should note that the density of the captured stars around the massive particle is somewhat less than that of the purely stellar encounter; however, the captured stellar mass is roughly the same. Figures 5 125.13 show the evolution of the gas component of the disk. As the encounter occurs and the massive particle reaches perigalaxion one arm of the spiral structure is strongly disrupted and a large portion of the gas in this region is captured by the particle. Additionally, there is a strong amplification of the other spiral arm during the encounter that relaxes soon after the
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121 massive particle leaves the immediate disk region. As the particle moves away from the disk, it pulls with it a characteristic gas bridge as has been previously modeled (Toomre and Toomre 1972, Howard and Byrd 1990). Additionally, a small percentage of the gas is scattered into escaping trajectories. As this is seen to be a feature in all simulations we ran, it is seems likely that much of the intergalactic gas that has been observed was once part of galaxies that have undergone an encounter since their initial formation. The effects of the encounter are also seen in the amount of gas that is in or near the gas bar that forms early on. Initially, this bar comprises 31.5% of the total gas mass in the disk. By the end of the encounter, this has climbed to 46.7% of the gas in the system and 55.2% of the gas that has not been captured by the massive particle. A large jump is seen in the gas mass near the bar shortly after the massive particle reaches perigalaxion at t ::: 7tdyn. This instreaming takes place over a roughly three dynamical time period, before and after which the inflow gas mass rate is relatively low and constant. During this time, the stellar mass in the central disk region remains roughly constant, though, as in the purely stellar case, the excitation of a weak bar mode is noted. The particle is able to capture approximately 7.8% of the disk's stellar mass and 15.5% of its gaseous mass. This would indicate that in a positive energy and angular momentum encounter that grazes the large disk, a smaller, gaspoor system may be able to strongly enrich its gas component at the expense of the larger system. As can be seen, the gas spirals into the region around the massive particle and is initially prohibited from reaching the center of this region by a centrifugal barrier, thus suggesting a limit on the captured gas inflow rate in the dwarf system.
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encbt 191_0040000.Clmp N I .. I 2 enctst 1gs...0120000.dmp Ste U or D isk t = 4.000 St ellar Disk t = 12.000 o~~~~~~ ,.. 0 "' I \"; ~ \ ;it{~~{'. ;i:Wt' ~ : ; 122 e nct s t 1 gs_0080000.dmp N I .. I 2 enc t 1 t 1 gs...01 60000.dmp Stellar Disk 1= 8 .0 00 St ellar D isk t 16 .000 0~~~~~ ,.. 0 "' I ~~i ~ ~: ~r,, Figure 5.11: A plot of stellar particle positions for a massive particle encounter
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Go, Partic les enct,t 1 g,_0000000.dmp t ~ 0.000 Gas PorticlH encht 1 t;1L0060000.dmp t 6.000 123 enctst 1 gs_0040000.dmp N I Go, PorticlH ts 4 .000 1 _~.~2~~0~_, enct,t 1 gs._0080000.dmp N I Go, Particles t:a: 8 .000 1 ~,~2~~0~_, Figure 5 12 : A plot of gas particle positions for a massive particle encounter prior to perigalaxion.
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enctst 1 gs_O l 00000. d m p >0 "' I 5 encts t 1 g s_Ol 40000.dmp >0 "' I 5 Gos Particles Gos Particl es ; : ~ ,, ... 0 t=10.000 .J(' t1 4 .000 1 2 4 enctst 1 gs O 1 20000.dm p "' I 5 e nctst 1 g s_0160000.dmp >0 "' I 5 Gos Particl es t = 1 2 .000 Gos Particle s 16.000 .?f. .. Figure 5 .13: A plot of gas particle po s itions for a massive particle encounter after perigalaxion
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125 Dwarf/Disk Encounter Figures 5. 145 .16 show the evolution of the stellar and gas disks of an encounter between a Toomre n= 1 disk with gas and a KH disk, also containing gas. As before, the gas comprises 5% of the total mass of each system and is distributed according to an exponential density law in both disks. The values of the various equation of state and stability parameters are the same as in simulation enctstlgs In the KH disk, unlike the stellar particles that are mainly supported by dispersion, the gas particles have a rotational component so as to study the development of structure in the gas component of this system. The global properties of this simulation are very similar to those discussed in the aforementioned simulation, enctstlgs. These include the gas and stellar capture masses, gas inflow rates and global development of the Toomre disk This seems to bear out the assertion by Howard and Byrd ( 1990) that there is little difference in the global evolution of the system whether one uses a massive particle or a tightly bound dwarflike system as the perturber. While we find this to be true for the large disk, there are some differences in the behavior of the dwarf type disk that deserve some further investigation. As can be seen in Figure 5 .16, the detailed plot of the location of the gas particles for each disk individually, the particles from the Toomre disk are prevented from spiraling into the center of the KH disk after capture. There are two mechanisms responsible for this behavior. The first is the centrifugal barrier mentioned above. The second, however, is support caused by gas pressure. As can be seen, a portion of the KH disk's original gas is pushed into the center of the dwarf by the inflowing large disk gas This should lead to
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126 an increase in nuclear star formation rates and an angular momentum segregation between the two gas flows. As the acquired gas must shed its angular momentum, it may, through viscous forces, transfer some of its angular momentum to a portion of the dwarf's original gas or through gravitational interaction, spin up some of the stellar mass captured from the large disk. A somewhat smaller captured stellar mass is found within 0 6Lut of the dwarf center than was the case in the massive particle encounter. Closer investigation shows the captured stellar component to be more loosely bound as would be consistent with gravitational heating of the stellar component of the system. Also, the boundary between the two gas regions will almost certainly develop areas of strongly shocked gas. It may be possible to observe rings of very active star formation sites in dwarves that have recently undergone encounters that have removed gas from another system Finally, two brief notes. A lower star formation rate may be observed in the outer portions of the dwarf due to high shear in the system. The effect of this shear would be to disassociate star forming regions before they can reach densities that are Jeans' unstable, either due to mass accumulation or velocity compression. Secondly, near perigalaxion, the dwarf develops a short lived gas tail. This tail is reassumed by the dwarf after it leaves the strong tidal influence of the primary disk. If the encounter were either longer lasting or bound, or if the dwarf's selfgravity was less, it is reasonable to assume that this tail would be captured and begin to form a stream around the outer edges of the primary disk. It is likely that such a scenario is responsible for the presence of the Magellanic Stream around the Galaxy.
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encbtJgs_.0050000.dmp N I ~ ; ., St e lla r Disk t: 5 .000 ~~~~~~~~~ ' 2 encbt.3gs_Ol JOOOO.dm p "' I 5 Stellar D i sk tz U.000 0 127 e ncbtJg9_0090000.dmp ,.. 0 N I .. I 2 enc b t 3 gs_0170000. d m p "' I 5 Stellar Disk \: 9 .000 Stellar Disk \ : 17 .000 F i gure 5 .14: A plot of s tellar particle positions for a dwarf/disk encounter
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enctst3gs_0050000.dmp N I Gos Parti cles t = 5 .000 "'~ ~~~~' ' 2 enctst3gs..0110000.dmp >0 "' I 5 Gos Particl es t::11: 11.000 128 enctst3gs_0080000.d mp >0 N I Gos Part i cles : 8 .00 0 1 _',~_'2~'~ '__J encts t3gs_01 '40000 dm p >0 "' I 5 Gos Parti c l es t = 1 4 .000 Figure 5 .15: A plot of gas particle positions for a dwarf/disk encounter
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Go, Part i c le, encbtJgs_OOSOOOO. dmp >0 N I \ J ,. 'i}~}i .,.( ,Jt),'. t : 8 .000 1~,~2~o~~~ encbtJgs_0140000.dmp >0 "' I 5 Gos Part icles t1 4 .000 l ...... ,,, ... ;, ..': i 0 129 Go, PArticles enct tJgs_ OOBOOOO. d mp 0 > ,. '' N I "' I \ 0 Gos PArti cle s enct,tJgs._0140000 dmp 2 5 Figure 5.16: A detailed plot of gas particle positions for each disk in the dwarf/disk encounter shown in Figure 5.15 t = B .000 t:z 14.000 3.5
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130 Models with Star Formation Massive Particle Encounter Figure 5.17 shows the evolution of the gas in a massive particle simulation where star formation has been allowed to take place. The density distributions for this simulation are as described for simulation enctst 1 gs. For the equation of state and local stability parameters, 1 = 1.5 and Q = l .5. These values have been selected so that any gaseous aggregations in the system would be marginally unstable to gravitational collapse. For this simulation, we choose not to display a plot of stellar particle positions as the evolution of the stellar component of the system follows the dynamics established in earlier simulations. The most notable feature of this simulation is the lack of structure that develops in the bar region of the disk. This is due to the energy input from star forming particles in this region of the disk. Figure 5 .18 shows the star formation maps for four times during the simulation. The main limitation of the critical density criterion can be clearly seen. Star formation activity is confined mainly to the bar region of the disk and to a small region surrounding the massive particle. Only during the time near perigalaxion is there activity outside of these regions. This activity is due to the strong amplification of the spiral density waves caused by the perturber. Additionally, there is a small amount of star formation during inflow around the massive particle. A better criterion is obviously needed to more accurately model the star formation processes in this type of system.
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enctst 1 sf_OOSOOOO. dmp N I Gos Particles l= 5 .000 1 3 1 enctst 1 sf_0080000.dmp N I Co, Porticle9 t= 8.000 1~ ,~2~o~~ T _~.~2~~o.._ ___ _, ____ _, X Gos Partic l es Gos Particles e n ctst 1 sLO 110000.dmp t 11.000 enctstl sf_Ol 40000.dmp >0 "' I 5 0 >0 "' I / 5 Figure 5 .17: A plot of gas particle position s for a ma ssiv e particle encounter w here s tar formation is allowed tz 14.000
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encts t 1 sf_OOBOOOO.dmp >0 N Star For m ation M op t ~ 8.000 .., _L.~.i..2~'Q'' enctst 1 sf_O l 20000.dmp N Star Formation l.lop t,s 12.000 ~~~~ 2 132 enctst 1sf_O100000.dmp >0 N Star Formation I.lop t:: 10 000 ... c_ ____ ..__ ___ _,_ ____ _._ ____ ...J ' 2 enctst 1 sf_Ol 40000.dmp >0 N I 0 S t a r Formati o n Mop t = 1 4 .000 "''''...l....' 2 0 Figure 5 .18: A star formation map for the si mulation shown in Figure 5.17
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133 Dwarf/Disk Encounter The time evolution of the gas component of a dwarf/disk encounter incorporating s tar formation i s shown in Figure 5 .19. In this simulation, the disks were generated in the sa me way as those in simulation enctst3gs, but using the gas parameters from enctstlsf. In addition to the now familiar bridges and tails we can see that in the middle of the simulation, the dwarf disk develops spurs in its gas distribution This is due to the fact that the input energy from star formation causes the gas component to be more loosely bound to the system as a whole and thereby more susceptible to tidal forces. As such, the streaming tail is long lived and, in this simulation, not bound to the dwarf system. This indicates that physical systems may lose mass due to active star formation episodes incited by encounters. The behavior is shown in Figure 5.21 The star formation maps are shown in Figure 5.20. As can be seen, star formation early in the encounter when gas accumulation into the center of both systems is the greatest. Additionally until the addition of star formation in simulations of this type, the dwarf has been relative free from the development of global instabilities. However, as the gas component of the dwarf is more susceptible to tidal forces, a weak bar mode is induced by the encounter. Hence, we can say that to accurately model interactions of this type, some form of star formation must be included in the algorithm. The energy input through supernovae and OB winds may be responsible for a number of the features seen in real systems, including long bridges and tails in encounters that would otherwise be too long ranged to excite such structure.
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e nct st.Js f _0060000.dmp >0 N I Gos Part i cles t = 6 .000 134 e nct st3 sf_D0 90000. dm p N I I Cos Particles t 9 .000 i _~.~2~~o~~~ 2 e ncts t 3sf_O 1 15000. dmp >0 ,n I 5 Gos Part icles Gos Particles t 11.500 en cbtJsL0150000. dmp ~ 1 5 .000 0~~ 0 >0 ,n I Figure 5 .19: A plot of gas particle positions for a dwarf/disk encounter where star formation is allowed
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enctst3sf_0060000 .dmp N I Ster For m a tion Mop t = 6 .000 : ~~ .. ..; .. :.. ~~~~~~~ ' 2 enctst3sL0115000.dmp >0 "' I 5 Ster Formati on Ir.lop t = 11.500 ,..,,..,. 135 encbt3sf_0090000.dmp >0 N I I 2 enctst3sL0150000.dmp Star Formati on Mop t = 9.000 r, . Star Formation Mop t = 15.000 0.... "' I Figure 5.20: A star formation map for the simulation shown m Figure 5.19
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e nct st3s f _OOB0000.dmp N I Gos Partic les t8.000 1' ~ .~~2~o~~~ X Go1 P orticte1 e ncts t 3sf_ 0160000. d m p t 1 6 .000 o.~~~,,,,,,,,, >0 "' I 1 3 6 e nct s t3sf_0080000.dmp N I Gos Partic les t: 8.000 1~ .~ 2 ~~0 ~~ Gos Part i c les enct s t3 sf_O 160000 .dmp t = 1 6.000 0 rrrrr~~~,,,,,,,,, "' I Figure 5 .21: A detailed plot of gas particle posit i on s for each di s k for the dwarf/disk encounter s hown in Figure 5.19
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CHAPTER 6 COUNTERROTATING ANGULAR MOMENTUM Introduction In chapter 4 of this dissertation we have discussed the basic considerations of evolving disk systems numerically from a set of initial conditions to study the global properties of such systems. We now ask "What is the effect of including counterrotating (CR) angular momentum in numerical models of disk systems?" As mentioned previously, this sort of question was, until recently, held to be merely a scientific curiosity. However, with the recent spate of observations of such systems (Rubin et al. 1992, Rubin 1994a,b Rix et al. 1992 1995 sage and Galletta 1994, Bettoni et al. 1991, 1990 Braun et al. 1992 1994, Walterbos et al. 1994, van Driel and Buta 1993, Kuijken and Merrifield 1994 Kuijken 1993, Wang et al. 1992), this question has become newly relevant. The investigation of this problem has comprised a major portion of the modeling effort we have undertaken. Specifically we will discuss the effect of including CR angular momentum on the global dynamics of a model disk. No attempt shall be made to comprehensively discu ss how the systems that have been modeled might have come about though some discussion shall be devoted to the fundamental ideas held in consideration at this time While the models presented will begin with initial conditions that are unlikely to be developed exactly from real dynamical considerations, they provide us with insight into what may be occurring in observed disk systems (a bit of a "s pherical cow" approach, if you will). 137
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138 This is especially true for those models using what we refer to as a "s tep function" distribution of CR angular momentum that are discus sed below as they may illuminate s uch phenomena as slow bars, nuclear bars and longterm stable bar/bulge combinations. In the following sections, we will discuss the stabilizing effect of CR angular momentum on disk systems that would be otherwise unstable to bar formation. This discussion will include an analysis using both the OstrikerPeebles (OP) and Christidoulou stability criteria discussed in chapter 3. Following this, we discuss the formation of novel m=2 (bar) structures from density profiles that lacking CR angular momentum distributions, would evolve quite differently Conclusions from these studies, comparison with recent observations by Friedli and Wozniak (Wozniak et al. 1995, Friedli et al. 1995) and a discussion of further work to be done will be discussed in chapter 7 of this dissertation CR Angular Momentum Inclusion in the Initial Conditions Counterrotating (CR) angular momentum is included in the disk systems to be modeled in two basic ways. The first method of inclusion is to reverse the azimuthal component, v
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139 reversed. This type of model is schematically shown in Figure 6 1 In the slope function s ubcase, there is a transition region between the fully retrograde and fully direct regions of the disk. In this ring, the percent of CR angular momentum is smoothly varied, using a linear function, from 100 % to 0%. A schematic diagram showing the percentage of angular momentum as a function of radius is shown in Figure 6.2 ::: Figure 6 .1: Schematic diagram illustrating the distribution of angular momentum in the step function case. % Jc~ 100,___ 0 ~Ro Ro Figure 6 2 : A graphic showing the function of percent CR angular momentum versus radius in a slope function model.
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140 Models for the Development of Systems with CR Angular Momentum While disk s ystems with CR angular momentum have been observed in nature, it is unclear how such systems might have formed in their environments. Several ideas have been put forth and will be discussed here. The strengths and weakness of these ideas will be considered but no attempt will be made to reach a conclusion as to their validity. Also applications of how such scenarios of formation might lead to a configuration similar to the initial conditions described above will be discussed The most widely held view regarding the formation of disk systems with significant CR angular momentum is that of an encounter or merger (Rubin 1994, Rix et al. 1992). Several versions of this idea have been considered with the most prevalent being the inclusion of a satellite system through merger and the stripping of gas from another system via an encounter. Such an encounter need not end in the consolidation of both systems, but rather merely provide the necessary material to construct the counterrotating component. One attractive model to explain the occurrence of NGC 4550, NGC 7217 and NGC 4826 is based on a polar ring galaxy model. A polar ring galaxy is an SO Hubble type galaxy that has a large ring of gas orbiting the center of mass of the system nearly perpendicular to the plane of the stellar disk. Several authors have shown (Sparke 1986, Sparke et al. 1998, Amaboldi and Sparke 1994, Tohline 1990, Christodoulou and Tohline 1993, Mahon 1991) that such a ring, over time, will precess into the plane of the disk. Depending on the direction of the precession, the gas ring will settle into a configuration in the plane of the disk that counterrotates with respect to the stars in the SO galaxy. As this gas begins to spiral in towards the center of the system, it will resemble
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141 NGC 4826. During this process, the system will form stars, presumably through a twostream instability (Sweet 1963), that will be counterrotating with respect to the original disk. If sufficient timeis availible, most of the gas in the system will be converted into stars and a configuration similar to that of NGC 4550 will have benn realized. While an attractive evolutionary theory, this view suffers from the difficulty that the masses needed to form such systems should have the effect of disrupting the disk system Nevertheless, high mass polar rings (""' 109 1010 M0 ) are known to orbit some SO systems and the counterrotating mass of NGC 4550 is estimated to be of this same order. Perhaps, if the gas were acquired from more than one encounter, or if, gas acquired in a higher inclination encounter has a smaller perturbing effect on the stability of the capturing disk system, this difficulty could be avoided. Another possibility is the convers10n of chaotic orbits to regular orbits by the destruction of the bar in galaxy. The argument, put forth by Athanassoula (1995), can be simply illustrated by using the analogy of a simple pendulum When gravity is "turned on" for the pendulum, it will execute simple harmonic motion. As the gravity is "switched off', the pendulum will traverse the complete circle around its pivot point. The direction of rotation of the pendulum will be determined by when, in the pendulum's oscillation, the gravitational force is removed. Hence, it would be equally probable, if gravity is removed at some random time, for the pendulum to rotate in either a positive or negative sense. In a similar way, chaotic orbits created by the bar potential will be converted with equal probability into direct or counterrotating orbits when the bar potential is removed. It has been shown in numerical simulations (Shlosman and Noguchi 1993, Athanassoula
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142 1992, Sellwood 1995) that the bar can be destroyed by an accumulation of mass into the center of the disk. An encounter with a small system through the center of a barred disk system will destroy the bar as well (Weil and Hemquist 1992, Hemquist and Weil 1993). The most plausible theory to produce nonmixed initial CR angular momentum has been put forth by N. Vogl is (Voglis et al. 1991, Voglis 1994 ). He has shown that, during the protogalactic formation process in the early universe, torques exerted by other forming systems at long range can reverse the angular momentum vector of the inner portions of a protogalaxy during the expansion phase of the early universe. Voglis reports that this feature is reproduced in Nbody simulations of a forming protogalaxy The protogalactic mass of interest is allowed to evolve in the presence of other tidally torquing masses and develops regions of segi:egated angular momentum that are slow to mix. Such a scenario might be approximated by a step function model in an idealized way. As observed now these systems would have extensively mixed their angular momentum components. However, it is possible that the products of such a set of initial conditions might survive until the present time. Also, the merger with a counterrotating dwarf system will also smear out and form a ring of counterrotating material (Hemquist, Quinn and Fullager 1993, Mihos et al. 1995) similar to that of the slope function model. Given these two scenarios, it seems that it is very possible to produce systems very similar to the initial conditions used in step/slope function cases. Davies and Hunter ( 1995b) have shown that given these sets of initial conditions, it is possible to produce a number of different observed phenomena that are difficult to account for otherwise.
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Fully Mixed Case Results 143 Numerical Simulations In investigations of fully mixed cases for consistently truncated Toomre n=O and n= l disks, we initialize the disks with increasing amounts of CR angular momentum and measure the evolution of the bar mode and pattern speed When parameterizing the dis ks, we have chosen k=0.7 (t=0.245) which is clearly unstable according to the OP parameter. What we have found is that when the percentage of CR angular momentum reaches a critical value, the bar mode is suppressed in both n=O and n= 1 disks in clear contradiction to the prediction of instability made by the OP parameter. Table 6.1 shows both the mode strength and pattern speed as a function of the percentage of CR angular momentum. As can be seen, there is little dependance of the pattern speed on the amount of CR angular momntum present as long as the m=2 mode is present. Also, there is a slight weakening of pattern stregnth due to the increasing amount of CR angular momentum. Additionally, we note that the pattern establishes itself more slowly as the percentage of CR angular momentum is increased. Above the critical percentage, all nonaxisymmetric instability is quenched, as determined by a Fourier analysis of the azimuthal disk modes. The m=l instability predicted by Kalnajs ( 1972) and observed by Zang and Hohl (9187) is avioded because we have choosen k < 0. 7071, the value where the onset of such an instability would occur (Araki 1987).
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144 Table 6 1 : The mode strength and pattern speed of the m=2 mode for Toomre n=O and n= I disks as a function of percent CR angular momentum in the fully mixed case. Toomre n=O Toomre n=l % counter Pattern Speed m=2 mode Pattern Speed m=2 mode rotating strength strength 0 0.54 0.41 1.05 0.30 5 0.44 0.44 0 90 0 29 10 0.48 0.44 0 .81 0.29 15 0 .51 0.10 0.98 0.26 20 0.04 0 95 0 23 25 1.12 0.10 30 0.05 As a test of the Christodoulou parameter (equation 3.97), we calculated the a parameter for each case run and determined whether the numerical model was in agreement with the prediction made. Table 6.2 shows the results for both n=O and n= l stellar disks. As can be seen, while the Christodoulou stability parameter does predict the global behavior of the n= 0 system, it fails rather badly in predicting the stability of the n= l system This is likely due to the difficulty of calculating D.1 for a differentially rotating curve with a velocity maxima. Further investigations will need to be done to determine the actual values of a. The error in the values of a as calculated for a system of 16000 particles as compared to an analytic calculation is on the order of 0.4% as compared to the analytic result derived from the Toomre disk formulae.
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145 Table 6 2 : Results of the Christodoulou stability criterion check for Toomre n=O and n= I disks % Toomre n=O Toomre n=l JcR O'. Prediction Actual O'. Prediction Actual Behavior Behavior 0 0.3972 unstable unstable 0 2406 stable unstable 5 0 3576 unstable unstable 0.1732 stable unstable 10 0 2860 unstable unstable 0.1212 stable unstable 15 0.2002 stable stable 0.0727 stable unstable 20 0 .1201 stable stable 0 0364 stable unstable 25 0.0601 stable stable 0.0145 stable marginally stable 30 0.0240 stable stable 0.0044 stable stable Step/Slope Function Case Results Toomre n=O disks As was shown in chapter 4, when a Toomre n=O disk is constructed to be bar unstable the majority of the initial disk matter is swept into a rapidly growing, slowly rotating, long bar. After this mode has been saturated there is very little material left that is populating the disk. When initial disks are built with a significant portion of their angular momentum counterrotating in a step function model, we find that we are able to strongly modify the evolution of this type of system. Table 6.3 shows the parameter space investigated in this
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146 Table 6 .3: Toomre n=O models step function models investigated as a function of k and J c Rk\JcR 100 % 80 % 67% 50 % 33% 20 % 0.4 stable stable stable stable stable stable 0 5 slowly stable forming bar 0.6 bar/spiral disk/bar disk/bar stable stable 0 7 bar/spiral disk/bar 0.8 frag./bar frag./disk /bar 0.9 frag./ frag. frag./bar frag./bar collapse study. As is shown by the bold table cells, we are able to produce long lasting bar/disk configurations in these models if the percentage of CR angular momentum in the center of the disk is sufficiently high. The pattern of the bar in these models is counterrotating and is approximately 1/3 the pattern speed of the fully prograde disk. Such systems may account for a portion of the 'slow bar' systems that have been observed. The evolution of this type of system can be seen in Figure 6.3. Additionally, as fully direct Toomre n=O disks (and presumably other systems whose rotation curve constantly rises) lack the shear necessary to truncate bar growth inside the disk radius, the presence of two separate regions provides the means to accomplish this. Thus, the existance of a truncated bar in a system with a rising rotation curve may be due to the presence of CR angular momentum early in a galaxy's lifetime.
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147 St.1or 01. SWor oi. s1 .. 1or o... l 0.000 crt.U t ~ .000 crutl t 10.000 4 1 0 2 4 4 1 0 2 4 4 2 O 2 4 4 2 O 2 4 2 0 2 4 4 J O 2 4 4 2 O l 4 1 0 2 4 2 0 2 4 Figure 6 .3: Evolution of an unstable Toomre n=O step function case with 50% of the angular momentum counterrotating.
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148 Toomre n= l disks Similar to the Toomre n=O disk cases discussed a bove we found that we were able to produce a counterrotating bar of fractional p a ttern sp eed if the initial conditions were constructed to have a s ufficiently large portion of the inner di s k counterrot a ting. However for this type of initial condition, we found that if we start with only I 0% of the initial angular momentum rever s ed the global evolution of the sys tem proceed s in a markedly different fashion. Table 6.4 shows the models evolved in this series of tests. Of particular interest is Model # l, the step function case. Here we see the development Table 6.4: Step/Slope function models for a Toomre n=l dis k with J c R = O.lJy. M# Model Type Inner CR Bar Outer Bar Comments tcty n l Step Function yes yes inner bar 40 reverses 2 Steep Slope yes yes inner bar 40 destroyed 3 Shallow Slope Develops Core yes core persists 40 of a s hort, rapidly rotating inner counterrotating bar On a longer time s cale a larger directly rotating bar forms Over time, we find that the inner bar's pattern speed is reduced and eventually its pattern motion stops and reverses. After this time, the inner bar oscillates with decreasing amplitude around the outer bar (when viewed in the outer bar's frame) until the inner bar lines up with the outer bar and rotates cospatially with it for the remainder of the simulation. The isodensity plots for this simulation are s hown in Figures 6.4 and 6.5 For N=32000 particles the time elapsed before the bars align
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149 permanently is ,....., 1.9 Gyr. Due to relaxation effects we expect that in a physical syste m this proce ss will take so mewhat longer an d therefore, s uch sys tems may be a ble to be observed now Further discussion of this can be found in the final chapter of this dissertation. For Model #2, a steep s lope function model we find that while the two counterrotating bars form as in the first case, they do not reverse their direction over the s imulation time but rather the inner bar is destroyed. In Model #3, a s hallow s lope function model, a long term stable core is developed along with a large bar. This core is s table to the perturbation of the bar potential and lasts throughout the simulation. Isodensity plots for this simulation are shown in Figure 6.6.
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150 'i I "''~______.j I 0.5 0 0..5 I I 0. 5 0 0 $ 1 1 o., 0 0.5 1 s,...,.o..~ s:i: .... oi..........ity sw.o..~ c,1.Mt 1 2 4 800 1 2 S.200 c,t..t 2'. 500 I i '''~u I 0.5 O o I I 0.5 0 O .!I I 1 0. 5 0 0..5 1 Slelel'Ollit......it, SW.DWI~ St..._.Dlllll..........,. .,.,1 :n.1'00 crt..t 1 21.100 ct'llllt t Zl .00 '"''~= I 0.5 o 0..5 I I o...5 0 0 .5 I 1 0,,..5 0 0.5 I SW.Olllt......., Sl .... Ola..... St.lllarOla~ ct'llllt t 21. 700 ert.t t .. 27 .000 .,.,_ I 27.300 ' : I y ,...._______.___~ 1 0.5 0 0.5 1 1 0.., 0 0 5 l I 0.5 0 0 .5 I Figure 6.4 : Isodensity plots of Toomre n=l step/slope function model #1 prior to inner bar reversal.
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15 l ''., ...,..0 5 0,.,.0,~ 1 t I ..(l, S O 0 S \ I''., <)...,... ~ .,', St .... ow.~ Sl ..... OWi...., ~ow.~ Ottl 28.SOO c,rt..t I 21.900 ..,.. I 11. 100 I 0.5 0 0 S I j 1 o.J O 0. S 1 i ''. <>...,._.,~ 0 ~ 0 .5u, St ..... OWi........... !,t .... OW,........, st .... ow.......,. ..t 21 ..00 C11a.t I 21.XIO crtt J0.000 ''_ 0...,.., 0,.,.0,~1 I I o.J Cl ().5 I I~.. ....~. .~ ~ . .... a........., Sl ... ot.11........., SW.OW.~ t JQ..lCIO ort..t I l0'00 ort..t t J\.000 1 o.5 0 O.S I j 1 (J.S O O S 1 j ''., ...,_,...,..,,.,.,.,~, Figure 6 5 : Isodensity plots of Toomre n=l step/slope function model #1 following inner bar reversal.
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152 o..5 o o.5 1 o..5 o o.5 1 _, 0.5 0 0.5 1 St ..... OWI~ st ..... Ol.i......, st.._.oi..~ ort.u I J2.!00 ort..U I J.J.100 crt..u l l4. 100 1 0..5 0 0.5 1 I 0.5 0 0.5 I I 0. 5 0 0. 5 1 1 0..5 0 0 5 1 I o.5 0 0..5 1 I 0.5 0 o.5 I Shihro.11:........... SW.Dia......,.., St.llarOllli~ artaat2 I l7.IOO tatl tJl.100 art.tl I 31 .IOO I 0..5 0 0..5 I I ().5 0 0..,, I I 0. 5 0 0. 5 I Figure 6.6 : Isodensity plots of Toomre n=l step/slope function model #3.
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CHAPTER 7 DISCUSSION OF RESULTS AND FUTURE WORK General Conclusions Introduction The final chapter of this dissertation is devoted to the summing up of the work reported herein and stating conclusions that may be drawn from it. We will discuss implications on the present understanding of barred spiral systems and propose a direction for further research. Moreover, we will discuss where our present modeling program is likely to proceed and what other issues that may be investigated in view of the findings here. As in all numerical work some caution should be applied to the results reported While this research provides insight into what may be taking place in galactic disk systems on a global scale, it is not a substitute for reliable observations and solid, predictive theoretical work The research reported here has centered on the investigation of the global properties of the systems of interest. Application of these results to the local dynamics of the system should be viewed cautiously and any conclusions drawn about individual particle evolution may only be correct in a statistical sense. Much of the research done for this dissertation involves the evolution of initial conditions to systems in which m=2 (bar) modes are prominent features. This work has been done as part of a long term program to understand the behavior of such systems. Research done by other student investigators may be found in the dissertations of Ball (1984 ), England (1986), Moore (1992) and Kaufmann (1993). Barlike structures are 153
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154 ob s erved in nearly twothirds of all disk systems and therefore, the understanding of the global dynamics of these sy s tems their formation and their evolution, is vital for an understanding of galactic evolution as a whole A good review on the subject of barred spiral systems has been written by Sell wood and Wilkinson ( 1993). Numerical Integration of Toomre Disks While the dynamical properties of the Toomre family of disks have been known for some time. We present results here regarding their global development in regimes well removed from their analytically stable, initial equilibrium configurations. In the absence of a stabilizing halo, we find that both Toomre n=O and n=l disks evolve away from their initial equilibrium configurations towards a density distribution resembling that of an exponential disk This result may be of considerable importance because although many authors assume that disks constructed from Toomre or Toornrelike initial conditions, such as Mestel disks (Mestel 1963) and Miyamoto disks (Miyamoto and Nagai 1975 Nagai and Miyamoto 1976) will remain in roughly that configuration throughout a simulation, we find that without some stabilizing component, such as a bulge or halo these disks will not remain in their initial state, but rather evolve towards an exponential configuration. This is significant since observations seem to show that most disks follow this sort of a density profile. Why nature seems to prefer this configuration will obviously have something to do with the minimization of the free energy of the system, but we have been unable to find a theoretical explaination. While the results here apply only to the disks studied, perhaps a more comprehensive study of the evolution of these families
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155 of disks in the absence of halos 1s needed to better understand what mechanisms are responsible for this behavior. Encounters and Local Group Dynamics A number of simulations were performed to investigate the effects of nonmerging encounters between a variety of systems and to gauge the accuracy of some of the assumptions used by Valtonen and his collaborators for their Local Group dynamics simulations. While we did not model merging systems, many of the results from this study are applicable in that regime. The major results of the study are the discovery of the increase of gas inflow rate towards the center of a disk shortly after the point of closest passage and the effects of the inclusion of star formation during these simulations The first result implies that shortly after a nonmerging encounter, a strong burst of star formation should begin in both the bar and the undisrupted arm of the disk The former will be due to the increased inflow rate that takes place over 35 dynamical times of the system and the latter as a result of the strong amplification of the mode due to the passage of the perturber. Therefore, observations of a comparatively young stellar population might provide constraints on the time of passage for a nonmerging encounter. Observations by Combes et al. (1994) show that those systems that are interacting or have recently undergone interaction are observed to have greater CO luminosities and star formation rates. While the specific results may differ somewhat for merging encounters, a period of higher star formation activity should still be expected. Finally, it should be noted that this effect takes place regardless of the composition of the perturber. Gas poor systems will cause inflow and starburst activity after an encounter with a gas rich system
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156 nearly as well as a gas rich system will. Likely of greater importance are the particulars of the interaction. These will include the sense of the interaction angular momentum with respect to the spin of the larger system, the impact parameter and the total energy of the incounter between the two systems. The second effect implies that for these encounters to be modeled properly, it is likely necessary to include star formation in the algorithm While this has little impact on the global evolution of the disk system, it may play a large role in accurately modeling the details of spiral arm development and gas capture. As was shown, star formation in a gas rich perturber had the effect of thermalizing the gas component, thereby making it easier for tidal forces to strip gas from the dwarf system. Therefore, for those systems that are less tightly bound than the one used here, significant gas loss will likely occur to the more massive system in the form of bridges, tails and streams. Therefore, a number of the smaller, gas poor systems observed may have undergone an encounter with a larger system that has robbed the dwarf of its gas. Again, this event may be attested to the presence of a relatively young population of stars left over form the starburst activity initiated by the passage. As O and B stars are short lived, observations to determine this would need to focus an finding A and F populations and distributions. Initially, these stars may form near the center of the smaller galaxy and may still be found there in relatively greater numbers. Limitations of this study are the inability to adequately model star formation processes and the lack of tracing of Population I objects The first difficulty is likely to be a subject of study in the field of numerical simulations for some time. Clearly, however the
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157 criteria used here are unable to produce star forming regions in all of the areas in a disk system where they are observed to take place of are expected to take place i e., the spiral arms and shock regions produced by the perturbing particle. A proper study of the second issue will discussed in the section titled "Future Work to be Done" However, to answer the questions posed by Valtonen, the evolution of the orbits of these Population I objects must be studied in a statistical way, as conclusions drawn from individual orbits may be inaccurate. Counterrotating (CR) Systems The primary result of this portion of our modeling effort is the fact that counterrotating angular momentum can significantly alter the global evolution of disk systems It may do this either by stabilizing a normally unstable system, creating greater unstable structure or a combination of the two. By including the angular momentum in differing distributions, the systems will take a markedly different paths to their final "equilibrium" states. In the fully mixed case, if the amount of CR angular momentum is sufficient, the disk will be stabilized against the formation of nonaxisymmetric modes. Note that ring and other axisymmetric modes may arise if the k is large enough. Also, while Araki predicts the formation of an m= l nonaxisymmetric mode if k is too large, we have only seen the axisymmetric instabilities when k> 0.75. This seems to point to a need to include the system's total angular momentum in the determination of a stability parameter as that is the only change that has been made in the global parameters of these systems G. Byrd claims to be unable to stabilize a disk system against nonaxisymmetric modes through the use of CR angular momentum in contrast to our results (Davies and Hunter l 995a,b)
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158 and those of Sell wood and Merritt ( 1994). We believe that this is due to the use of a Generalized Mestel Disk with b=O as initial condition s for his s imulations Thi s rather pathological choice of initial conditions gives one a flat rotation curve over the entire disk, but requires an infinite density at the disk's center. Consequently his results may be an artifact of his choice of disk model and initial conditions. However it may also be due to k being greater than Araki s 0.7071 critical value for m=l mode formation in portions of the initial disk. In step/slope function models, we are able to construct initial conditions that lead to bar/disk and bar/bar combinations that are not otherwise achievable in a single component model. Friedli and Martinet ( 1993) have been able to construct bars with differing pattern speeds using two components and Friedli ( 1995) has constructed counterrotating bars using the same method These results show the importance of where the CR angular momentum is distributed in the initial disk. As these modes seem to be global in nature and form on time scales that are dependant on the structure s size, it is unlikely that they are the result of local perturbations. Future Work to be Done Improvements Two fundamental improvements need to be made to the modeling efforts to increase their usefulness in the study of barred systems. The first of these is to extend the modeling program into the third dimension. One of the great strengths of the algorithm used herein is its ability to easily model systems of any geometry As we have confined
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159 ourselves to studies in two dimensions, much of this strength 1s lost. Paramount in this is the construction of three dimensional initial conditions. It would be beneficial to be able to extend the Toomre family of disks either through the use of Miyamoto disks or by assuming an isothermal z distribution. Most efforts that use these types of disks also incorporate a halo and bulge and, therefore, are more stable against the formation of nonaxisymmetric modes We would like to use a model that does not require these components to be vertically stable, allowing us to more easily investigate those mechanisms that lead to global instability Another possibility is to use the disk discussed by Fall and Efstathiou ( 1980) which models the observations in a fairly systematic fashion. The difficulty with these two approaches arise from the difficulty in exploring the parameter space leading to instability. The second major improvement to be made is to use a star formation algorithm that more accurately reproduces the observation of active star forming regions in the spiral arms of a disk system. While there is a desire to make use of as much physics as is understood in concocting a 'recipe' that attempts to model such processes realistically, our fundamental lack of understanding of crucial processes involved in star formation in the ISM (i.e. weak magnetic coupling) will limit such an approach Clearly however, any method of this sort must include the considerations of local density (via the Jeans' mass or some such), local velocity fields (thus incorporating negative velocity divergence fields and local shear), and local temperature (thus modeling efficient cooling in molecular clouds). While using the local Toomre Q parameter takes into account local density and velocity fields (and possibly local shear as well) in a fashion that allows for
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160 easy adjustment it is a local linear theory that may not describe the physics of a region dominated by global nonlinear perturbations well. Another possibility is to look at the local time scales for collapse, divergence and shear and assign a 'score' to each particle based on inverse of each time scale. If the score is above a certain value, star formation is initiated. While this is again a strictly local measure it avoids the difficulty of having to calculate the local epicyclic frequency using local estimates of the gravitational potential. Continuing Work In the area of modeling nonmerging interactions, there is obviously a need for further examination of the encounter parameter space. The most promising avenue is varying the mass and radius of the dwarf disk to produce models that are less tightly bound Additionally, models of varying gas percentage in the larger disk should be investigated to determine the effect of gas mass on the passage of the dwarf system. Finally, the ratio of dwarf mass to disk mass should be varied, along with the impact parameter, to produce a range of simulations that may gauge the encounter limits at which the satellite may be destroyed, even though the encounter is nonmerging. For the counterrotating angular momentum study, continued work needs to be done to investigate systems constructed of two superimposed Toornre n family disks having differing scale lengths that counterrotate with respect to each other. Friedli ( 1995) has reported that such initial conditions are able to produce primary and nuclear bars that rotate at different pattern speeds. Additionally, we will investigate varying the value of k as a function of the radius in a single component disk to attempt to produce models similar to those we have already studied.
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161 On the i ss ue of Toomre n family disks evolving axisymmetrically to exponential disk s we will run s table models for families higher in the Toomre s equence. Model s wherein the value of n 2 2 have rotation curves that drop off more rapidly after the v e locity maximum than does that of a n exponential disk having the s ame initial velocit y maximum and radius. We wish to s ee if these disks evolve to an exponential configuration with greater or lesser central condensation, Such a s tudy would allow us to inve s tigate if their is a single perferred exponential profile for all disks or if each disk eveolves to its own final s tate that is independant of the rest. Questions Still to be Answered The main question le f t unanswered by this study is that of the evolution of Population I objects in disks that have undergone encounters Valtonen claims that s uch object s s hould be placed on highly eccentric plunging' orbits by a merger event. We feel that to treat such a question without the benefit of a three dimensional model would be to neglect too much important physics. Therefore, this question will have to wait until s atsifactory initial conditions for a three dimensional disk have been constructed successfully. We plan to investigate this claim by symmetrically distributing several hundred particles in a halolike distribution around the center of the galaxy and then running the encounter simulation. We hope that by following the orbits of many particles, we will be able make a statistical claim as to the behavior of these objects and what one might observe in these types of systems A second issue that is still unresolved is whether the Christodoulou parameter i s able to accurately predict the stability of the CR disk systems investigated herein. At this point
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162 we are unable to use the o: parameter to accurately predict the behavior of an unstable Toomre n= I disk. Whether this is a believable result or a difficulty in the calculation of the various quantities that are used to determine the value of o: is unclear. If the former is true, this study will represent the first known counterexample. While we expect to answer this question soon for the fully mixed disks, the application of the methods to the step/slope function models is a great deal more complicated as these systems are not simply connected topologically Therefore f (the connectedness parameter given in chapter 3) is not simply calculated and a more complicated method of calculating o: will have to be used. Questions Raised The most interesting question raised during this study is that of the mechanism underlying the axisyrnrnetric evolution of Toornre disks to exponential disks While work still needs to be done to determine the nature of this change, the more fundamental issue is determining the reason for the change. The Toornre densitypotential pair is analytically stable and Evans and de Zeeuw ( 1992) have shown that any axisymmetric disk may be analyzed into a superposition of Toornre n=l (Toornre 1963 Kuzmin 1956) disks with differing central densities and scale lengths. This being the case, any exponential disk may be constructed out of n= 1 disks, which we have shown evolve to exponential disks. Obviously, this is an area in which more work is needed. An additional question is whether counterrotating bars can be formed in physical disk systems and, if so, will they persist. Observations by Friedli and Wozniak (Friedli et al. 1995, Wozniak et al. 1995) have shown that several barred systems possess misaligned
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163 inner bars. As the pattern speeds for these nuclear bars has yet to be determined, it is possible that one or more of them may counterrotate with respect to the outer bar. Also important is the consideration of the effects of such an inner bar on the orbits in barred systems. It is possible that such a disk will introduce a substantial amount of chaos into the system Finally, what are the mechanisms leading to the eventual destruction or assimilation of the nuclear bar by the large outer bar? While dynamical friction certainly has an effect on the nuclear bar, it seems that other mechanisms, such as torque or orbit stripping due to the larger bar, will act on shorter time scales and exert a dominant influence on the nuclear bar's evolution.
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APPENDIX PROGRAM GEN STD 1 C C This program generates a Toomre n= 1 disk of specified mass MD, C radius RD, shape parameter B, with NTOT particles of equal mass. C C======================================================== IMPLICIT NONE INTEGER NRMAX,NUM NBODMAX NINTERP,SMINDX PARAMETER (NRMAX=200,NBODMAX=64000,NINTERP=5000) REAL *8 RD MD B K R,COSA SINA, VT, VR VX VY ,DX,DY EPS,XO RAD, & SMASS,PI,R2,C2,THETA RAN3,SR2(NRMAX),RR(NRMAX),YO RO, & TMASS,DENS,DR,K 1,K2 K3,K4,SF2(NRMAX),G, VRG VTG,RC RINV & RCMl, FLNVO FVO,YY TINC YYOLD SMASSl,E,A,SM R3INV,RNIN, & ACCSM, TINY V02(NRMAX),ACC(NBODMAX,2),ACCX ACCY DACCX, & X(NBODMAX), Y(NBODMAX) ACSMOOT(O : NINTERP+ 1 ),XW3,XW 4, & XW,G2,RP,XW2 DACCY,NM,DRSM,DELDRG,MH INTEGER I,NTOT,IDUM,NC,NINC,L TOT,J,N(NRMAX),DUM COMMON/COM1/RD,MD,PI,B C2 ,DR, MH,RO COMMON/VEL 1N02(NRMAX) OPEN(UNIT=l0,FILE= 'tlk7crl. out',STATUS='UNKNOWN') 164
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165 OPEN (UN IT=20 FILE= t I k7crl dat STATUS= 'UN KNOWN ') WRITE (6,*) 'INPUT THE RADIUS OF THE DISK :' 50 FORMAT(lX ,' ') READ ( 5 *) RD WRITE(6,*) RD WRITE ( 6 50 ) WRITE(6,*) INPUT THE MASS OF THE DISK (XlOEll)' READ(5 ,*) MD WRITE(6,* ) MD WRITE ( 6,50) WRITE(6 ,*) 'INPUT THE SHAPE PARAMETER: READ(5 ,*) B WRITE(6, *) B WRITE(6,50) WRITE(6 ,*) 'INPUT HEATING PARAMETER KAPPA:' READ(5 ,*) K WRITE(6,*) K WRITE(6 50) WRITE(6,*) INPUT THE NUMBER OF S. SHELLS :' READ(5, *) NC WRITE(6 *) NC WRITE(6 50) WRITE(6,*) 'INPUT THE TOTAL NUMBER OF DISK PARTICLES:' READ(5 *) NTOT
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WRITE(6 *) NTOT WRITE(6,50) WRITE(6 ,*) INPUT EPSILON:' READ(5,*) E WRITE( 6,50) WRITE(6,*) 'INPUT A:' READ(5,*) A TINY=l.OD20 PI=3.1415926535893 EPS=0.05 IDUM=1 35 SMASS l=MD/NTOT DR=RDINC 166 C2=MD*PI*B C2=C2/(2.D0*(DATAN(RD/B)(RD*B)/(RD*RD+B*B))) TOT=NTOT NTOT=O CALL INTEG(SM,A) NM=O.DO WRITE(6,*) 'CALCULATING TOTAL NUMBER OF STELLAR PARTICLES' DO 52 L=NC, 1,1 RC=L*DR
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167 IF (L.EQ. l) THEN RCMl=O.DO ELSE RCM I =(L1 )*DR ENDIF IF (L.EQ. l) THEN TINC=TMASS(RC) ELSE TINC=TMASS(RC)TMASS(RCM 1) WRITE(6, *) TINC,SMASS I ENDIF IF (RC.LE. I .DO) THEN NM=NM+TINC ENDIF NINC=TINC/SMASS 1 NTOT=NTOT +NINC 52 CONTINUE WRITE(6, *) NTOT WRITE(6,*) 'IF NTOT IS CORRECT INPUT l: READ (5, *) DUM IF (DUM.NE. l) THEN WRITE(6,*) 'INPUT NEW NTOT : READ(5, *) NTOT GOTO 35
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168 ENDIF SMASS=MD/NTOT SMASS=SMASS/NM SMASS l=SMASS 1/NM MD=MD/NM C2=MD*PI*B C2=C2/ (2. DO*(DATAN(RD/B) ( RD*B)/ ( RD*RD+B B) )) WRITE(6 *) C2 YY=O.DO WRITE( 10, *) NTOT,2,0.DO,O,O WRITE(6,*) BUILDING STELLAR DISC',NM,MD C ****************************************************************** C C BUILD STELLAR DISC C C ****************************************************************** C C INITIALIZE VARIABLES AND SMOOTHED ACCELERATION TABLES CWRITE(6 *) SMOOTHED ACCEL TABLE
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169 DELDRG=2.DO/NINTERP DO l=O,NINTERP/2 XW=l*DELDRG XW2=XW*XW XW3=XW2*XW XW4=XW2*XW2 ACSMOOT(l)=XW3*(4./3 6. *XW2/5.+0.5*XW3) ENDDO DO l=NINTERP/2+ l ,NINTERP XW=l*DELDRG XW2=XW*XW XW3=XW2*XW XW4=XW2*XW2 ACSMOOT(l)=1./15 +8 *XW3/3.3 *XW4+ & 6 *XW3*XW2/5 .XW4*XW2/6. ENDDO ACSMOOT(l)= 1.0 DO l=l,NRMAX RR(l)=O.DO V02(l)=0.D0 X(l)=O.DO Y(l)=O.DO
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SR2 ( I )=O.DO SF2(I)=O.DO ENDDO DO I= l ,NBODMAX ACC(I, 1)=0.DO ACC(I,2)=0.DO ENDDO WRITE(6,*) PUT PARTICLES DOWN' NUM=O DO 56 L=NC 1,1 RC=L*DR IF (L.EQ. l) THEN RCMl=O.DO ELSE RCMl=(Ll)*DR ENDIF RR( L)=(RC+RCM 1 )/2 .DO IF (NC.EQ.l) THEN TINC=TMASS(RC) ELSE TINC=TMASS(RC)TMASS(RCM 1) ENDIF 170
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NINC=TINC/SMASS I N(L)=NINC DO I=l,NINC NUM=NUM+I R=RCDR *RAN3(IDUM) R2=R**2 THETA=2.0*PI*RAN3(IDUM) COSA=COS(THETA) SINA=SIN(THETA) X(NUM)=R *COSA Y(NUM)=R*SINA ENDDO 56 CONTINUE WRITE(6, *) NUM C 171 C SUM OVER ALL PARTICLES FOR EACH PARTICLE TO FIND ACCELERATION CWRITE(6, *) FIND ACCELERATIONS' DO J=l,NTOT1 XO=X(J) YO=Y(J)
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172 ACCX=O .DO ACCY=O .DO DO l=J+l,NTOT DX=X(I)XO DY=Y(I)YO RAD=DSQRT(DX*DX+DY*DY) RINV= l .DO/(RAD+ TINY) R3INV=RINV/ ( RAD*RAD+ TINY) RNIN=RAD*NINTERP/(2.DO*EPS) SMINDX=RNIN SMINDX=MIN(NINTERP,SMINDX) DRSM=MIN( l ,RNINSMINDX) ACCSM=( l .DODRSM)* ACSMOOT(SMINDX)+ & DRSM* ACSMOOT( 1 +SMINDX) R3INV=ACCSM*R3INV DACCX=R3INV*DX DACCY =R3INV*DY ACCX=ACCX+DACCX ACCY=ACCY+DACCY ACC(I, l )=ACC(I, l )DACCX*SMASS ACC(I,2)=ACC(I,2)DACCY*SMASS ENDDO
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ACC(J, l)=ACC(J l ) +ACCX*SMASS ACC(J ,2)=ACC(J ,2)+ACCY* SMASS ENDDO C C CALCULATE VELOCITIES C .173 WRITE(6 *) CALCULATE VELOCITIES' NUM=O WRITE(6,*) VEL. PER RING' DO J=NC,l,1 DO I=l,N(J) NUM=NUM+l V02( J)= V02(J)+ACC(NUM, l )*X(NUM)+ACC(NUM,2)*Y (NUM) ENDDO IF (J.EQ.NC) THEN V02(J)=FVO(RD )* *2 ELSE V02(J)=V02(J)/FLOAT(N(J)) ENDIF IF (V02(J).LT .O.D0) THEN WRITE(6,*) V LESS THAN ZERO FOR RING:',J WRITE(6, *) V02(J),RR(J)
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174 V02(J)=FVO(DR)**2 ENDIF ENDDO DO 58 L=NC ,2,l RC=L *DR YYOLD=YY Kl=G2(RC,L,YY,0) K2=G2(RCDR12.,L YY(DR/2.DO)*K 1, l) K3=G2(RCDR12.,L YY(DR/2.D0)*K2, l) K4=G2 (RCDR,Ll, YYDR *K3,0) YY=YY(DR/6.D0)*(Kl+2.*K2+2.*K3+K4) IF(L.EQ.NC) THEN SR2(NC)=0 D0 ELSE SR2(L)=(YY /DENS(RC)) ENDIF IF (RC.LE.(RD*.9)) THEN SR2(L)=( l D0+E)*SR2(L) ENDIF SF2(L)=(0.50*SR2(L)*( l.DO+FLNVO(RC))) 58 CONTINUE SF2( 1 )=SF2(2)+(SF2(2 )SF2(3))
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175 SR2( I )=SR2(3)+(SR2(2)SR2(3)) C C ASSIGN VELOCITIES AND WRITE PARTICLES CWRITE(6,*) ASSIGN VELOCITIES' NUM=O DO J=NC,1,1 DO I=l,N(J) NUM=NUM+l CALL GASDEV(VRG,VTG,A,IDUM) COSA=VRG/SM SINA=VTG/SM VT=K*DSQRT(V02(J))+SQRT(SF2(J))*SINA C IF (VT.LT.0.00) VT=VT VR=DSQRT(SR2(J))*COSA VX=VT*Y (NUM)/RR(J)+ VR *X(NUM)/RR(J) VY=VT*X(NUM)/RR(J)+VR*Y(NUM)/RR(J) RP=SQRT(X(NUM)*X(NUM)+ Y(NUM)*Y(NUM)) WRITE(lO,*) X(NUM),Y(NUM) WRITE(lO,*) VX,VY WRITE( l 0, *) SMASS WRITE(20, *) 'MD=' ,MD,'B=' ,B
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176 ENDDO ENDDO WRITE(6,*) 'DISCS BUILTPROGRAM END' STOP END **************************************************************** FUNCTION RAN3(IDUM) C C Returns a uniform random deviate between 0.0 and 1.0. Set IDUM C to any negative value to initialize or reinitialize the sequence. C If running on machine that is weak on integer arithmatic, then use C commented lines. Any other large value of MBIG and MSEED will C work. but the current C version is from Numerical Recipes by Press et al. C C======================================================= IMPLICIT NONE REAL *8 FAC,RAN3 INTEGER MBIG,MSEED,IFF,IDUM,MJ,MA(55),MK,II,MZ,K,I, & INEXT,INEXTP PARAMETER (MBIG=lOOOOOOOOO, MSEED=161803308,MZ=O,FAC=l.DO/MBIG) DATA IFF/0/
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IF(IDUM.LT O.OR.IFF EQ.O)THEN IFF=l MJ=MSEEDIABS(IDUM) MJ=MOD(MJ MBIG) MA(55)=MJ MK=l DO l=l,64 Il=MOD(2 l *1,55) MA(Il)=MK MK=MJMK IF(MK.LT.MZ) MK=MK+MBIG MJ=MA(II) ENDDO DO K=l,4 DO l=l,55 MA(l)=MA(l)MA( 1 +MOD(I+ 30,55)) IF(MA(l).LT.MZ) MA(l)=MA(l)+MBIG ENDDO ENDDO INEXT=O INEXTP=31 IDUM=l 177
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ENDIF INEXT=INEXT + I IF(INEXT.EQ.56) INEXT= I INEXTP=INEXTP+ l IF(INEXTP.EQ.56) INEXTP= l MJ=MA(INEXT)MA(INEXTP) IF(MJ.LT.MZ) MJ=MJ+MBIG MA(INEXT)=MJ RAN3=MJ*FAC RETURN END 178 C*************************************************************** C C FUNCTION TMASS C C*************************************************************** FUNCTION TMASS(RR) IMPLICIT NONE REAL *8 RR,Al,A2,A3,A4,RD,MD,PI,B,C2,DR,TMASS,MH,RO COMMON/COMI/RD,MD,PI,B,C2,DR,MH,RO Al=DATAN(RD/B) A4=DSQRT(RD*RDRR *RR)
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179 A2=(B*(A4RD))/(RD*RD+B*B) A3=(B1DSQRT(RR*RR+B*B))*DATAN((A4)/DSQRT(RR*RR+B*B)) TMASS= (2.DO* C2/ ( PI B ) )*(A I +A2A3 ) RETURN END C ************ *************************************************** C C FUNCTION DENS C C*********************** *********** *** *************************** FUNCTION DENS ( RR) IMPLICIT NONE REAL 8 RR,Al A2 RD ,M D PI B C2 ,DR, DENS MH RO COMMON/COM l/RD,M D,PI B C 2, DR,MH RO Al= l .DO/(( RR*RR+B*B )** 1.5) Al=Al *DATAN(SQRT(RD*RDRR*RR))/DSQRT(RR*RR+B*B) A2=DSQRT(RD*RDRR*RR)/((RD*RD+B*B)*(RR*RR+B*B)) DENS=(C2/PI**2.0)*(A 1 +A2) RETURN END C************* **** *********************************** ****** ***** C
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C FUNCTION G C 180 C********************************** ****** **** ******************** FUNCTION G(R l ,X ) IMPLICIT NONE REAL *8 R l ,X,A l ,A2 ,A3,RD,MD,PI B C2 ,DR, FLK FLNVO DENS & FVO,G,FK,MH,RO COMMON/COM l/RD,MD PI B C2,DR,MH,RO Al=FLK(Rl) A2=FLNVO(Rl) A3=DENS(R 1)*( 1 FK(R 1)**2.DO)*FVO(Rl )**2.DO/R l G=(X/(2 DO*Rl)) *( 1.A 1A2)A3 RETURN END C*************************************************************** C C FUNCTION G2 C C**************************************************************** FUNCTION G2(Rl,J,X,FLAG) IMPLICIT NONE INTEGER J,NRMAX,FLAG
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181 PARAMETER ( NRMAX=200) REAL *8 R 1,X A 1,A2,A3,RD,MD,PI,B C2,DR,FLK,FLNVO,DENS, & G2,FK,MH RO, V02(NRMAX), V2 COMMON/COM l/RD,MD,PI B C2 DR,MH,RO COMMONNEL 1N02(NRMAX) Al=FLK(Rl) A2=FLNVO(R 1) IF (FLAG.EQ.O) THEN V2=DSQRT(V02(J)) ELSE V2=(DSQRT(V02(J))+SQRT(V02(J1)))/2.D0 ENDIF A3=DENS(Rl)*( 1FK(R 1)**2.DO)*V2**2.D0/R 1 G2=(Xl(2 D0*R 1 ))*( 1.A 1A2)A3 RETURN END C***************************************************************** C C FUNCTION FK C C**************************************************************** FUNCTION FK(RR)
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182 IMPLICIT NONE REAL *8 RR RD,MD,PI B,C2,DR,FK,MH RO COMMON/COM l/RD,MD,PI,B,C2,DR,MH,RO FK=.7 RETURN END C**************************************************************** C C FUNCTION FVO C C**************************************************************** FUNCTION FVO(R 1) IMPLICIT NONE REAL *8 Rl,RD,MD,PI,B,C2,DR,Al,FVO,MH,RO COMMON/COM1/RD,MD,PI,B,C2,DR,MH,RO A l=(C2*Rl *Rl)/(B*(Rl *R 1 +B*B)** 1.5) FVO=DSQRT(Al) RETURN END C**************************************************************** FUNCTION FLK(RR) IMPLICIT NONE
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183 REAL *8 RR,RD,MD PI,B,C2 DR,FLK,MH,RO COMMON/COM l/RD,MD,PI,B,C2 DR MH,RO FLK=O.ODO RETURN END C***************************************************** C C FUNCTION FLNVO C C**************************************************************** FUNCTION FLNVO(RR) IMPLICIT NONE REAL *8 RR,A l ,A3,A4,MH,RO,RD,MD,DR,PI,B,C2, & FVO,FLNVO,PARAN,ARITH, & Cl COMMON/COM l/RD,MD,PI,B,C2,DR,MH,RO Al=RR**2.DO+B**2.DO C 1 =DSQRT(C2) A3=(A 1 )**(3.D0/4.DO) A4=(A l)**(7.D0/4.DO) ARITH=C 1 A3/DSQRT(B) PARAN=Cl *RR*RR*A4/DSQRT(B)
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184 FLNVO=(RR/FVO(RR) )*(ARITH(3 ./2.) *PARAN) RETURN END C************************************************************** C C SUBROUTINE GASDEV C C**************************************************************** SUBROUTINE GASDEV(VR,VT,A,IDUM) IMPLICIT NONE INTEGER IDUM REAL *8 Vl,V2,R,FAC,VR,VT,RAN3,A R=l. DOWHILE((R.GE. l .).OR.(R.LE.A)) V l =2. *RAN3(1DUM)l. V2=2. *RAN3(1DUM)l. R=Vl *Vl+V2*V2 ENDDO FAC=DSQRT(2 *LOG(R)/R) VR=Vl*FAC VT=V2*FAC RETURN
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1 8 5 E N D C* *************** ************************************ C C SUBROUTINE INTEG C C *** ********* *** **** *** ********* *** ************ ********** SUBROUTINE INTEG ( SM ,A ) IMPLICIT NONE REAL *8 X Y YT A,FMAX STEP SM X=O. Y=O. YT=O. SM=l.DO IF ( A.EQ 0 ) GOTO 410 FMAX=SQRT(2 *DLOG ( A ) / A) STEP=FMAX/10000 DOWHILE(X.LE.FMAX) Y= ( X**2.D0)*(2.7l828** ( (X* 2 D0 ) /2 .DO)) *STEP YT=YT+Y X=X+STEP ENDDO SM=SQRT(SQRT(2.D0/3 l4 l59) *YT )
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410 RETURN END 186
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BIOGRAPHICAL SKETCH Born in Portland Oregon on August 18, 1966, the author grew up and graduated from high school in Medford Oregon in 1984 During this time the he was active in s everal community organizations including the Boy Scouts of America wherein he earned the rank of Eagle Scout. Upon his graduation, the author continued his education in the s mall idyllic community of Ashland Oregon, at Southern Oregon State College. While there he served in several roles including Student Senator, president of several student organizations, student manager and Executive Vice President of the Student Body In recognition of his service and scholarship, the author was named in the 1987 edition of Who's Who in American Colleges and Universities. In 1989, the author finis hed the requirements for graduation and so received the degrees Bachelor of Science in Physic s and Bachelor of Science in Computer Science After being accepted for graduate study at the University of Florida, the author moved to Gainesville and embarked on a course of study that would lead to the earning of the Doctor of Philosophy degree. On March 20, 1994, coinciding with the Spring Solstice, the author was married to Kathy Sue Holt in Salt Lake City Utah, before loved ones and friends. The author is a committed Christian who dedicated his life to the service of Jesus Christ in 1981 and is a member of a congregation affiliated with the independent Christian churches. 200
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ja esH. Hunter Jr., Chai t Adjunct Professor of Physics Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Henry. Kandrup Associate Professor of Astronomy Adjunct Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosop y. Jame Ipser Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy Associate Professor of Astronomy This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences, and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1995 Dean, Graduate School

