Citation

## Material Information

Title:
Self-consistent models of barred spiral galaxies
Creator:
Kaufmann, David Eugene, 1964-
Publication Date:
Language:
English
Physical Description:
xviii, 186 leaves : ill. ; 29 cm.

## Subjects

Subjects / Keywords:
Corotation ( jstor )
Density ( jstor )
Galaxies ( jstor )
Galaxy rotation curves ( jstor )
Parametric models ( jstor )
Periodic orbit ( jstor )
Spiral arms ( jstor )
Spiral galaxies ( jstor )
Stochastic models ( jstor )
Velocity ( jstor )
Astronomy thesis Ph.D
Dissertations, Academic -- Astronomy -- UF
Spiral galaxies -- Mathematical models ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

## Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 180-185).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by David Eugene Kaufmann.

## Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Resource Identifier:
030277338 ( ALEPH )
30608391 ( OCLC )

Full Text

SELF-CONSISTENT MODELS OF BARRED SPIRAL GALAXIES

By

DAVID EUGENE KAUFMANN

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

1993

Eric, Elizabeth, and Carol

ACKNOWLEDGMENTS

With this dissertation my education has, in two very different senses, both drawn to a close and started anew. Although there will be no more classes, exams, or theses,

I have just begun to learn.

Many special people have helped to bring me this far.

Dr. George Contopoulos, the de facto advisor of my thesis, not only has taught me essentially all that I know about the fascinating topics of nonlinear dynamics and chaos,

he has also provided an unmatched role model for this young scientist.

know of no other person who strives harder to find truth.

I personally

To this goal I also dedicate

myself.

Drs. James Hunter and Stephen Gottesman helped me greatly during

the course of

my research, especially during the periods of Dr.

Contopoulos' absence.

Dr. Hunter,

who served as chairman for this dissertation, instilled in me the value of balancing purely

computational results with the underlying theory.

value without the other.

He taught me that one is of limited

Dr. Gottesman, who acted as cochairman for the dissertation,

impressed upon me the need for theoretical results to address the observations and kept me from getting absorbed in the purely abstract. After all, there is the real world.

Dr. Neil Sullivan,

the final member of my committee,

and Dr.

James Fry,

member until the Fall of 1992, allowed me the freedom to conduct this research as I

deemed fit.

For that I thank them.

a

iii

I would also like to thank several colleagues whose technical assistance was invaluable. Dr. Nikos Hiotelis generously loaned me his smoothed particle hydrodynamics code and very ably instructed me in the basics of its proper use. Drs. Martin England and Elizabeth Moore provided essential information concerning the neutral hydrogen observations of NGC 1073 and NGC 1398, respectively. Also, Dr. Preben Grosbol helped me understand some of the more subtle details of his original codes. Dr. Robert Leacock, Jeanne Kerrick, Darlene Jeremiah, Debra Hunter, and Ann Elton have my deepest gratitude for leading me around the pitfalls of academic bureaucracy.

Many others have provided me with steadfast friendship and moral support during the course of my graduate study. Dr. Jim Webb and Tom Barnello thankfully showed me that quality basketball does indeed exist outside the state of North Carolina! Dr. Gregory Fitzgibbons kept me laughing instead of crying while I studied for the written qualifying exams. Dirk Terrell and Dan Durda finally persuaded me, to my benefit, to stick my head underwater and learn scuba diving. I enjoyed many exciting games of backgammon with Jer-Chyi Liou and Damo Nair, even though they defy the laws of probability with their dice-rolling skills! Chuck Higgins put up with me for one year as a roommate and kindly slowed down to allow me to keep up during our cycling excursions. Bryan Feigenbaum, who just may be able to beat me (occasionally) in oneon-one basketball, tolerated me as a roommate for three years and made some trying times bearable with his humor. My good friends Jaydeep Mukherjee and Billy Cooke have always been there for me through the good times and the bad. Thanks guys! And to all those whom I have not explicitly mentioned here, please accept my apologies and my sincerest gratitude.

iv

Finally and most importantly, I would like to thank my mother Betty, my father John, my brothers Greg and Eric, and their respective wives Elizabeth and Carol. They have always stood behind me with unwavering support and encouragement. To them I dedicate this dissertation.

V

ACKNOWLEDGMENTS LIST OF TABLES 0@ LIST OF FIGURES ... ABSTRACT ........

* 0 0 0 0 0

* 0 * 0 0 *

* 0 0 0 0 * 0 0

* . 0 * 0 0 * 0

* 0 * 0 0 0

* 0 0 0 0

* 0

* 0

* 0

0 0 0 0 0 0 * 111i

0 0 * 0 0 0 0 0&

* 0 0

* 0 0

viii

*0 . 0 *x

* 0* 0XVii

CHAPTERS

1. INTRODUCTION .......

2. MODELING TECHNIQUES.

* 0 0 * 0 * 0 0 * * 0 * 0 * 0

* . 0 * 0 0 0 * 0 * 0 * 0 * *

The Method of Contopoulos and Grosbol (CG Method)

The CG Method Modified for the Case of Barred Spiral Galaxies . . . 0 . . 31 Gas Response Using Smoothed Particle Hydrodynamics (SPH) . . . . . . . 43

3. PROGRAM GALAXIES

NGC 3992 .:**

Observations Best Model. NGC 1073.:0

Observations Best Model. NGC 1398 .:*

Observations Best Model.

0

* 0 0 0 0 0 *0 0 * 0

* 0 * 0 0 * * * 0 0 * 0 0 0 0

* 0 0 0 0 0 * * 0 0 * * 0 * 0

* . 0 * * * * . . 0 * * * * .

* . . . . . . . 0 0 0 0 0 * *

. . 0 0 0 0 * * 0 0 * * * 0 0

* 0 0 0 * 0

* 0 ~ * * 0

. . . . . .

0 * ~ * * .

. . . . . .

0 0

0 *

* .

* 0

* .

* .

* 0

* 0

* .

* .

* . .

* . 0

* 0 *

* . .

* 0 0

* 0 *

* . 0

* 0

0.50

0 60 0* 0 0 5

* 0

* 0

0 0

0 0 * ~ * 0

. . . . . .

0 57

. 61

*.72

*.72

. 77

*.82

* 0~ .82

* 0* .86

4. VARIATION OF PARAMETERS

Variation of A..

Variation of Ar

Variation of io.. Variation of Es. Variation of Q Variation of A.. Variation of 02.-

0. 0 * 0 * . 0

0 0 * 0 * 0 0 *

0 0 00 * 0 & 0

0 0

* .

* .

* .

. 0 0 0 0 * 0

* . . . . 0 ~

* . 0 0 ~ * *

0 0

. . . . . . . ~ 0 0 0 0 0 ~

. . 0 0 0 0 ~ 0 0 ~ 0 ~ 0 ~

* 0 0 0 * 6 0 0 0* 0 0

& 0

0 0 0

0 0

0 094

0 0 0 0 0*0 094

0 0 0

0 0 00* 0 0 0 97

0 0 0

* 0

0 0 * 0 0

0 * 0 * 0 *0 0 0 *0 *0 *0 0 0 0* *0 0 0 0

0 0 0 * 97

98

0 0 0 0 0 0 0

0 0

.* 0 0 . 0 0. * * * 102

0 0 0 0 0* * * .0.* . * . 103

Variation of rl, r2, K 1, K2 and A4 *.... Variation of the Bar Semiaxes a, b, and c

Variation of the Bar Mass MB

0 0 0 0 0

0 0 0 0. 0. 4. 0 0.0*0. 0 0*0 0. * 0 . 0 . 0 105

. . 0 0 0 ~ 0 0 ~ 0 0 ~ 0 * 0 * 0 0

0 0 ~ * ~ 0 0 ~ ~ ~ ~ . 0 0 ~ 0 ~ ~

108 108

Variation of co and sFo

0 0 0 * 0 . 0 0 0 0 0

0 0 0 0 0 0 0 0 0 *0 * * 0 0

*o 00109

vi

* 0 *

* . 0

* .1

* 11

* 0

0 0 0 0 0 0 0 0 11

0 0

0 0

Variation of co and ar

0 * 0 0 * a 0 0 0 0 0 0 0 *0 0 0* 6S 0 0 S *

Interpretation of Results. . . . . . . . . . . .

5. STOCHASTIC ORBITS

0 * 0 0 0 0 0 0 0

0 * * 0 0 * 0 * 0 * 0 0 6 0 0 0 0 0 0 0 0 0

Surfaces ofSectio n .. .. .. .. .......
Individual Stochastic Orbits. . . . . . . . . .

* . 0 * 0 0 * 0 0
* 0 0 0 0 0 0 0 0

* 0 0 0 0 0 * *
* 0 0 0 0 0 0 0

* . 0 0 0 0 * 0
* 0 0 0 0 0 0 0

S 0 06 0

0 0 0136

Proportions of Ordered and Stochastic Orbits in the Models

0

0 0

0 0 * 0 0 * 0 6 141

00 * a 0* 0 0 * 0 0 0 0 145

7. SUMMARY

0 0 0 * 0 * * 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 a 0 0

Self-Consistent Models of Barred Spirals The Role of Stochastic Orbits. . . . ..

The Gas Response

6 0 0 0 6 0 0 0 0 0 0 0 0

Directions for Future Research......

APPENDIX: MODEL BAR QUANTITIES

0 0 0 0 0 0

0 0 000 0 00 * * 0 * * 0 171

* 0 0 0 0 0 0
* 0 0 0 0 0 0
* 0 0 0 0 0 0

*0 0 0 0 0 0 173
* 00 0 0 1714
* 00 0 0 175

* 0 0 0 0 0 0 0 0 0 0 * 0 0 0 0 0 0 * 17 7

BIBLIOGRAPHY

6 0 0 0 & * 0 0 0 6 0 0 * 0 0 0

BIOGRAPHICAL SKETCH

0 0 0 0 0 * 0 0 0 0

* 0

* 0

* 0 0 0 0 0 0

* 0 0 0 0 0 0

* 00 0 0 180J

* 00 0 0 186

vii

112 113 118 123

0 0 0

00 9 0

171

6. GAS RESPONSE ...............

LIST OF TABLES

2-1 Initial conditions of the periodic orbit and the eight nonperiodic orbits used to
calculate the effect of the velocity dispersion . . . . . . . . . . . . . . . . . . . 24

2-2 The values of the xiand wviof Table 2-1.*. . .. .. .. . .. . . ... . . . . 24

2-3 Adjustable parameters in the axisymmetric rotation curve fitting procedure. . 39

3-1 HI rotation velocity data for NGC 3992.

.0 0* 0 9 0 0 0 0 * * 0 0 * 0 0 0 * 0 0 0*

58

3-2 Selected global and disk parameter values adopted for NGC 3992. The errors
given are simply the formal errors of a least-squares analysis of the observed velocity field and do not imply that these quantities have been determined to

this level of precision.

0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 0 0 0 0 0 0 0 0 * 0 0 0 0 0 0

3-3 Photometrically derived parameter values for NGC 3992. .. . . . . . . . . .

60 62

3-4 Parameter values for the best self-consistent model of NGC 3992.

0 0 0 0 0

3-5 Resonance locations of the best model of NGC 3992 .. . . . . . . . . . . .

3-6 Parameters used to calculate the surface density response for the best models

63 66

of 67

3-7 HI rotation velocity data for NGC 1073.

3-8 Selected global and disk parameter values adopted for NGC 1073. Here again,
the errors listed are simply the formal errors of a least-squares analysis of the

75

velocity field....... . . . . . . . . . . . 0 0 . . . . . . . . . . . . . . . . .

3-9 Photometrically derived parameter values for NGC 1073. . . . . . . ..0. *.*. 76

3-10 Parameter values for the best self-consistent model of NGC 1073 . .. . ... 77 3-11 Resonance locations of the best model of NGC 1073 . .. .. .. .. .. . .. 79

viii

0*060000*000*0000000073

3-12 Parameters used to calculate the surface density response for the best model

3-13 HIlrotation velocity data for NOC 1398 . . ... . .. .. .. ..........

3-14 Selected global and disk parameter values adopted for NGC 1398 ... . . . . 3-15 Photometrically derived parameter values for NOC 1398 ... . . . . . . . .

3-16 Parameter values for the best self-consistent model of NGC 1398 .. . . . . .

3-17 Resonance locations of the best model of NGC 1398 .. . . . . . . . 3-18 Parameters used to calculate the surface density response for the best

0 0 0 0

model
0 # 0 0

5-1 Summary of basic stochastic orbit types and their behaviors . . . . . . . . . . 137

5-2 Estimated breakdown of the mass-weighted orbit populations comprising the
models of Chapter 3 according to type (trapped versus stochastic) and location
(bar, disk, or both). The errors in the cited figures are somewhat uncertain,

but are estimated to be of the order of 5%.

ix

79

84 85 87 87

89

90

0 0 0 0 * 0 0 9 0 * 0 0 * 0 0 0 0 * 0 143

LIST OF FIGURES

2-1 The effect of varying~b on the shape of the model rotation curve given by Eq.
(2-1): (1) b = 1OEd; (2) 5b =55ad; (3) Eb = 26d; (4) cb = Ed- In all cases fb = 1. 13 2-2 The effect of varying fb on the shape of the model rotation curve given by Eq.
(2-1): (1) fb = 0; (2) fb = 0.5; (3) fb = 1; (4) fb = 2. In all cases ~b = 5d14 2-3 Rotation curves of the exponential disk (solid curve) and a point with the

same total mass (dotted curve).

.0 0 0 0 * 0 0 0 0 * 0 * 0 . 0 0 0 * 0 * 0 0 * 0 0 0

33

2-4 Rotation curves of the Plummer sphere (solid curve) and a point with the

same total mass (dotted curve).

35

0 6 0 0 0 0 0 0 * 0 0 0 6 0 0 * 0 0 0 0 . 0 *~ 0 0 0 *

2-5 Three normalized "axi symmetric" bar rotation curves for cases where a =1
and c = 0.1: (1) b = 0.15, (2) b = 0.3, and (3) b = 0.6. The dotted line
represents the Keplerian rotation curve for a point mass equal to MB.

3-1 NGC 3992 (NASA Atlas of Galaxies Useful for Measuring the Cosmological
DIistance Scale 1988) . . . . . . . . . . . . * * * *... ...

3-2 NGC 1073 (NASA Atlas of Galaxies Useful for Measuring the Cosmological
Diiistance Scale 1988) . . . .* . . . . . . . . . . . . . . . . . .

3-3 NGC 1398 (Sandage 1961, The Hubble Atlas of Galaxies).

0 0 0 ~0 0 00 ~0

56

3-4 Comparison of observed and theoretical rotation curves for NGC 3992. The
theoretical curve is derived from our most successful model of NGC 3992.
Also shown are the contributions of the separate components of this model to

the total theoretical rotation curve.

0 0 0 0 6 0 0 0 * & * 0 0 0 0 * 0* 0 * 0 0 0 0

3-5 Characteristics of the orbit families included in the model. Each characteristic
plots x, where a given orbit crosses the minor bar axis b, as a function of
Jacobi constant, as parameterized by rc .. . 00..00.00.. 00.. .. ..

3-6 The 2/1 family of periodic orbits in the model of NGC 3992. The darker
circle represents corotation at5.5Skpc .... . . . . . . . . . . . . . . . . . .

59

63

65

x

37

52

54

3-7 The 4/1 family of periodic orbits in the model of NGC 3992 . . . . . . . . . . 65

3-8 The -2/1 family of periodic orbits in the model of NGC 3992. The darker

curves represent the minima of the bar and spiral potentials.

0 0 0* 0 0 0 0 0

66

3-9 Grayscale image of the unprojected surface density response of the best model

69

3-10 Grayscale image of the surface density response of the best model of NOC
3992 projected to the galaxy's actual orientation ... . . .. .. .. .. . . . . 70

3-11 The response-to-imposed 20 component amplitude ratio R* for the best
model of NGC 3992. The positions of the major outer resonances

(corotation, -4/1, and outer Lindblad) are noted.

0 *0 0 0 ~0 0 0 0 0 0 0 0 ~ 0 0

3-12 The phase difference AO (in radians) between the response and imposed 20
components of the best model of NGC 3992. The positions of the major
outer resonances are noted as in Figure 3-11 . . . . . . . . . . . . . . . . . . . 72

3-13 Comparison of observed and theoretical rotation curves for NGC 1073. . . . 75 3-14 Characteristics of the orbit families included in the best model of NGC 1073. . 78 3-15 Representative orbits of the three main periodic families in the best model of

0 078

3-16 Grayscale image of the unprojected surface density response of the best

3-17 The response-to-imposed 20 component amplitude ratio R* for the best
model of NGC 1073. The positions of the major outer resonances

(corotation, -4/1, and outer Lindblad) are noted.

0 080

0 081

3-18 The phase difference AXO (in radians) between the response and imposed 20
components of the best model of NGC 1073. Again, the positions of the
major outerr esonn nessarantenoted.. . . .. . . . .. ... .. . . . . . .

3-19 Comparison of observed and theoretical rotation curves for NGC 1398. The
contributions of the separate model components are also shown .. . . . . . .

3-20 Characteristics of the orbit families included in the best model of NGC 1398. .

xi

71

82

84 88

of NG C 3992 . ....................................

0 0 6 0 0 0 0 0 * 0 * 9 0 *

3-21 Representative orbits of the three main periodic families in the best model of

3-22 Grayscale image of the unprojected surface density response of the best

3-23 The response-to-imposed 20 component amplitude ratio R* for the best
model of NGC 1398. The positions of the major outer resonances
(corotation, -4/1, and outer Lindblad) are noted................ 92

3-24 The phase difference AO (in radians) between the response and imposed 20
components of the best model of NGC 1398. Again, the positions of the
major outernnesoante, . enot*ed0 0 0 * * .0 *.0 0 0 0 0.* 93

4-1 The ratio R* and phase difference AO in models of NGC 3992 where A=
2000 km 2s-2kpc41 ("best" model, denoted by filled circles), A = 1000
km 2s2kpc4I (open circles), and A = 4000 km 2-2 kpc1 (plusses). . * & . . . 95

4-2 The ratio R* in models of NGC 1073 where the residual amplitude Ar = 0
km 2s2kpc-1 ("best" model, filled circles), 1000 km2s-2kpc-' (open circles), and 2000 km 2s-2kpc-1 (plusses). For comparison, the primary amplitude A=
9000 km2 s'2kpc'..........-.I.... . 96

4-3 The ratio R* and phase difference AO0 in models of NGC 3992 where io
-10' ("best" model, denoted by filled circles), io= -5' (open circles), and io
-~15' (plusses). . .*.0.0.*.0.0.*.0.0.0.4.0.*.0.4.0.0.0 . 0.*.0.0.**0* * 0*4*6* 0 6.*.* 0* 99

4-4 The ratio R* and phase difference AO~ in models of NGC 3992 where ES = 0.2
kpc-I (open circles, A = 1213 knM2s2kpc1), 0.4 kpc'1 (filled circles, "best"
model), and 0.8 kpc-1 (Plusses, A = 5437 km2s2kpc1). . * * . . * * . . * . 100

4-5 Amplitude ratio R* and phase difference AO in models of NGC 3992 where
=p 34.7 km s-lkpc1 (corotation= 4a/3, open circles), Qp= 43.6 km
s 1kpc-1 ("best" model, corotation= a, filled circles), and Qp= 55.7 km
s 1kpc' (corotation = 3a/4, plusses). . .0.*.*.*.0.*.*.*.*.0.*.6.* .0.0.*.0.0.0.*.0.101

4-6 The ratio R* in models of NGC 3992 where A = 0.01 kpc (open circles), 0.1
kpc (filled circles, "best" model), and 1.0 kpc (plusses). . a . * # * . . . * . . 102

4-7 The ratio R* and phase difference AO for models of NGC 3992 in which 02 =
0' ("best" model, filled circles), 150 (open circles), and 30' (plusses). . 0.4. 104

xii

*0
4-8 The ratio R* in models of NOC 3992 where ri = 1.5 kpc ("best"" model, filled

circles), 0.5 kpc

(open circles), and 2.5 kpc (plusses) ... . . . . . . . . . .

4-9 The ratio R* in models of NGC 1073 where rl = 2.95 ("best" model, filled
circles), 2.0 kpc (open circles), and 3.5 kpc (plusses) ... . . . . . . . . . .

4-10 The ratio R* and phase difference AO9 for models of NGC 3992 in which

106

MB = 1.5 x 1'0.Aic,( )("be circles), and 7.5 x 109AL).-

st" model, filled circles), 1.5 x 101A.10

(plusses).

(open

6 0 * 0 0 0 0 0 0 0 * 0 0 0 0 0 0 0 *0

4-11 The ratio R * in models of NGC 3992 where co = 750M0O/pc ( "best"
filled circles), 500A10./pc 2 (open circles), and 1000.A'I/pc2 (plusses).

4-12 The ratio R* in models of NGC 3992 where eo = 0.235 kpc-1 ("best"
filled circles), 0.157 kpc-1 (open circles), and 0.353 kpc-1 (plusses)..

model, 0* il11

model, 16 * 112

4-13 The ratio R* and phase difference A O in models where uro = 100 km s-1,-) r
= -7.0 km s'1kpc-1 ("best" model, filled circles), uo= 30 km S-1,Orr = -1.0
km s-1kpc-I (open circles), and cro = 150 km S-1, 0r= -11.0 km s-1kpc-1

(plusses) . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-1 Surfaces of section in the model of NOC 3992. Top: Jacobi constant equals

-213438 km2s-2

2 2 A
Bottom: Jacobi constant equals -202713 km2 2s-

5-2 More surfaces of section in the model of NGC 3992.

Top:

114 124

Jacobi constant

equals -198084 kmn2 s-2. Bottom: Jacobi constant equals -194202 km2 S-2.. 125

5-3 More surfaces of section in the model of NGC 3992.

Top:

Jacobi constant

equals -191307 kmn2 s -2 . Bottom: Jacobi constant equals -189692 km2 S-2. . 126

5-4 Surfaces of section in the model of NGC 1073.

Top: Jacobi constant equals

-34143 km2 S-2

Bottom:- Jacobi constant equals -32393 km 2 S-2.

0 00 *127

* 0

5-5 More surfaces of section in the m
equals -30871 km2 S-2 Bottom:

iodel of NGC 1073. Top: Jacobi constant Jacobi constant equals -29746 km2s-2*.

5-6 More urfaces of section in the model of NGC 1073. Top: Jacobi constant
equals -29405 km 2 -2* Bottom: Jacobi constant equals -29265 km2 S-2. .

128

0.129

5-7 Surfaces of section in the model of NGC 1398.

Top: Jacobi constant equals

-314204 km 2 S2. Bottom Jacobi constant equals -306600 km 2 S-2*

xiii

105

110

130

5-8 More surfaces of section in the model of NGC 1398. Top: Jacobi constant
equals -299940 km2 S-2. Bottom:- Jacobi constant equals -294526 2m s2 . 131

5-9 More surfaces of section in the model of NGC 1398. Top: Jacobi constant
equals -290724 M2 S-2. Bottom: Jacobi constant equals -288987 km2 -2. . 132

5-10 Schematic drawing showing how ioops in the orbits outside of corotation
generate consequents on the surface of section. A portion of an orbit containing such a loop is shown, together with arrows indicating the
direction of motion along the orbit. The drawn circle represents corotation,
and the directions of the general motion are also indicated with arrows. . . 134

5-11 Stochastic orbit trapped within the bar. The value of its Jacobi constant (EJ =
-191307 k 2 S-2) is slightly less than that of the Lagrange points L, and L2

(Ej = -191188 km2 S-2). The darker circle represents corotation at 5.5 kpc.

5-12 Stochastic orbit confined to the outer disk. The value of its Jacobi constant
(EJ = -194202 km2 S-2) is also less than that of Ll and L2. The bar and

137

spiral potential minima are drawn for reference.

5-13 Stochastic orbit which traverses both the bar and the outer disk. The value of
its Jacobi constant (EJ = -189692 km 2 S2) is greater than that of L, and L2.

but less than that of L4 and L5 (Ej = -187300 km 2 S2).

5-14 Stochastic orbit, whose Jacobi constant (EJ = -189476 km s-1) is
approximately the same as the orbit of Figure 5-13, calculated in a spiral

potential of ten times greater amplitude than that of Figure 5-13.

* 0 0 0 0 140

5-15 Stochastic orbit whose Jacobi constant (EJ = -183000 km2 -2 ) exceeds that

of L4 and L5 and is energetically unconstrained.

6-1 The gas response in the best model of NGC 3992 with the bar/spiral imposed.
Top panel: 1.5 pattern rotations. Bottom panel:- 2.4 pattern rotations. . . . . 149 6-2 The gas response in the best model of NGC 3992 with the bar/spiral imposed

(continued). Top panel: 3.3 pattern

6-3 The gas response in the best model
(continued). Top panel:, 8.1 pattern

rotations. Bottom panel: 5.6 pattern

of NGC 3992 with the bar/spiral imposed rotations. Bottom panel: 10.4 pattern

xiv

0 0 0 0 0 0 0 & 0 * 0 0 0 * 0 138

0 * 0 0 0 0 0 0 6 0 139

0 0 0 0 6 0 * 0 0 * 0 * 0 * 0 141

rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6-4 The gas response in

the best model of NGC 3992 with only the Ferrers bar

present. Top panel:- 1.5 pattern rotations. Bottom panel: 2.5 pattern

6-5 The gas response in
present (continued).
pattern rotations...

6-6 The gas response in
present (continued).
pattern rotations...

6-7 The gas response in

the best model of NGC 3992 with only the Ferrers bar Top panel: 3.5 pattern rotations. Bottom panel: 5.3
0 0 * * * 0 0 * 0 0 0 0 0 I * * * 0 6 0 0 4115 3

the best model of NGC 3992 with only the Ferrers bar Top panel: 7.6 pattern rotations. Bottom panel: 10.6

the best model of NGC 1073 with the bar/spiral imposed.

Top panel: 1. 1 pattern rotations. Bottom panel: 2.8 pattern rotations. . . - . 155 6-8 The gas response in the best model of NGC 1073 with the bar/spiral imposed
(continued). Top panel: 4.5 pattern rotations. Bottom panel: 5.4 pattern

6-9 The gas response in the best model of NGC 1073 with the bar/spiral imposed
(continued). Top panel: 6.8 pattern rotations. Bottom panel: 8.3 pattern

6-10 The gas response in the best model of NOC 1073 with only the Ferrers bar
present. Top panel: 1.1 pattern rotations. Bottom panel: 2.6 pattern

rotations.

158

6-11 The gas response in the best model of NGC 1073 with only the Ferrers bar
present (continued). Top panel: 4.5 pattern rotations. Bottom panel:- 5.5

6-12 The gas response in the best model of NGC 1073 with only the Ferrers bar
present (continued). Top panel: 7.3 pattern rotations. Bottom panel: 8.3

160

6-13 The gas response in the best model of NGC 1398 with the bar/spiral imposed.
Top panel: 1.4 pattern rotations. Bottom panel:- 3.3 pattern rotations. . . . 161

6-14 The gas response in the best model of NGC 1398 with the bar/spiral imposed
(continued). Top panel: 4.1 pattern rotations. Bottom panel: 5.0 pattern

162

rotations.

xv

pattern rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 * 0 0 0 0 0 0 0 4 0 0 0 * 0 * 0 0 0 0 0

6-15 The gas response in the best model of NGC 1398 with the bar/spiral imposed
(continued). Top panel: 7.2 pattern rotations. Bottom panel: 11. 1 pattern
rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6-16 The gas response in the best model of NGC 1398 with only the Ferrers bar
present. Top panel: 1.9 pattern rotations. Bottom panel: 3.1 pattern

6-17 The gas response in the best model of NGC 1398 with only the Ferrers bar
present (continued). Top panel:, 4.2 pattern rotations. Bottom panel: 5.3

6-18 The gas response in the best model of NGC 1398 with only the Ferrers bar
present (continued). Top panel: 7.5 pattern rotations. Bottom panel: 11. 1

xvi

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

SELF-CONSISTENT MODELS OF BARRED SPIRAL GALAXIES By

David Eugene Kaufmann

August, 1993

Chairman: James H. Hunter, Jr.
Cochairman: Stephen T. Gottesman
Major Department: Astronomy

Self-consistent models of barred spiral galaxies based on the observed properties of NGC 3992, NGC 1073, and NGC 1398 are constructed and analyzed. The method of model construction is a slight modification of the technique developed by Contopoulos and Grosbol for the case of unbarred spirals. The main factors which influence selfconsistency are the amplitude, pitch angle, scale length and z-thickness of the spirals, the mass of the bar, the angular velocity of the bar/spiral pattern, the central surface density and scale length of the disk, and the central value and slope of the velocity dispersion.

Stochastic orbits whose Jacobi constants lie between the values at the Lagrange points L, and L4 are found to play a significant role in supporting self-consistent spiral structure, especially in the regions just beyond the ends of the bar. Stochastic orbits whose Jacobi constants lie below this interval tend to fill more or less uniformly either rings in the outer disk or ovals in the bar region, depending on the regions to which they are confined. Stochastic orbits whose Jacobi constants lie above that of L4 also tend not to support any imposed structure. The model bars are predominantly comprised of elongated orbits trapped around the x, family and terminate close to corotation.

xvii

The response of gas to the forces of the most successful models is calculated using a two-dimensional smoothed particle hydrodynamics code. The results confirm that a bar alone is not sufficient to drive a strong spiral response in the gas of the outer disk. An underlying spiral structure in the more massive stellar component appears to be required. If stellar spirals are present, strong gas spirals may persist for long times.

xviii

CHAPTER

1

INTRODUCTION

Spiral structure in galaxies was first observed by

William Parsons, Third Earl of

Rosse, in several nebulae that had been catalogued earlier by William Herschel.

Because

this discovery, made in the mid-nineteenth century, preceded the establishment of these nebulae as separate galaxies external to the Milky Way, they became known simply as spiral nebulae. In addition, their spiral shapes seemed to suggest that the systems rotate

about an axis perpendicular to the plane of the spiral.

Confirmation of this hypothesis

was provided in 1914 by Vesto Slipher, whose spectroscopic observations of a number of these spiral nebulae revealed the Doppler shifts produced by rotation. The nature of

the spiral nebulae, one of the

subjects of the so-called " great debate" between Huber

D. Curtis and Harlow Shapley at the National Academy of Sciences in April 1920, was

firmly

established in the early 1920s by

Edwin Hubble.

In 1923,

Hubble,

using

the 100-inch telescope at

Mount Wilson, was able to

resolve the outer parts of the spiral nebulae M3 1 and M33 into multitudes of apparent

point sourc individual

.es.

Although at first unable to determine

stars, he soon found that some were inde

whether these point sources were ,ed Cepheid variables. Using the

Cepheid period- luminosity relation, originally discovered in

1912 by Henrietta Leavitt,

Hubble was able to show conclusively that M3 1 and M33 are at very large distances,

and therefore galaxies external to the Milky

Way.

I

2

In 1926, Bertil Lindblad, working within Shapley's concept of the Milky Way (i.e. that the Sun's position is generally in the galactic plane but relatively far from

the galactic center), developed a mathematical model for its rotation.

This model

depicted the galaxy as consisting of a number of subsystems, each rotating with its own characteristic angular velocity and degree of flattening about a common axis. Jan Oort, in 1927 and 1928, demonstrated the basic correctness of Lindblad's model and showed

that the galactic disk is in a state of differential rotation.

That is, stars closer to the

galactic center orbit around it with higher angular velocities than those farther out in the disk. At about the same time Lindblad began to try to understand the nature of spiral structure, a problem he worked on until his death in 1965.

Perhaps the simplest interpretation of galactic spiral structure is that it represents a pattern of material arms that rotate as a rigid body about the galactic center. Differential rotation, however, presents this interpretation with a serious difficulty called the winding dilemma. Consider, for example, a galactic spiral arm of uniform pitch angle i0 = -30' at time t = 0. Also, suppose that the galaxy's angular rotation rate Q depends on radius r (i.e. the galactic disk rotates differentially). The azimuth of the arm at radius r and time t is given by

o(r,t0 = 0(7-,0) + Q(r-)t. The pitch angle i at radius r and time t is defined by

(1-1)

ta~n I=

7 r )

=___I&QI

(1-2)

For a typical galaxy with a flat rotation curve, Q(r)r 200 km s-I

r 10 kpc, and

t 1010 yr. In this time the pitch angle i would have decreased from the original -30'

3

to a value of -O.' 28. Therefore, any material arms would have wound up and rendered the spiral structure unrecognizable on timescales comparable to the age of the galaxy.

Binney and Tremaine (1987) summarize four possible solutions to the winding

dilemma.

Firstly, spiral structure may represent a statistical equilibrium in which

the age of any given arm is quite young.

The idea here is that clumpy features are

continuously produced in the galactic disk and are sheared off into spiral segments by differential rotation. A spiral arm, then, would simply represent an aggregation of

these shorter spiral segments.

While this chaotic spiral arm hypothesis is believed to

be applicable to flocculent galaxies (e.g. Toomre 1989), it has difficulty accounting for the striking global coherence exhibited by the spiral patterns of grand design galaxies. Secondly, spiral structure may be the temporary result of a recent tidal interaction with a companion galaxy. In this case the companion excites a transient global spiral density wave in the primary galaxy. The nature of density waves is discussed in more detail below. Models of this type are able to reproduce many features observed in grand design galaxies, such as dust lane location and strength as well as the main properties of the

neutral hydrogen distribution and velocity field.

According to Binney and Tremaine

(1987, p. 394), however, "they cannot account for all spirals because encounters with massive companion galaxies in favorable orbits are not common enough." Thirdly, spiral structure may represent a detonation wave of star formation, driven by supernovae

explosions or expanding HIL regions, that propagates around the disk.

The wave is

sheared into a trailing spiral by differential rotation and ultimately settles down with a fixed shape and pattern speed. This self-propagating star formation hypothesis, though, requires finely tuned star formation rates and does not adequately address the problems

4

of the broad arms of the older disk components, the very regular patterns seen in some grand design galaxies, and the streaming observed along spiral arms. Finally, spiral structure may represent underlying waves in the density and gravitational potential of the disk itself. The orbiting stars and gas adjust their motions such that they tend to linger near the minima of the potential; moreover, the gas in these regions is compressed, thereby inducing rapid star formation that generates the bright young stars and HI

regions which delineate the visible spiral pattern.

While the material that constitutes

the density waves is constantly changing, the waves themselves are neutrally stable and represent a quasi- stationary spiral pattern that rotates as a rigid body with angular velocity QP Bertil Lindblad (1961, 1963) originally proposed such a scenario, but his approach, which emphasized the role of individual stellar orbits, was not well-suited

for quantitative global analysis.

C. C. Lin and Frank Shu (1964) first provided the

mathematical formalism of the spiral density-wave theory.

Therefore, the idea that

spiral structure represents a quasi- stationary density-wave has become known as the

Lin-Shu hypothesis.

Since the present thesis concerns the structure of barred spiral

galaxies, we must also consider those phenomena associated with the presence of a bar.

Bars are found in a large fraction of disk galaxies. In fact, depending upon the reference catalog cited, between 25% and 35% of all disk galaxies are classified as

strongly barred (Sandage and Tammann 1981; de Vaucouleurs et al.

1973).

1976; Nilson

classified as possessing weak bars or oval distortions.

The class of barred galaxies

is very heterogeneous. In it we find galaxies that span the entire range of Hubble types. In addition, "other properties, such as the size of the bar relative to the host galaxy, the

5

degree of overall symmetry, the existence of rings and numbers (and position relative to the bar) of spiral arms in the outer disc, the gas and dust content, etc., vary considerably from galaxy to galaxy" (Seliwood and Wilkinson 1992, p. 1).

Observations of bars generally concern either the light distribution or the velocity field. Analyses of the light distributions reveal that bar radii (i.e. semimajor axes) are always less than the disk radii (defined, for example, by the radius, R25, at which a galaxy's surface brightness falls to 25 mag arcsec-2) of the host galaxies. Also, bar radii are typically shorter relative to disk radii in late-type galaxies than in early-type galaxies. In the rather diverse sample of barred spirals considered by Elmegreen and Elmegreen (1985), bars provide anywhere from -20% to -40% of the total luminosity within the radius of the bar, and from a few to -20% of the total luminosity within R25- Moreover, bars in early-type galaxies are generally stronger,) more rectangular,

and possess flatter major axis luminosity profiles than bars in late-type galaxies.

The

latter tend to be more elliptical in shape and possess light profiles that are centrally peaked and fall off exponentially along the bar major axis.

Observations of the velocity fields of barred galaxies indicate that there exist

significant noncircular streaming motions along the bar major axis (e.g.

1983; Bettoni et al. 1988; Jarvis et al. 1988).

Kormendy

This result is consistent with the idea,

discussed in more detail below, that bar structure is dominated by orbits elongated along the bar. Comparison of bar radii with the azimuthally averaged rotation curves of their host galaxies indicates that early-type galaxies tend to have bars that extend beyond the rising parts of the rotation curves, while bars in late-type systems tend to end at or before the rotation curves peak or level out (Elmegreen and Elmegreen 1985). Typical

6

values of the central stellar velocity dispersion are from 150 to 200 km s-I, with radial profiles of the velocity dispersion varying from flat to sharply falling (Sellwood and Wilkinson 1992). Finally, estimates of the angular rotation rates of bars can be made by matching the observed gas flow patterns with hydrodynamical models in which the bar tumble rate is varied (e.g. Athanassoula 1992).

Although much work remains to be done concerning the structure and evolution of barred galaxies, our understanding of the origin of bars seems to be quite advanced. Some of the earliest N-body simulations of collisionless, rotationally supported stellar disks dramatically exemplified the bar instability (e.g. Miller and Prendergast 1968; Hockney and Hohl 1969; Hohi 1971). Shu (1970) and Kalnajs (1971) formulated the problem of the stability of axisymmi-etric stellar disks in terms of normal modes. The complete solution, however, is rather lengthy, and the shapes and eigenfrequencies of the unstable modes have been calculated only in a small number of cases; nevertheless, N-body simulations of a few of these cases have yielded dominant unstable modes

in close agreement with the theory (e.g. Sellwood and Athanassoula 1986).

Toomre

(198 1) proposed that the bar instability is driven by the positive feedback of swingamplified waves. The idea here is that any initial leading disturbance will propagate out to corotation, where it will unwind and become a trailing disturbance. As it unwinds, it is swing-amplified by the coordination of the local epicyclic frequency with the

rate of unwinding of the leading wave.

The amplified trailing wave then propagates

inward, and if it is able to reach the center, it will be reflected as an outgoing leading

wave, thereby starting the amplification/reflection process over again.

Wave action is

conserved at corotation by the generation of a trailing wave which propagates outward

7

toward the outer Lindblad resonance (OLR). Frequencies which cause the phases of the waves to match are associated with standing waves. The interference of equal-amplitude swing-amplified leading and trailing standing waves leads to bar formation.

There are several ways to control the bar instability. One, noted by Hohl (1971),

is simply to increase the velocity dispersion.

This tends not only to stabilize a

stellar disk against local axisymmetric instabilities by raising the value of Toomre' s Q parameter,' but also to inhibit the bar instability by reducing the effectiveness of the swing amplification mechanism. Another way is to immerse the disk in a massive halo. This idea stems from the work of Ostriker and Peebles (1973), who found empirically that their N-body models were stable against bar formation if the ratio of the rotational kinetic energy of the disk to the total absolute potential energy was less than a critical value of 0.14 0.02. A third way, suggested by Toomre (1981), is to interrupt the feedback loop. This will occur if the ingoing waves are prevented from reaching the

center.

The existence of an inner Lindblad resonance (ILR), which damps ingoing

wavs iawae-article interactions, provides such a mechanism.

Toomre proposed

that strong spiral perturbations generated in an unstable galaxy shift material inward and raise the value of Q - until an ILR appears and shuts off the feedback loop.

Much work has been done concerning the structure and dynamics of bars.

A

major portion of it deals with the periodic orbit families that exist in model bar potentials. Contopoulos and Grosbol (1989) give a comprehensive review of this topic. Sparke and Sellwood (1987) and Pfenniger and Friedli (1991) have examined the orbital 1. Toomre (1964) found that the condition for stability of a differentially rotating stellar disk to local axisymmetric instabilities is Q = '1 ~ > 1, where (7,. is the local radial velocity dispersion, K~ is the local epicyclic frequency, G is the Newtonian gravitation constant, and -57 is the local surface density.

8

structures of numerically generated two-dimensional and three-dimensional N-body bars, respectively. While the work of Pfeniniger and Friedli presents some exciting new results concerning the behavior of orbits and the structure of bars perpendicular to the primary disk plane, we restrict our attention in the present thesis to the two-dimensional case. The main results of all this work are (1) the overwhelming majority of particles that make up the bar are on orbits trapped around the "central" or, in the notation of Contopoulos

(1975), xi

family (i.e.

the family that reduces to circles in the axisymmetric case) and

(2) bars end at, or slightly inside, corotation.

In their study Sparke and Seliwood found

a significant population of particles that had sufficient energy to be able to move freely between the bar and outer disk (i.e. their Jacobi constants exceeded the value at the

Lagrange points L1 and L2)-

they termed "hot") in

Pfenniger and Friedli also noted such a population (which

their study.

The existence of a bar in a galaxy

provides at least three

possible

mechanisms for the generation of spiral structure (cf.

Elmegreen and Elmegreen 1985):

a rotating,

growing

bar may

excite transient stellar spiral

arms (a phenomenon

observed

by James and

Sellwood

[1978]

and Sellwood

[1981]

in their numerical

simulations), (2) a rotating, static bar may excite spiral waves in the gas of the outer disk (a phenomenon observed, for exam-ple, in the hydrodynamical model of Sanders and Huntley [1976]), and (3) a rotating, static bar may drive stellar spirals in a strongly

self- gravitating disk (Julian and Toomnre

1966; Feldman and Lin

1973;

Lin and Lau

Goldreich and Tremaine

1978).

The blue and near-infrared

provided

by Elmegreen and

surface photometry

Elmegreen

of fifteen

barred

(1985) allows us to begin

spiral galaxies to discriminate

(1)

1975;

9

between these structure- generating mechanisms.

They find that, in general, the spiral

arms of their sample galaxies contain star formation regions superposed on density

enhancements in the underlying stellar disk.

Also, they find that the amplitudes of

the spiral arms, as measured by the near-infrared brightness difference between the arm and interarm regions, decreases outward for early-type galaxies (typically with flat bars, as mentioned before) and increases outward for late-type galaxies (which generally have exponential bars); moreover, they find that galaxies with strong bars

also tend to have strong spirals.

Regarding spiral arm morphology, they note that

all of their sample galaxies that have irregular spiral arms also have short bars Of exponential or indeterminate type. On the other hand, all galaxies with flat bars have

long, continuous arms or a smooth ring.

The Elmegreens suggest that this apparent

bimodal distribution of barred spiral types indicates fundamentally different origins and natures. According to this suggestion, early-type barred spirals have bars that extend close to their corotation radii and continuously excite spiral structure as they slowly

grow.

Late-types, on the other hand, have rather weak bars that extend only to the

ILR and excite correspondingly weak, often multiple, asymmetric and patchy spiral

arms.

These interpretations have received some recent support from the numerical

simulations of Combes and Elmegreen (1992).

Given the numerous possibilities concerning the nature of the spiral arms and the positions of the major resonances in barred spiral galaxies, it is of interest to ascertain

whether or not self-consistent models of these galaxies can be constructed.

Since

realistic models of galaxies are, in general, non-integrable, we must resort to numerical techniques in order to produce self-consistent models. The first effort of this type was

10

made by Martin Schwarzschild (1979), who succeeded in constructing a self-consistent model of a nonrotating, triaxial stellar ellipsoid via the method of linear programming. His approach consisted basically of computing a library of orbits in the potential derived from an assumed density distribution. The time-averaged density in each cell of a grid that spanned the volume of the model was calculated for each orbit. The technique of linear programming was then employed to derive the nonnegative population weights required in order for the ensemble of orbits to give back the imposed model density. Pfenniger (1984b) has used a nonnegative least squares technique to find the population weights for a self-consistent model of a rapidly rotating, two-dimensional Ferrers bar.

The most recent approach to the problem of self-consistent galactic models has been developed by Contopoulos and Grosbol (1986, 1988). Patsis (1990) and Patsis et al. (1991) have successfully applied this method to a sample of thirteen unbarred spiral

galaxies.

We extend that work in this thesis by applying the method of Contopoulos

and Grosbol to a sample of three barred spirals: NGC 1398, NOC 3992, and NOC 1073. In Chapter 2 we describe in detail the method of Contopoulos and Grosbol, our modifications of this method in order to apply it to the case of barred spirals, and the smoothed particle hydrodynamics (SPH) method used to calculate the gas response. In Chapter 3 we present the relevant observations together with our most successful models of the program galaxies. In Chapter 4 we present the results of varying the parameters of the most successful models. In Chapter 5 we detail the role of stochastic orbits in these models, and we give the results of the gas dynamical calculations in Chapter 6. Finally, we summarize the main results in Chapter 7.

CHAPTER

2

MODELING TECHNIQUES

The Method of Contopoulos and Grosbol (CG Method)

Before discussing the method of Contopoulos and Grosbol as applied to barred

spiral galaxies,

it is useful to

describe

the technique

as it was

originally

applied to

unbarred spiral galaxies (Contopoulos and Grosbol

1986,)

1988; Patsis

19901; Patsis et

al. 199 1).

The starting point of any modeling attempt is the observational data. Here the

relevant data are the axisymmetric rotation curve and the surface photometry, preferably in the red or near infrared. The former gives a rough estimate of the gravitational mass distribution in the galaxy, at least insofar as the dynamics of the disk are concerned. The latter gives important information not only about the axisymmetric distribution of

the stellar component of the disk, but also about the shape

and/or spiral perturbation which may

and strength of any bar

be present.

One kinematic result from observations of spiral galaxies is that the overwhelming majority of them exhibit rotation curves which, after steep initial rises, are flat or slowly

rising as far out as they can be determined (e.g. Rubin et al.

1978; Bosma 1981). Any

reasonable model rotation curve must exhibit

this behavior as well.

Secondly,

observed that some spiral galaxies have more prominent nuclear bulges than others.

In

fact, this feature has been used as a classification criterion for these galaxies (Sandage

1961).

Hence it is desirable to have a model rotation curve which can represent the

bulge and disk contributions separately.

Lastly, the underlying axisymmetric potential,

I1I

it is

12

Vo(r),

derived from the rotation curve should be expressible

using

standard analytic

functions.

The two model rotation curves employed by Contopoulos and Grosbol both

have these three features.

The one that has been used most extensively to date is

v(r) :--'miax v'fblbrCXI) (-Icb?-') + [ exp ,Fr]

where cb1

and cd1

are the scale lengths for the bulge and disk components, respec-

tively.

The quantity vmax,. is the asymptotic value of the circular rotation velocity

large r, and

the bulge fraction fb gives the importance of the bulge relative

to the disk.

Figures 2-1 and 2-2 show the effects of varying Eb and fb on the shape of the rotation

curve.

given

The axisymmetric

potential

Vo(r),

derived from the relation

VI /o~(r)
d'r

V2 (r)
r

by

/1o(r) = Viiax(tl) exp(\$tb?') - [Inr 1er))

where E, (x) is the first exponential integral.

In this case the angular velocity Q(r)

goes to infinity for r = 0, thereby giving exactly one inner Lindblad resonance (ILR). For this reason a similar, but slightly m-odified, rotation curve was developed which allows either zero or two ILRs. The circular rotation velocity for this model is given by

V(r*) = Vnax V.I-b rE,'(-cbrl) + [-(1 + IEdrxp (-Edr)]

(2-3)

while the corresponding axisymmetric potential is

71~r =-t'y,,,{fb(lI+ ":b 0 Cxl)(-Qbr_) - [lnr + El(c~lr) + exp (-Ed"-)]}

(2-4)

(2-1)

for

is

(2-2)

v(r)
r

13

I .8

.6

E
~ImI

.4

.2

0

0

.5

I

1.5

2

2.5

3

3.5

4

red

Figure 2-1: The effect of varying Eb on the shape of the model rotation curve given by Eq.

(2-1): (1) Eb = 10cd; (2) Eb = 55d; (3) c.b = 2Ed; (4) 5b = Ed. In all cases fb = 1.

The first step in applying the CG method to a real spiral galaxy, then, is to fit one of these model rotation curves to the observed rotation curve. Since the observations heretofore have not been detailed enough to allow one to determine which model curve is more appropriate for a given spiral galaxy, the first model has been preferentially

applied due to its simplicity.

The fitting can be done in any of several ways. Patsis

(1990) has used a least squares method.

Once the axisymmetric rotation curve v(r) is determined and the corresponding potential VO(r) derived, the next step is to utilize the available surface photometry in order to find the parameters associated with the spiral perturbation. It is preferable, in the present context, to use red or near infrared data because these wavelength ranges best isolate the gravitationally dominant stellar components of galaxies. Examples of

E

14

the types of parameters that can be derived from the analysis of surface photometry can be found in Grosbol (1985), Elmegreen and Elmegreen (1985), Kennicutt (1981),

and Ohta et al.

(1990).

~Wp11-T'-T-TVT7VVT74fV [7 1 WV FpT FTTT

H

I
I I

-111F11lIIYT F

0

.5

1

1.5

2

2.5

3

3.5

4

re4d

Figure 2-2: The effect of varying fb on the shape of the model rotation curve given by Eq.
(2-1): (1) fA = 0; (2) fb = 0.5; (3) fb = 1; (4) fb = 2. In all cases Eb = 55d.

A cursory examination of the images of spiral galaxies indicates the general form that the spiral perturbation must take. The spiral perturbation must increase from zero outward from the center, reach a maximum somewhere in the disk, and then die off

to zero again at large distances. It must also reproduce the shape of the arms.

The

model perturbation originally used by Contopoulos and Grosbol (1986) to investigate nonlinear dynamical effects at the 4/1 resonance has these properties. It is a two-armed

1

.8

.6

.4

.2

0

I I

-------------- -

15

logarithmic

spiral perturbation given

by

'Ir~ l - r) Cos ( t~ - o - 0

where i0 is the pitch angle of the spirl, 1& is a measure of the amplitude, Also., the pe

,.is the spiral scale length, and A, a constant, trturbation is assumed to be time -independent

and rotating rigidly with a constant angular velocity QP. Hence, implicit in this analysis is the assumption of quasi- stationary spiral structure (QSSS), a hypothesis first put forth

explicitly by Lin

and Shu

(1964).

The final form of the spiral perturbation that Contopoulos and Grosbol (1988) used,

and later Patsis

(1990) and Patsis et al.

(1991), was altered by their results of 1986.

They found that nonlinear effects near the 4/1 resonance in galaxies with strong spiral perturbations limit the extent of any possible self-consistent spiral structure, specifically to within the 4/1 resonance. Beyond this resonance the orbits change their orientation

abruptly

and the density response maxima do not lie along the imposed spiral.

necessitated a change in the form of V, (r,0) to allow for a cutoff of arbitrary abruptness at a specified radius. Also, the formula was generalized to allow for 40 and 60 components

in addition to the 20 component. be written most generally as

The final form of the imposed spiral perturbation can

Vi(r,0) =

1n=24,6

(_1, lII r 0> A.~x()r X)(_Fm,?1).7Cos t-an '10+11. - 771,

where

and A,

I
- -,.,~ -~(1 I-tan h [ sn (2n

Arm., K2m, and r2,, are all constlants.

- 7-)],) -+A rm}i

Am, mneasures the maximum

(2-7)

amplitude

achieved by the mO component, A,,, gives any residual amplitude of the mO component

(2-5)

This

{(A
Am('t) = in

(2-6)

16

that extends beyond its cutoff radius r2,r, and K2.n defines the steepness of the cutoff. By analogy with equation' (5), s is the scale length and iomthe pitch angle of the mO component. Also, differences ini the phases of the various components are allowed with the corresponding values of 0,,,~.

Analysis of the surface photometry can yield (at least rough) estimates of most or all of the parameters needed to specify the spiral perturbation. It is not necessary at this stage to find the specific values of the parameters which yield the most self-

consistent model.

Variations of and corrections to the parameters are made after the

results of subsequent modeling attempts are known. In fact, this is the essence of the

CG method.

The observations do, however, place limits on the ranges of acceptable

parameter values, as well as provide good initial guesses.

After the parameters of the spiral perturbation are determined, at least roughly, from the surface photometry, the next step is to calculate the periodic orbits. For this the equations of motion are needed. Since these models are restricted to two dimensions (i.e. the disk plane), the equations of motion can be derived from the two-dimensional Hamiltonian

+ '2 / - /QJo + I.-r + Vjr )=

(2-8)

where i is the radial component of the velocity, JO the angular momentum in the inertial frame of reference, Q,, the angular velocity of the reference coordinate system (i.e. the frame corotating with the spiral pattern) with respect to the inertial frame, and h the numerical value of H. Also, VO(r) and V, (r,O) are the axisymmetric and perturbation

potentials, respectively.

From Eq.

(2-8) the necessary equations of motion can be

17

derived.

They are, in polar coordinates in the rotating frame,

(10
(11,

_ Jo
-Q
"-I

(ii
dt

di- ail dt Or

These equations can be

solved numerically by any of the standard integration techniques.

The one that has been used by Contopoulos and Grosbol (1986, 1988) is a fourth-order Runge-Kutta method with a variable stepsize, which allows the relative error in each

variable to be maintained below a preset level (typically one part in

controls part in

10~)

The error

The periodic orbits of particular interest here are the ones comprising the "central,"

or in the notation of Contopoulos (1975), xi

the purely axisymmetric case.

family, which reduce to circular orbits in

The introduction of a bar or spiral perturbation breaks

this central family into an infinity of fam-iies by gaps at all even resonances between the

epicyclic frequency Kc(?), defined by t;"(r) =

+ 3(1" (0 4Q ) r) + 2rQ(r) d,

and the orbital angular frequency measured in the frame corotating with the perturbation

Q (r) - Q

(Contopoulos 1983).

Here Q(r) = That is, where

Q(r) - QJ)--

a = 1,2,....

(2-10)

Also, regions of instability are produced in the central family near the odd resonances, where

Q(r) - ~i~1) - 2u. -

12 = 1,2,

(2-11)

From these unstable regions bifurcate the odd resonant families.

dJo
(It

DII

(2-9)

allow for energy conservation along the integrated orbit to the order of one

for a typical orbit integration.

109

18

Since the available dynamnical evidence (Contopoulos and Grosbol 1986, 1988; Patsis 1990; Patsis et al. 1991) indicates that the spiral structure of normal spiral galaxies extends from the ILR (2/1 resonance) to either the 4/1 resonance (for strong spirals) or the corotation resonance (for weak spirals), one can get a reasonable estimate of the pattern speed Qi4. This parameter is notoriously difficult to determine observationally, and therefore must be left adjustable in the fitting procedure. Nevertheless, observations of the extent of the spiral structure, together with the axisymmetric rotation curve, can produce reasonable initial values.

The significant branches of the central family, then, that exist in the fully perturbed model between the inner Lindblad resonance and corotation are found. In this region there is a one-to-one correspondence between the radius r, of the circular periodic orbit of the axisymmetric case and its value of the Hamiltonian h. This allows for a rather simple algorithm to find the periodic families of the fully perturbed case. For example, consider the branch of the central family which exists between the ILR and the 4/1

resonance.

The radius of the periodic orbit halfway between these resonances in the

axisymmetric case will have the value r0 = =r, (7rIjR. + r4/1), where rILR and r41 are the positions of the inner Lindblad and 4/1 resonances, respectively. The orbit will cross the 0 = 0 axis with a radial velocity ?o = 0. Now let the "amplitude" of the perturbation potential be increased from zero by some small fraction of its maximum value. Also, let the value of the Hamiltonian for the new orbit remain h = h(r,). If this new orbit is started at radius r0 with zero radial velocity, however, it will not, after being integrated for one revolution, cross the 0 = 0 axis at radius r0 with zero radial velocity. That is, it will not be periodic due to the imposed perturbation. But since the

19

applied perturbation is small, the orbit will be close to a periodic orbit, thus the actual periodic orbit can be readily determined using a two-dimensional Newton method.

Consider an arbitrary orbit of fixed "energy" h, close to a periodic orbit, starting with initial conditions (ro, ?0) along the 0 = 0 axis. After one revolution it will cross the 0 = 0 axis again but with different radius and radial velocity (rl, ?1). Now vary separately r0 and ?o by A, and A?, respectively, and integrate these new orbits for one revolution each, producing the following mappings of initial to final conditions: (ro +

Ar, ?o) --+ (r2, ?2) and (ro, io + A?) -* (r3, 13).

These quantities are now

used to

compute the partial derivatives of the final radius and radial velocity, rf and respect to the initial radius and radial velocity, ri and ?j. Specifically,

&Vf r,)- rl
Or, A

&r'f
C)~

r3 - ri

(2-12)

r2 ~1
Ar

Cwt A3

These numerically determined partial derivatives are used to find the first-order corrections to ro and O necessary to adjust ro and O more closely to the initial conditions of the periodic orbit. The corrections, 6r and 5i, are determined by solving the following equations:
rj + 67- +&6 a - -0ro++6

(2-13)

i+ "r+ a 5 + 0

These corrections are added to ro and ?o,

ro --*ro+ 6r

(2-14)

0o + S ,

rf, with

20

thereby

improving the estimates of the initial conditions of the periodic

procedure

level.

is then repeated until

There

16rI

and I 6 I fall

below

is usually no problem with convergence

some predefined

of this

method if the

tolerance starting

initial conditions are not too "far" from the periodic orbit.

In this manner it is possible to "step up' its full value in small increments, finding acc

'' the perturbation amplitude from zero to 6urately (typically to one part in 1010) the

periodic orbit at each step.

After the

first step,

better

starting guesses at the initial

conditions

of the periodic

orbit of the next step

are provided by an extrapolation

procedure.

This whole process

is done for

only

one orbit between

any given

of even resonances inside corotation.

The orbit "energy" h,

parameterized by

chosen to be that of the circular orbit (in the axisymmetric case) halfway between the positions of the resonances in question, as in the example of the ILR and 4/1 resonance

above.

Once this periodic orbit is found in the fully perturbed case, the whole family

of orbits between the resonances is calculated by varying the energy r, both positively

and negatively.

Again, after the first step each way, an extrapolation procedure is used

to provide better first guesses at the initial conditions of the periodic orbit.

In models of unbarred spiral galaxies, the resonance ranges probed for branches of

the x, family are usually limited to:

2/1 (ILR) to 4/1, 4/1 to 6/1 , and 611 to 8/1.

This is

because in a realistic model of a spiral galaxy, the positions of the resonances bunch up near corotation and most of the galaxy inside corotation is also inside the 8/1 resonance.

For example, in the standard model for NGC Grosbol (1986), corotation is roughly at 23 kpc.

5247 considered by

Contopoulos

and

In comparison, the ILR is around 1.5

kpc, the 4/1 resonance around 12 kpc, the 6/1 resonance around 16 kpc, and the 8/1

orbit.

This

pair

is

21

resonance around 17.5 kpc. And, according to Contopoulos and Grosbol (1986, p.155), ".the congestion of infinite resonances n/i near corotation produces a large degree of stochasticity there." Still, in cases where significant stable periodic families exist beyond the 8/1 and/or 10/1 resonances, orbits in these regions are generally included.

Now that the families of periodic orbits to be included have been determined, the next step, undoubtedly the most important step in the method of Contopoulos and Grosbol, is to generate the density response map. For this a two-dimensional polar grid is employed; moreover, since the entire problem is symmetric, only half of the grid is actually used in order to save computational time and storage space. Use of a polar grid permits both a simple way of analyzing the self-consistency of the density response, and higher resolution near the center where the density and density gradient become large. The radial range of the grid, as well as the radial and azimuthal widths of the grid cells, can be varied in the analysis. The only restriction is that the number of azimuthal cells should be a power of 2. The reason for this will be outlined later in the discussion of the analysis of the density response.

The density response map, then, is constructed by computing a representative sample of the periodic orbits, and a set of orbits dispersed in radial velocity around them, and incrementing the value of the density of a given cell by the product of the weight of the orbit, to be discussed in detail shortly, and the amount of time spent by

the orbit in the cell.

Each periodic orbit family included in the analysis is sampled

at fixed intervals of "energy,"9 as parameterized by the radius r, of the circular orbit

of equivalent energy in the axisymmetric case.

The sampling step in the '"energy"~ so

parameterized is taken to be NOR times the radial width of a grid cell, where NOR=

22

1, 2, 3,

is an input parameter. In all of the models considered in this thesis we have

taken the value of NOR to

be 2.

Every periodic orbit chosen by the sampling process to be included in the density map is then integrated for one half period (since only half of the polar grid is used) in order to have its contribution to the density map added and to generate initial conditions

for the dispersed orbits.

The periodic orbits,

as originally

determined,

all have their

initial conditions (r, ?)

specified along the 0 = 0 axis.

It is desirable, however, in order

to estimate more accurately, and

with reduced integration times per orbit, the actual

density

response

of orbits

dispersed around the periodic orbit, to start orbits for the

density response calculation away from this axis but along the trajectory of the periodic orbit. Therefore, during the integration of the periodic orbit, its radius r, radial velocity ?, azimuth 0, and angular momentum JO are stored every fraction 1O of its half period,

where NOA =

1) 25 3, .

is also an input parameter.

After the density contribution of each periodic orbit is added to the density map,

the orbits dispersed about the periodic orbit are computed.

Whereas the periodic orbit

is calculated for its full half period, it is not really necessary to calculate the dispersed orbits for this length of time due to the increased numbers of dispersed orbits calculated

per periodic orbit.

The actual time

of integration for the dispersed orbits is controlled

by a parameter PN such that

integration

1 PZ time-t=t0 PN x O x2

(2-15)

where PZ is the period of the base periodic orbit. The value of PN is typically chosen to be of order NOA, empirically determined to be a good compromise between the need to generate an adequate representative density response for the orbit and the need to

23

minimize computational time per orbit.

The dispersed orbits

are started at the NOA

stored points along the periodic orbit.

The distribution of dispersed orbits is assumed to

be Gaussian with velocity dispersion a, thus the orbits must be dispersed and weighted appropriately relative to the central periodic orbit in order to estimate best, using a finite

number of orbits, a continuous Gaussian distribution.

Consider a periodic orbit,

characterized by

(r, 0, f4, JO), a point along the trajectory

of the orbit.

The problem is to compute the surface response density of a continuum of

orbits all starting at this point, but distributed in radial velocity about i in a Gaussian

fashion with dispersion a-.

Let the surface response density of the orbit started at (r, 0,

and calculated for time

to be given by

S(7-10 0,l~ Jol to)*

The problem, then, can

be stated mathematically as

S(r, 01Jo Ito) -

70

-00(

exp (-.I 9

s (r, 0, r, Jo, o) dtV

(2-16)

where S(r, 0, JO, to) represents the

total surface response density of all orbits distributed

about the period orbit and calculated for time to. In order to estimate the above integral,

scheme employing the Hermite

polynomial H,,(x)

is used (in

considered in this thesis we have taken n = 9).

1965,)

Specifically

(Abramowitz

and Stegun

890),

00

I exp (-X)
-00

f(x)dx =

wvtf (xz) + R,,,

(2-17)

i=1

where xi is the jith zero of H,,(x), the weights

w are given by

Y~ -172! W

(2-18)

and R,, is a small remainder.

i , JO)

all

cases

24

Since we use a nine point (n = 9) Hermite integration to calculate the effect of velocity dispersion on the density response, a total of nine orbits, the one periodic orbit

four on either side in space,

are integrated,

with the

initial conditions

given in Table 2-1.

Since

00
f exp (-X 2) dx the weights wvi
-00

must be normalized by dividing by

Table 2-1:

f/-r before the integrations are done.

Initial conditions of the periodic orbit and the eight

nonperiodic orbits used to calculate the effect of the velocity dispersion.

Orbit Number
1 (periodic orbit)

2,3
45
6,7 819

Initial Conditions

7")071 I

Orbit

oTx1,I Jo

7? 07 r~X-,IJo 7,1 0) i oX3,JA

r, 0,

OrX4, J

,) 0)r 9X5,J

Table 2-2: The values of the xi and wvi

xi

i
1
2
3
4
5

0.00000
0.72355...
1.46855...
2.26658...
3.19099...

WV1
WV2 WV3 WV4 WV5

of Table 2-1.

wvi

7.20235...
4.3265 1...
8.84745...
4.94362...
3.96069...

In addition to the Hermitian weight mentioned above, we must also include some further weighting factors to measure properly the contribution of a particular orbit to the response density map. These weights are introduced to account for (1) the relative

abilities

of different periodic families coexisting at energyg"

rc to trap matter around

(2) the population of a particular orbit of "energy" r, according to the imposed

Weight

x 10-1 x 10-1 X 10-2 x iO-1 x10.

them,

25

axisymmetric surface density at radius r, (3) the "grid" effects of the parameters NOA and NOR, and (4) the time of orbit integration.

Sometimes, in the course of calculating the periodic orbit families to include in the model, we find more than one stable family at the same "energy" r, Therefore the question arises as to the relative importance of each orbit family in the energy intervals where there is overlap. Contopoulos (1979) has given an estimate of the relative amount

of trapping done by a periodic orbit of a given energy r, .

for the mth

This estimate is measured,

orbit family, by the weight

(2-19)

whereK '(?-,)is the epicyclic frequency at radius r,, xm is the maximum deviation of the periodic orbit from r,, and is the square of the RMS radial velocity dispersion. Since the velocity dispersion a- mentioned earlier is assumed to be isotropic, = a-2.

Therefore the relative weight wm, of the mth family

_Wm
Wm nl /TT

at "energy" r, is given by

(2-20)

where n is the total number of stable orbit families coexisting at energy r,. In energy intervals where only a single orbit family exists, this weighting factor is obviously equal to one.

The next weighting factor to be considered is the most obvious. Orbits of lower energy r, which exist nearer to the center of the galaxy, should be weighted more than

orbits of higher energy, which exist in the outer regions of the galaxy. reflects the fact that the matter density in the disk decreases outward.

This simply Therefore a

.6d
exp K 4-) (rc) x
9 < * 9
A-d 7'

26

weight wE, of the form

WE= Z(rc),

(2-21)

is used. quantity

Here E (r,) is cannot simply

the surface density of the unperturbed disk at radius r,.

This

be derived from the axisymmetric rotation curve, because it is

not clear that the majority of the gravitational mass in spiral galaxies is concentrated in or near the observed disk. In fact, observed disk luminosity profiles typically decrease

exponentially outward (Freeman 1970).

Unless the actual mass-to-light ratio varies

wildly across the disk, it is plausible to infer that the matter density in the disk also exhibits an exponential decrease with radius, and that the excess gravitational mass responsible for the observed rotation curve is distributed in a spherical or oblate

spheriodal distribution, but away from the disk in any case.

Given these facts, the

surface density of the unperturbed disk is taken to be exponential, given by

Z(r) = co exp (-Eo7-),

(2-22)

where co is the central surface density and EO-1 the scale length of the disk. Thus the weight factor applied for an orbit of energy rc is

wE -co exp (-EO7rC).

(2-23)

The final weighting factors are introduced in order to eliminate the particular choices of NOR, the number of sampling steps in "energy" r,

effects of the per grid cell,

NOA, the number of different starting positions for dispersed orbits along the periodic

orbit, and to, the time of integration.

The grid effects are taken out by a weight of

27

the form

Wgrid -NOR xNOA

(2-24)

Also, since the contribution of a particular orbit to any one grid cell is the previous weight of the orbit multiplied by the amount of time spent by the orbit in the cell, the orbit weight must be divided by the total time of orbit integration. As noted before, the integration time is 1 PZ for periodic orbits and PZ x P N xNA for orbits dispersed

Again, PZ is the period of the base periodic orbit and PN

the parameter which controls the length of integration for nonperiodic orbits through

Eq. (2-15). these times.

Therefore the corresponding weights applied are simply the inverses of

9)
Pz,
2x NO A
PNxPZ' The total weight applied to a given orbit,

periodic orbits

(2-25)

nonperiodcorbits then, can be written as

Wtot- Wv )X Wrn X WE X Wgrid X Wtime.

The complete density response map, then, is produced by (1) specifying the minimum (RMIN) and maximum (RMAX) radial extent of the grid, as well as the radial (DRS) and azimuthal cell widths, (2) specifying the parameters NOR, NOA, and PN described above, (3) specifyingT the central value, o~o, and slope, a,, of the velocity dispersion (the form of the velocity dispersion is assumed to be uT(r) = 070 + rur, a linear relation being the simplest functional form possible which allows a(r) to vary across the disk), and (4) stepping in radius r, from RMIN to RMAX in increments of

(2-26)

28

DRS
NO-R and integrating all orbits existing at each value of r, weighted as described above.

The results from this raw

map are then preprocessed slightly before being stored for

later analysis.

Each semiannulus of the grid is processed in turn.

Thus,

for a fixed

semiannulus, the density value of a each azimuthal cell is "smoothed" with of the two adjacent cells according to the smoothing formula

the values

1
DEN(i) - -DEN~i
2

+ - (DEN(i - 1) + DEN(i 1I)),
4

(2-27)

where i

is the azimuthal cell index. At the same time,

the total "density,"1

RHO, in

the semiannulus is computed,

RHO =

j
i=1

DEN(i),

(2-28)

where n is the number of azimuthal cells.

The value of each cell is then normalized by

the "average" cell value of the semiannulus and finally

decreased by one, such that

DEN(i) -

nx DEN(i)
RHO

-1I

(2-29)

represents directly

from the

the fractional deviation of the i th azimuthal cell of the semiannulus

average.

The final

step

of the method

of Contopoulos

and Grosbol

is

to analyze

density

response for consistency with the imposed density.

In other words, the

consistency of the model is measured.

separately.

Here again, the analysis is done on the semiannuli

Two basic quantities are used to estimate the self- cons istency of the model.

The first compares the amplitude of the 20 component of the response density

corresponding amplitude of the imposed density.

to the

The second measures the difference

between the phases of the response and imposed density maxima in the semiannulus.

the

self-

29

First, a Fast Fourier Transform (FFT) with respect to azimuth is performed on the equally spaced data in array DEN. The reason for requiring the number of azimuthal

cells to be an integer power of 2 now becomes clear.

While the FFT can be made to

accomodate data sets of other lengths, the algorithm is applied most easily to a set of length n~ = 2 k, k a nonnegative integer. The power and phase of each positive frequency component fi = = 0, 1, 2, ..., n12, are computed. For the i = 1 frequency component (i.e. the 20 component), the difference AO between the response density maximum and imposed potential minimum is then calculated. This angle difference is measured with respect to the imposed potential minimum rather than the imposed density maximum because these two coincide almost exactly and because the angle of the the imposed

potential minimum is more readily ascertain should be zero for all values of the radius.

ed. In a fully self-consistent model, AO Therefore AO constitutes one measure of

the self-consistency of a given model.

The amplitude of the i = 1 frequency component corresponds to the quantity

27,-r)
(70 resp

(2-30)

where 972,,,sp i S the actual amplitude of the 20 component of the response density in the semiannulus under consideration and ao,resp is the mean surface response density in the semiannulus, or RHO/n. It is of interest to compare this with the corresponding quantity of the imposed density, or

go imp

(2-31)

The mean imposed density at the radius of the semiannulus is given by the formula for the exponential disk [Eq. (2-23)]. The amplitude of the 20 component of the imposed

30

density

is derived from

the imposed potential

via Poisson's equation.

To this end

Vandervoort (1970;

and Grosbol

1988) has

given the following

formulae, valid in the case of a tightly

three-dimensional potential V,

wrapped spiral density wave, relating the full

and density p, of the spiral: ~V Vosechlk-,'(z/ZXA)j

(2-32)

P1~-

IkAI(IkA + ') 17l e2IkAI (I/A)
47GA2 Osc-z,

where V10 is the spiral potential formula in the disk plane (z = 0), A is the z thickness of

the spiral and k is the wave number.

The parameter A, then, allows the vertical thickness

of the spiral to be controlled and must be included in the adjustment of parameters.

For

the form of spiral potential used by Contopoulos and Grosbol,

Mn

(2-33)

r t an io

where m is the number of spiral arms. From the above relations Or2, imp can be computed.

Specifically,

0'2, zmp

00

pi(z)(lz -9

00

I
0

p(z)clz.

(2-34)

Here A is

taken to be constant over the entire disk of the

galaxy.

At this point the

quantity

00,respJ 0, i p

(2-35)

can be formed.

In a fully self-consistent model R *

should be,

in principle, equal to

unity for all values of the radius.

In practice, however, it will not be so because cao~resp

and Uo,imp are calculated in the different ways described above. The actual value of R * can be scaled over a rather wide range by varying co, the central surface density of the

I
0 (kr)'

31

''imposed'' disk.

The maximum value of co is limited only by

the total axisymmetric

potential. Therefore, practical self- cons istency requires that R *

simply be constant with

This is the second measure of the self-consistency of a given model.

The whole procedure outlined above is then repeated for different sets of model

parameters, and a "best," model achieved by a sort of manual

iteration.

It would be

desireable to automate the whole procedure, but at this point too much human decisionmaking is required at intermediate steps for this to be feasible.

The CG Method Modified for the Case of Barred Spiral Galaxies

In this section are described the modifications made to the CG method in order

to apply it to the case of barred spiral galaxies.

is the nature

Perhaps the most significant change

a

simple function is fit to the

observed axisymmetric rotation curve,

thereby

allowing

the total axisymmetric potential to be derived in a correspondingly simple form, model

components are used.

This is necessitated by

the fact that the

full complement of

nonaxisymmetric structure cannot, in this

modeled as

a simple

perturbation of the axisymmetric disk.

This approach was tried initially, by replacing the

quantity j'77AO + "A,, with mO in the formula for V, (r,O) for values of radius less than the length of the bar, with somewhat limited success. It was found that even with the inclusion of significant higher harmonics the imposed bar density distribution could not adequately simulate realistic bars, even qualitatively. Therefore a three component model is employed, with components introduced to represent separately a bar, a disk,

and a halo.

of the assumed model. In contrast to the original CG method where

32

The axisymmetric

component is modeled as

the superposition of disk and halo

components.

In view of the fact,

mentioned

above,

that the observed

brightness

profiles of disk galaxies are typically exponential, an exponential disk is used. The corresponding potential in the disk plane is readily derived (cf. Binney and Tremaine

1987, p.77)

and can

be written

VD (r) = -w7rGco r[I0o(y) Ki (y) -

I, (y) Ko (y)] I

(2-36)

where

G is the Newtonian gravitation constant',

CO

is the central

surface

density of

the disk,

= 1
~or,

and the I,~ and K,, are modified

Bessel functions

of the first and

second kind, respectively.

Again, EO- is the scale length of the exponential disk.

circular rotation velocity VD (r)

corresponding to this

the relation

2 (r) = d- r

(2-37)

using the fact (Abramowitz and

Stegun

1965, p.376)

d10 (x)
dx

=Ii11

clKo (x)clx

- K1x)

(2-38)

dI ( x)
dx

=I (0

and

dK1 (lx

- Ko~x

I -Kix)
x

The result is

VD (r) 1Gco~or2EIo(y)Ko(y)-

The general form equal to the mass

-1(Y)

JYi(y)].

of this rotation curve, along with the Keplerian curve

(2-39)

for a point mass

MN~D - ., of the exponential disk is shown in Figure 2-3 (essentially

Figure 2-17 of Binney and Tremaine

1987, p

78).

The

that

I I I I

33

I I I I I I I

I I I . I

I I I I

I I I I

2

I I I I

4

I I I I

6

I I -L

8

10

I I I I

Figure 2-3: Rotation curves of the exponential disk (solid curve) and a point with the same total mass (dotted curve).

As can be readily seen from Figure 2-3, the rotation curve of an exponential disk declines after approximately two disk scale lengths and rather quickly approaches the

Keplerian value.

This, as described above, is in contradiction to the observed rotation

curves of most disk

galaxies.

Hence a separate halo component is included in order

to "hold up" the rotation curve at large radii.

The density distribution chosen

for this

component is

that of a Plummer sphere:

PH(V 3M4I )

__5

(

(2-40)

Here MH

is the total halo mass and b is

a parameter which

controls

the central

condensation.

The potential

corresponding to

this density

distribution is

particularly

simple:

- GMIH
1. TuP-1 -

(2-41)

.8

.6

0
C-,

.4

.2

0

0

-r--T I

34

From the potential the circular rotation velocity is derived:

GMHI
VII (r)- r3
F-r2 + b)2)

Figure 2-4 shows Keplerian rotation

(2-42)

the rotation curve of the Plummer sphere and the corresponding

curve for a point mass equal to MH.

At first glance it seems as

though nothing has been gained by including the halo component. The rotation curve of the Plummer sphere declines and approaches the Keplerian curve also at large radii. There is a crucial difference, however, between the halo and the disk. Since the disk is luminous, its exponential scale length can be observed. The halo, though, is made up of dark matter which is not accessible to direct observation. The totality of knowledge about the matter distribution in the halo must be inferred by its gravitational effects. The end result is that the parameter b, unlike the exponential disk scale length E0, is free, constrained only by the observed rotation curve. Hence the value of b can be near or beyond the observed disk (optical or HI), thereby placing the disk completely within the rising part of the halo rotation curve.

For modeling the bar, the triaxial homeoidal density distribution given by

p(x, Y,z) {PC( -7)

m<1 112 > 1,

(2-43)

where m27 an+a+>o > c > 0 are the long, intermediate, and short bar axes, respectively, is used. This is the n = 2 Ferrers potential (Ferrers 1877; see also Binney and Tremaine 1987, p.61). This particular bar component was chosen not only because it represents many features of observed bars rather well-Hunter et al. (1988) and Ball (1984, 1992) found that bar components with a Gaussian brightness profile, similar to that of the n = 2 Ferrers bar, best fit the infrared observations of the bars

35

of NGC 3992 and NGC 3359, respectively-but also because its dynamics has been

the subject of a number of previous investigations

(e.g.

1972; Papayannopoulos and Petrou 1983; Athanassoula et al.

de Vaucouleurs and Freeman

1983;

Pfenniger 1984a,

1985). The strict inequalities in the bar axes' lengths are only required for mathematical

reasons in the derivation of the associated potential and forces.

They do not represent

a real restriction from essentially

spherical, oblate or prolate figures.

Exact

solutions

for these

special

cases

I I I.a

I I I I

I I I I

T- I I

I I I I

I I I I I I I I

2

4

r/b

6

8

Figure 2-4:

Rotation curves

of the Plummer sphere (solid

curve) and a point with the same total mass (dotted curve).

The central density p, of the bar can be expressed in terms of the total mass MB

and its

axes' lengths as

PC -10 5 MB Pc 32-x abc

(2-44)

exist.

.8

.6

C" ~0 N.
0 N.
Ld

.4

.2

0

0

10

36

The projected

surface density

of this figure is obtained

by integrating

p(x)y)z)

respect to z over the range where p is nonzero:

YB(X,Y) -

Zi n ax

-I
r-n~ a x

p(x,y, Z)dz,

(2-45)

where Zmax

=cV~1

ZB(XY

a2 b 16 p

The result is expressed by

y2x2
a2 V2

5
2

7 7MB(
2wTab1

5
2

y 2
a2

(2-46)

The potential in tl

VB (xIY) -

disk plane can be written as

c{13(21Vx2 y2 6y

- TV12OX 4 Y2

- VV21OX 2 Y 4 _'ViOy

+I"Vodox +IV~y

- VVroi o 2

(2-47)

+W/OOO - W0O30X6

- I'Vl30y},

where C = 27r GP, abc L 5GI~iJ, and the Wi k are coefficients dependent

on both the

axes' lengths a, b,

and c and the position

(xy).

Complete derivations

of the

three-dimensional form of the potential, the Cartesian components of the force,

coefficients Wj are given by Pfenniger (1984a, appendix).

general and the

A summary of all the bar

quantities needed for this study is included in the appendix of this thesis.

Despite the fact that there is no "axi symmetric" rotation velocity associated with

this bar potential, an estimate

of this quantity

can be obtained.

By analogy

with the

axisymmetric

disk and halo rotation velocities, the "circular"

velocity

VB(r, 0

solely to the bar potential is

given by

(2-48)

Above, the velocity and potential have been taken in terms of the polar coordinates (r,0)

instead of the Cartesian ones (x,y).

The "axisymmetric" rotation velocity VB(0, then, is

Idr

0 V

(2-49)

with

due

VB(0" 0) r&VB(?,0
j117

VB (r)

x 2 b2

37

In a similar manner, all other "axisymmetric" quantities of the bar can be estimated. Figure 2-5 gives representative "axi symmetric" rotation curves for three sets of bar dimensions.

01 fI I I I I I I I I I I I I I I I I
1.5 33
1/

Fiue25 he onaie aiymtic a oaincre o ae hr
1anc=0.1(1b=0.1,()b=03an(3b=0..Tedte

F ige -:eTreenormaeKli axiymtca rotation curvefoapinmss foruasestwhreMa.

As in the original CG method, the observed angle-averaged rotation curve is fit by the model rotation curve. The model curve in this case, though, is derived from the component rotation curve by the relation v(r) =v' (r) + v' (r) + v' (r), (2-50)

where vT(r) is the total rotation velocity. The method of curve fitting is somewhat more crude in our modified method than in the original CG method. In particular, the use of a least squares fitting routine is forgone in lieu of simple visual fitting. There are a couple of reasons for this. First, in barred spiral galaxies the noncircular motions can be

38

much more pronounced than in normal spirals (e.g. Duval and Athanassoula 1983).

The

uncertainty in the "correct" axisymmetric rotation curve, then, grows in proportion to the magnitude of the noncircular motions. Large uncertainties, in themselves, however,

do not preclude the use of a mathematical fitting scheme.

problem was that

A second and more serious

least square fits in which all input parameters were allowed to adjust

freely resulted, in some cases, in physical implausibilities, such as near-zero bar masses. For these reasons the following procedure for fitting rotation curves was implemented.

since the component contributions

to the rotation curve are somewhat

independent of each other (the bar component dominates the inner region of the Observed rotation curve, the disk component the central region, and the halo component the outer region), it was decided to fit visually one component at a time, starting from the center

outward.

The first major contributor to the rotation curve is the bar.

The bar that was

used was the one with the maximum allowable mass under the constraints of the rotation

curve. This component typically fit the steep initial rise of the rotation curve.

component included was the exponential disk.

the rotation curve was used.

The next

Again, the maximum disk allowable by

The addition of the disk component permitted good fits to

the observed rotation curve over most of the radial range.

The Plummer sphere halo

component was added last in order to "hold up" the rotation curves at large radii. parameter values obtained this way were adjusted slightly in order to achieve the

visual fit.

The parameters that were adjusted in order to provide the fits to the rotation

curve are given in Table 2-3.

While the bar contribution to the rotation curve certainly

depends on the axes' lengths a, b, and c.

these parameters, as we will see, are fixed

by the surface photometry.

First,

The best

39

Table 2-3: Adjustable parameters in the axisymmetric rotation curve fitting procedure.

Component
Ferrer s B ar

MB

Exponential Disk

Plummer Sphere Halo

co and Eo MHand b

The method of Stark (1977) is used to derive the lengths of the bar's axes. A complete explanation of the method can be found in Stark (1977) and in Ball (1984), thus the derivation will not be included here. In short, Stark's method assumes that the volume brightness of the bar is constant on similar, nested, triaxial ellipsoidal surfaces. Contopoulos (1956) has shown that the projected isophotes of such a bar, from any

viewing angle, are similar, concentric ellipses.

Assuming that the triaxial bar figure

has one of its principle axes normal to the disk plane, one can combine the observed axial ratio of the bar isophotes, &3o the inclination angle of the galaxy, i, and the angle between the line of nodes of the disk and the major axis of the bar isophotes, 04, to give a one parameter family of triaxial figures as the solution. The parameter q\$, which controls the family, is in some sense a measure of the true azimuth of the triaxial figure in the disk plane referred to some fiducial direction. In the present analysis 05 is taken to be the angle measured in the disk plane between the long axis of the bar and the line of nodes of the disk. A useful property of this family of solutions is that there is a one-to-one correspondence between q\$ and the axial ratios of the bar. Specifying, say, the axial ratio c/a is equivalent to specifying 0 directly, and that is the actual procedure used here. The value of c/a is taken to be 1/10 after Pfenniger (1984a [see also references therein, de Vaucouleurs and Freeman 1972; Kormendy 1982], 1985). In all of the cases considered in this thesis the long and intermediate bar axes a and b are

40

assumed to lie in the principle disk plane of the galaxy, and the short axis c is assumed to be perpendicular to the disk plane.

After the determination of the lengths of the bar axes from the Stark analysis, and the visual fitting of the axisymmetric rotation curve, we derive starting values for QP and for the parameters which define the spiral perturbation. There is ample theoretical and numerical evidence (Contopoulos 1981; Schwarz 1981; Teuben and Sanders 1985; Sellwood 1980, 1981; Thielheim and Wolff 1981, 1984; Sparke and Sellwood 1987) that bars end at or slightly inside corotation, the distance at which the angular rotation rate of matter in the galaxy equals Qp. For a first estimate of the pattern speed, then, the bar is assumed to end at corotation. As before, the parameters characterizing the

spiral pattern beyond the bar are derived from the available surface photometry.

A

new feature is now added, however, to the form of the perturbation. Since the spiral is assumed to begin at or near the end of the bar, an inner cutoff is added to the formula for the spiral amplitude Amjr):

Arn~r) =

- Armnt)(l +I tanh [t,,ir - rimn)])
4

x (I + t anli EK2m,(V2m - r)]) + A,,m} . As we shall see later, the optimum positions for these inner cutoffs yield important

information concerningT the nature of the bars in the most successful

models.

some changes to the argument of the cosine in Eq. (2-6)

are required.

Namely,

argument -

________+ 11 I I
tam '10V1??V~, 771,

r >a r

(2-52)

At this point, the entire barred spiral model is specified.

As in the original CG method, the next step is to calculate the main families of periodic orbits. In the barred spiral models, periodic orbit families were searched

(2-51)

Also,

41

for between the ILR and the 8/1 resonance inside corotation, and between the -8/1 and -2/1 (outer Lindblad) resonance outside corotation. For bars of moderate to high strength, however, there are substantial regions of "energy" r, both inside and outside of corotation, where there are no stable, dynamically significant periodic families. In order to represent matter existing in these regions for the surface response density calculation,, we included orbits started here on "circular" trajectories. The initial radius of a "circular" orbit of energy r, was taken to be r, and the initial radial velocity was taken to be zero (hence the designation "circular"). Since the majority of orbits in these regions turn out to be stochastic, the surface density response they generate is not particularly sensitive to the particular choice of initial conditions. Thus we have given them the most simple initial conditions possible, the "circular" orbit initial conditions described above. Despite the fact that most of these orbits turn out to be stochastic, they yield interesting density responses nonetheless, as we will see later in Chapters 3, 4, and 5.

Next, the surface response density map is created.

The procedure is exactly the

same as in the original CG method, except for the handling of the "circular" orbits

described above.

The differences are (1) in the formula for the weight wi, which

gauges the relative amount of trapping done by the rnth family, x, the deviation of the orbit from r, is taken to be 1 kpc, a typical value for this type of orbit, (2) the starting points for dispersed orbits along the "'circular"~ orbits are taken to be every interval 27NO in azimuth, (3) in cases where the given "energy"~ and initial conditions of a circulara" orbit result in an imaginary angular momentum solution (i.e. where the initial conditions lie outside the limiting curve) the "energy" is artificially increased to the point

42

where the angular momentum becomes real (this procedure is justified by

noting that

both observations

of real barred galaxies

and N-body

simulations of barred

galaxies

show that these regions are populated with stars), and (4) all "circular" orbits and orbits dispersed around them are integrated for a fixed time T, chosen such that good estimates of the average density response are derived from these orbits and computation times are

held to reasonable lengths.

Typically T is chosen to be approximately

1 billion years.

These changes require some changes in the precise form of the weighting factors;

however, the meaning of each factor remains the same.

Specifically,

for all orbits,

wF -co exp (Eorc)+

2A-

7r

ZB(rc, 0)dO,

(2-53)

and for the "circular"

orbits,

1

Wtilme -

(2-54)

The resulting density map is preprocessed exactly as before prior to the analysis for self-consistency.

The essence of the analysis of the density response map remains unchanged in the

modified method

(i.e.

the application of the

CG method to barred

spiral galaxies);

however,

one detail

concerning

the

computation

of the amplitudes

of the

Fourier

components

of the imposed density has been altered.

Since the bar potential

is not

put into the model explicitly

in the form of its Fourier components, the amplitude of

the 20 component of the imposed density 20 component of the response density; spe

is derived in exactly the same way as the ecifically, the imposed density is calculated

as a function of azimuth, for a fixed semiannulus radius, at exactly the same azimuthal

resolution as the response density

map.

The imposed

density at a given position is

43

derived

from (1) the surface density

of the exponential disk and the projected bar,

and (2) the spiral perturbation according to the formula of Vandervoort.

A Fast Fourier

Transform is then performed on this array of imposed densities at the same time as on the array of response densities. At each radius, then, the amplitudes of the 20 components

are compared by the ratio of response to imposed, as in the original method.

difference calculation is

The phase

unchanged.

Gas Response Using Smoothed Particle Hydrodynamics (SPH)

To complement the self-consistent stellar models achieved by the modified method, the gas response of the ''best"~ model in each case has been investigated via the use of

a two-dimensional smoothed particle hydrodynamics (SPH)

code generously provided

to the author by Dr.

Nikos Hiotelis.

The SPH method was introduced independently

by Lucy (1977)

and Gingold and Monaghan

(1977)

and uses

a Monte Carlo method

to solve the equations of hydrodynamics.

The complete details of the SPH method are

given in several review articles (Hernquist and Katz 1989; Benz 1990; Monaghan 1985;

Steinmetz and MUller 1992)

and references therein, thus only a brief overview of the

general method (in two dimensions) will be given here, along with the relevant details of the specific SPH implementation used for the present calculations.

SPH is a Lagrangian method which does not require the use of a computational grid; therefore, unlike finite- difference schemes, which require a grid, no computational resources are wasted simulating large voids that may arise (Steinmetz and MUller 1992).

Since the SPH

method is not very complicated,

furthermore, it is quite robust.

it does not require a lengthy

Steinmetz and MUller (1992)

code;

mention also three new

developments which have greatly enhanced the potential of SPH:

(1)

the achievable

44

resolution has been improved dramatically

with the advent of the variable smoothing

length, (2) global time step restrictions have been alleviated via the use of a separate time step for each particle, and (3) the introduction of the hierarchical tree method to calculate the gravitational forces has reduced computation times for SPH with self-gravity from

the order of N 2 to NlogN without altering its original Lagrangian

formulation.

Finally,

the SPH scheme and hierarchical tree method can be vectorized to take advantage of

highly parallelized computing

architectures.

In the SPH method a continuous fluid medium is modeled as an ensemble of N

fluid elements

(in this thesis N is approximately 12,000 in all cases), each taken to be

a "particle" with position ri

and mass mi, smoothed out according

to some chosen

smoothing

kernel

W(r

- r', A)

The kernel

is

a function strongly

peaked

around

Ir- r II

-0

and the quantity

Ais called

the smoothing length.

The kernel

fulfull certain requirements. First, it must guarantee conservation of momentum, both

linear and angular.

An easy way to do this is to make the kernel spherically symmetric.

Secondly, it must have the properties

JW(r,A)d~r1

and

urn W,,(r, A) = 6(r). A --+O

The ensemble of particles together with the smoothing kernel can be used to estimate

the mean value of any spatially

varying physical variable at any arbitrary

position r.

The actual mean value at r of the spatially varying function A(r)

is given by

(A(r)>

I

A(r') WV(r - r', A) d'r'.

(2-56)

The estimate using the "smoothed" particles is given by the sum

(A(r)> =

N
S
j=1

p (rj) r - ri, A),

must

(2-55)

(2-57)

45

where, for the density p(rj), the similarly

estimated value

(p(r,)> -

is used.

N
S

?72~T4/(r~ -

The time derivative and gradient, respectively,

A)

(2-58)

of the quantity (A(r)) can be

written

(S teinmetz

and M Ull er

1992) as

dK(A > dA(r)>
cit dt

V< A (r))>--

I

A(r')7 VW

(r -r'A) dr'.

These

results

can be used

to derive

the averaged

hydrodynamic

equations

motion.

Specifically, they are obtained by multiplying each term of the exact equations

by the smoothing kernel and summing over all particles

(e.g.

Monaghan and Gingold

Monaghan

1985;

Benz 1990).

Including also the artificial viscosity

terms, the

following set of equations (Hiotelis et al.

N
Z ~-n
1=1

1
2

N
E 172J j=1

Li 1~
/; J

dr,
dt

[P1

199 1) are obtained:

(1+ fi~~)V~W

=-vj,

Above,

v1, Pi9

pig ui,

Vi, and VT4 are the velocity, pressure, density,

thermal

energy

per unit mass, gravitational potential, and gradient of the kernel W, respectively, at the

position ri of the

it particle.

The term TIq includes the effect of artificial viscosity.

The form of Hij is taken from Monaghan and Gingold (1983)

and is the same as that

used by

Hiotelis et al.

(1991):

H.-i - -CettL.i1+ /3,t 2

(2-61)

and

(2-59)

1983;

of

dv,
dt

du,
cit

(2-60)

46

Here, av and /3 are constant coefficients and ltij is defined by

= {(,-v,- - (r -r,)

(2-62)

otherwise.

Above , Aq and cqj are the average smoothing length and sound speed, respectively, between positions ri and rj. The parameter 7 2 = 0.01A 2 is introduced to avoid possible numerical divergences in pjt. In the expression for Elij, the term linear in juj produces a shear and bulk viscosity (Monaghan 1985), while the quadratic term is included to handle high Mach number shocks and is roughly equivalent to the Von NeumannRichtmyer viscosity used in fi nite- difference methods. In all calculations done here, a and /3 are taken to be unity. According to Hernquist and Katz (1989), these values represent a fair compromise for the typical range of Mach numbers encountered.

In order to integrate the hydrodynamic equations an equation of state must be specified. The gas in this case is assumed to behave like an ideal gas. Therefore, the pressure, density and thermal energy of the gas is related by

Pi = ('T - l)ptui,

(2-63)

where 7y is defined to be the ratio of specific heats. Since the calculations are done in two dimensions instead of three, the values of 7y corresponding to the adiabatic and isothermal cases are 2 and 1, respectively. The actual behavior of interstellar gas most probably is somewhere between these two extremes,, therefore a compromise value of ^/ = 5/3 is used. The wide variation present in the temperature of the interstellar medium

presents a problem concerning the appropriate choice of the thermnal energy uj.

An

initial value of ui corresponding to an average temperature of around 3500 K is used. We consider this a fair compromise between the larger expanses of very hot, low density

47

interstellar gas (T

106 K, n rr i, 0-1cnV3) and the smaller expanses of relatively

cooler and denser HI clouds (102 K T

6xi10'

1%, 20 cm-' >- n 0.3 crK)and

HII regions

(T

~8x

K,

?2 >0. 5 cm-)

It still remains to specify the exact smoothing kernel used.

The two-dimensional

version of the spherically symmetric three-dimensional spline-based kernel proposed by

Monaghan and Lattanzio

(1985)

and employed

by Hiotelis

et al.

(1991) is

is defined

I1-1u, +
_ 102
r -r,A) 7 12~(2 -a3
77A9 0,

3 u3
-T

0

(2-64)

>21

A *EI Also, the smoothing length is assumed to be variable in space and time,

-A(r, t). This is done to increase the resolution possible with the SPH method,

since A directly limits the resolution (all structure on scales smaller than A are strongly smoothed out). Whereas the introduction of a time-dependent A is straightforward, there

exist two possibilities to introduce a spatially variable

A

(Steinmetz and MUl*ler 1992).

These are the "gather" and "scatter" approaches (cf.

Hernquist and Katz 1989).

In the

''gather" method,

the smoothed average

value of the arbitrary physical quantity A(r)

is calculated by summing the contributions from the whole of space weighted by the smoothing kernel centered on r using the local value A(r) of the smoothing length:

A(r') WV(r - r', A (r)) d'r'.

(2-65)

Hence the contributions weighted appropriately'

to A at r are "gathered"~ in from

by the kernel at r.

the surrounding elements,

The "scatter" method, on the other hand,

derives its estimate for the same quantity by using the kernel local to the contributing

by

used.

It

where u hence A

I

(A(r))gather -

48

element and "scattering" its appropriate contribution to position r:

(r)scatter- j A(r')T'(r - r', A (r')) d2 r'.

(2-66)

Exclusive use of one or the other of these approaches does not guarantee the conservation of energy, linear momentum, or angular momentum. A combination is usually used. Benz (1990) has symmetrized the smoothing length by using Wij = W(ri - rj, (Ai + Aj) I 2), while Hernquist and Katz (1989) have symmetrized the kernel itself Wi1 = 0.5(W(ri

- rj, A) + W(ri

- rj, A)). In the present work, the first method of symmetrizing the

smoothing length is used.

Two further questions remain concerning A(r, t): (1) How

should A be defined? and (2) how should it evolve during the integration?

It seems natural to associate the smoothing length with the local density such that denser regions have smaller smoothing lengths in order to provide better resolution. One way to do this (Steinmetz and MUller 1992) is to let

A~prr))=

(2-67)

where C is a parameter of order unity. The problem with this definition is that A(ri) is needed in order to compute p(rj) in the first place. One way around this problem is to use the value of the density at the previous time step to calculate the smoothing length

(Miyama et al.

1984).

derivative of both sides

Another way, proposed by Benz (1990), is to take the time above and eliminate the density altogether using the equation

of continuity to yield

dA (r) =I1V cit 2

(2-68)

In this way the smoothing length can be considered to be just another hydrodynamic variable to be evolved in the course of the integration. Yet another method (Hemnquist

49

and Katz 1989) is to link the value of A to the number of neighboring particles (within a radius of 2A), and let A change in space and time so as to keep this number fixed. The method used here is the second one where A is given an initial value for each particle and is subsequently evolved according to Eq. (2-68) above.

The algorithm used to integrate the hydrodynamic equations of motion is a predictor-corrector time stepping rule. At any given point in the integration a single time step 6t is used to advance all the "Cparticles." The stepsize varies from one step to another and is given by the relation

6t mi(cN At , N At 009~T)

(2-69)

where T = 2w i S the estimate of the circular orbit period at position ri, and eN

(the Courant number) is an input parameter of order 0.5 utilized to suppress numerical instabilities. In all cases we take CN to be 0.4. Also, Fi and vi are the acceleration and

velocity of the ith particle.

The terms in the expression for S& are included to allow,

respectively, fast-moving "particles," highly accelerated "particles," and "particles" very close to the center to be followed accurately.

CHAPTER

PRO GRAM

GALAXIES

The galaxies chosen for this

study are NGC

3992, NGC 1073, and NGC

and are

shown in Figures 3-1,

3-2,

and 3-3.

These galaxies were chosen for several

reasons.

First, the neutral hydrogen emission from all three galaxies has been observed

at the Very Large Array (VLA) of the National Radio Astronomy Observatory (NRAO)

at high resolutions

and with

high signal-to-noise ratios.

These

observations

reveal

information about the kinematics, and therefore the total gravitational potential, of the

disk. Secondly, detailed surface photometry exists for each of the galaxies.

Elmegreen

and Elmegreen (1985) have provided both blue and near-infrared surface photometry of

NGC 3992 and NGC 1073.

Ohta et al.

(1990) have observed NGC 1398 in the B band.

In addition, Grosbol (1985) has scanned and analyzed the images of all three galaxies

on the red Palomar Sky Survey copy plates.

The surface photometry yields information

concerning the distribution of luminous matter in the disk.

Finally, these galaxies span

a moderately

wide range of barred spiral morphological types.

de Vaucouleurs et al.

(1976)

classify

NGC 1398 as

(R')SB(r)ab, NGC 3992 as SB(rs)bc, and NGC 1073 as

SB(rs)c.

This latitude allows us to look for possible correlations between the model

parameters of the most successful models and morphological type. Furthermore, since these galaxies show no gross peculiarities, they can be considered to be representative

examples of their respective

types.

3

1398

50

Figure 3-1:

NGC 3992 (NASA Atlas of Galaxies Useful for

Measuring the Cosmological Distance Scale 1988).

0

Co)

0

11

K

0

'p

i

,lt' P

*00
S
*

0

CMj

C.0)

C)

0

0

0

0

6

0

0

0

52

b
CMj

0

4

?i

I

1'

p4

4

0

9

0

C.0" #

0

(I)

~0. Cl)

v

Figure 3-2:

NGC 1073 (NASA Atlas of Galaxies Useful for

Measuring the Cosmological Distance Scale 1988).

8[C
0

*

11 (s4~o8s

8

v

4,

0

v

4

A

A'

0 0

-4A.

0

*

It*4

4. -

4

41

0 4

I

0

0

LOU

1~

9

09

N

=

I

lioz

Figure 3-3: NGC 1398 (Sandage 1961, The Hubble Atlas of Galaxies).

56

0

......................................................................................-'-~."

0

.9

V

4

0

*

(
0 .4

S

0

2 0

0

0 6

k

0

57

NGC 3992

Observations

Neutral Hydrogen

High-resolution

neutral hydrogen

observations

of NGC

3992

were obtained by

Gottesman, Ball, Hunter,

and Huntley

(1984, hereafter GBHH)

at the VLA between

September, 1980 and January, 1983.

The effec tive re solution obtained was 2 6. "1 x 20. "0,

with the major axis of the Gaussian beam at a position angle of -22.' 1.

In comparison,

the diameter of the HI disk is 504" and the position angle of the line of nodes is -'

111.05

0.06.

The full-width half-power

(FWHP) velocity resolution was 25.2 km 5-1

. The

reader is referred to GBHH for further details of the data reduction.

The angle-averaged rotation curve was derived, along with the following orienta-

tion parameters for the HI disk, via a least-squares fit to the velocity field:

major axis at

a position angle of -111L'5 0.'6, inclination of 53.14 0.19, and a systemic velocity

of 1045.8 0.6 km s-1

.The data points so derived are given in Table 3-1.

The quoted

errors are the formal errors of the least-squares solution. The angular radii of columns 1 and 4 in Table 3-1 have been converted to linear dimensions in columns 2 and 5 using

an assumed distance to NGC 3992 of 14.2 Mpc

(de Vaucouleurs

1979).

The data in

Table 3-1 are plotted as filled circles with error bars in Figure 3-4.

GBHH find that the spiral structure of NGC 3992 is only

weakly

discernible in

the neutral hydrogen distribution. The optical image, nevertheless (Figure 3-2), is seen clearly to reveal spiral structure which extends from an incomplete ring around the bar.

58

Table 3-1:

HI rotation velocity data for NGC 3992.

R (aremin) R (kpc) V (km S-1) R (arcmin) R (kpc) V (kms-1)

(1) (1) (2)1 (1) (1) (2)

141.8 37.8 123.6 52.7 170.9 32.7 195.6 16.5 222.5 10.9 240.0 9.1 254.2 7.3 258.2 5.8
262.2 4.7 265.5 4.2 267.3 4.4

2.89 3.13 3.36 3.63 3.86
4.09 4.36 4.63 4.89 5.10

11.9 12.9 13.9 15.0
15.9 16.9 18.0 19.1
20.2 21.1

269.1 5.5 270.9 5.5 267.3 7.1 265.5 7.5

258.2 +
249.1 + 247.3 + 240.0 +

10.9 11.1 12.5 10.9

0.14 0.37
0.64 0.84 1.14 1.37
1.64 1.90
2.14 2.36 2.63

0.58

1.53
2.64 3.47 4.71 5.66
6.77 7.85
8.84 9.75
10.9

18.2

(1) Radius measured from the center of the galaxy
(2) Rotation velocity

They also find a deficiency of HI inside this optical ring.

Two possibilities exist.

Either

the total gas surface density suffers a real depression in the central region of the galaxy, or most of the hydrogen in this region has been converted to molecular form and is

not being seen in the HI observations.

GBHH cite

12CO observations of NGC 3992

by Young,

which

set an

upper limit to

the H2

surface density of

only

a few

larger than the neutral hydrogen surface density, and conclude that the total gas surface density near the center appears to be less than in the disk. They add that "it is possible that dynamical effects associated with the bar may have exaggerated the phenomenon"

(GBHH, pp.

477-478).

As we shall see from the results of our gas calculations (Chapter

6), bars can indeed "sweep out" gas from large regions inside the bar radius.

240.0 20.0

238.2+

times

59

10

Figure 3-4:

Comparison of observed and theoretical rotation curves for NGC 3992. The

contributions of the separate components of this model to the total theoretical rotation curve.

As far as the kinematics of NGC 3992 are concerned, GBHH find that the motion of the gas is dominated by circular rotation, with only weak irregularities associated with the optical spiral arms. They interpret these irregularities as streaming of the gas

along the spiral arms. Also noted is a discontinuity

in the isovelocity contours of the

velocity field across the major axis of the galaxy, which the authors tentatively interpret

as an effect of the flow of gas around the bar of NGC 3992.

The uncertainty

in this

interpretation stems from the limited resolution in this region due to low gas densities.

Finally,

GBHH note some peculiarities in the neutral hydrogen

velocities

along the

(15.7 kpc).

Specifically, the

velocities

show a rather sudden

decrease at this radius, followed by

a m ore

decline thereafter.

GBHH identify this kinematical feature with a possible truncation

of the disk.

300 250

200

I I I II II I I I I III I I Ij I I I I

U
U Ci2
S

U
0

150 100

50

0

0

S

15

20

25

I I I

I I I I I

I I I I I

I I I I I

I I I ---F

total halo disk

bar

60

Table 3-2:

Selected global and disk parameter values adopted for NGC 3992. The errors

given are simply the formal errors of a least-squares analysis of the observed velocity field
and do not imply that these quantities have been determined to this level of precision.

Parameter

Value

Systemic velocity

(heliocentric, km s1 )a

1045.8 0.6

Distance

(MpC)b

Inclination angle (')'

Position angle, line of nodes (')'

'Gottesman, Ball, Hunter, and Huntley (1984)

bde Vaucouleurs (1979)

Surface Photometry

Elmegreen and Elmegreen (1985, hereafter EE)

have provided blue (B passband)

and near-infrared

(I passband)

surface

photometry

of fifteen

barred

spiral

galaxies,

including NGC 3992. They photographed these galaxies between 1979 and 1981 using

the 1.2 meter Palomar Schmidt telescope.

The blue images were

taken using baked

103a-O emulsions with a GG 385 filter, yielding an effective wavelength of 4350 A. The near infrared images were taken using hypersensitized IV-N emulsion with Wr 88A

filter, yielding an effective wavelength of 8250 A

Here again, the reader is referred to

BE for further details concerning the observations and data reduction.

Primary emphasis has been placed on the I passband images because they

describe the distribution of the older disk stars, the gravitationally dominant component

of the disk.

BE determine the near-infrared disk scale length of NGC 3992 to be 3.38

+ 0.52 kpc,

assuming a distance

of 11.3 Mpc

to the galaxy.

Converting

this scale

length to our assumed distance of 14.2 Mpc yields a value of 4.25 0.65 kpc, or an

inverse disk scale length of 0.235 +0.043
-0.031

kpc-1

B E also determined the slope of the

14.2

53.4 0.9
-111.5 0.6

better

61

radial dependence of the spiral arm-interarm surface brightness difference, from which we derive the inverse scale length of the spiral perturbation of 0.4 kpc-1.

Hunter et al. (1988) have analyzed the bar figure of NGC 3992 in the infrared image taken by BE. They determined the orientation and apparent extent of the bar from the positions of the extreme ends of its apparent major axis. Further, the apparent axial ratio N3 was estimated by fitting ellipses to the outermost isophotes of the bar, after

subtraction of the bulge component.

They found that the major axis of the projected

bar figure makes an angle 0b = 350 with respect to the line of nodes of the disk, that the semimajor axis of the bar is 62", and that the apparent axial ratio ,3O = 2.6. These three quantities were used along with the inclination angle i and the adopted distance to derive, via the method of Stark (1977, also see Chapter 2), the actual linear dimensions of the triaxial bar figure (again, assuming c/a = 0.1). The resulting long bar axis a is 5.5 kpc, the intermediate axis b is 2.1 kpc, and the short axis c is 0.55 kpc. Again, it is assumed that the long and intermediate bar axes lie in the disk plane. Finally, Kennicutt (1981) gives the spiral arm pitch angle of NGC 3992 as -11l' 20. Table 3-3 summarizes the quantities derived from the surface photometry that are used in this work.

Best Model

The first step in determining the most successful model of NGC 3992 was to fit visually the observed rotation curve with the theoretical curve comprised of contributions from the bar, disk, and halo. Next, the pattern speed Qp was fixed by the assumption that the long bar axis is equal to the corotation radius. Finally, initial values of the parameters characterizing the bar/spiral perturbation were set by the surface photometry.

62

Table 3-3:

Photometrically derived parameter values for NGC 3992.

Parameter

Value

Inverse disk scale length (kpc1 )a

Inverse spiral scale length

0.235 +0.043
- 0.031

(kpc-I)a

Angle between projected bar major axis and disk line of nodes (,)b

Apparent bar semimajor axis (arcsec)b Apparent bar axial ratio b Long bar axis length (kpc)c Intermediate bar axis length (kpc)c Short bar axis length (kpc)C

0.4 35

62

2.6 5.5
2.1
0.55

Spiral arm pitch angle (~

-11 2

aElmegreen and Elmegreen (1985) b Hunter, Ball', Huntley, England, and Gottesman (1988) 'derived using the method of Stark (1977) d Kennicutt (1981)

The periodic orbits of the model were then determined, and the surface response density

was calculated and measured for self-consistency with the imposed surface density.

The

free model parameters, primarily the adjusted and the model differentially

found.

Table

3992.

ones characterizing the spiral perturbation, were

corrected until the most successful model

was

The rotation

curve of this

model is

shown

in Figure 3-4.

Also shown

are the

contributions of the separate model components.

Figure 3-5 shows the characteristics

of the different orbit families

which

are included in the

model.

The most important

periodic families.

families

in the bar are

the 2/1 (Figure

3-6) and 4/1

(Figure

3-7)

resonant

The 3/1 resonant family exists over a shorter range of energy, and these orbits

3-4 summarizes the parameters of the most successful model of NGC

63

are found not to be very critical to the overall structure of the bar. Also included inside the bar radius a is a set of "circular" orbits extending from r, = 3.5 kpc to r, = 5.5 kpc.

Table 3-4: Parameter values for the best self-consistent model of NGC 3992.

Ferrers B ar

MA1B= 1.5 x 0 M

a =5.5Skpc b =2.1lkpc c = 0.55 kpc

QP= 43.6 km 5-1 kpc-1

Spiral

A =2000 km 2 s-2 kpc-1 A=0 km 25S2 kpc-1 'Es=0.4 kpc-1 ,io =-10' =, 1.5 kpc, r2= 10.6 kpc ,qliK2 =1, A 0.1 kpc

Exponential Disk

12 10

8 x 6

4 2 0

Plummer Sphere Halo

A'IH= 2.75 x 10"Mï¿½(

bH = 12kpc

0

2

4

6

8

rc (kpc)

Figure 3-5: Characteristics of the orbit families included in the model. Each characteristic plots x, where a given orbit crosses the minor bar
axis b, as a function of Jacobi constant, as parameterized by r,.

CO =750M04'pc=o 0.235 kpc-1

- / circular

7 -4/1

3/1/
2/1/ L2

10

12

64

Outside the bar the main family is the -2/1 resonant family (Figure 3-8).

These

orbits support the imposed spiral in the region between the -411 resonance (outside of corotation) and the outer Lindblad resonance (OLR). The -4/1 and -6/1 resonant families are also included, but they play a much less significant role than does the

-2/1 family in supporting the imposed spiral structure.

Two more sets of "circular"

orbits are included outside of corotation. The first, a continuation of the set inside of corotation, extends from r, = 5.5 kpc to 7.5 kpc. The second, included to sample the response of matter near and slightly beyond the OLR, extends from r, = 10 kpc

to 12 kpc.

The use of the "circular" orbits near corotation does not imply that there

are no stable periodic orbits in this region. However, as we shall see in Chapter 5, the available phase space in this interval of energy is dominated by stochasticity and

the trapping done by periodic orbits is small.

Therefore, given the magnitude of the

imposed velocity dispersion, it makes essentially no difference to the resulting density response map whether these orbits have the initial conditions of periodic orbits or not. Furthermore, despite the fact that the majority of these orbits are stochastic, they do provide significant enhancement of the imposed spiral structure and are instrumental in

achieving self- cons istency.

The behavior of these stochastic orbits will be considered

in more detail in Chapter 5. Table 3-5 summarizes the positions of the main resonances in the model. The epicyclic frequency Ki(r) and angular rotation rate Q(r), necessary for determining the resonance positions, are obtained by avrgigd.~, ), and d2rO in azimuth, where VB(r, 0) is the potential of the bar, and adding them to the corresponding

quantities of the disk and halo. K~(r), then, is simply

( 2T( r ff+ 3 dVT~r n )i dr2 r dT)adQri

1
(1d T~r'\2where VT(r) is the total axisymmetric potential.

65

6

4

2

ci

0

-2

-4

.6

-4

-2

0
kpc

2

4

6

Figure 3-6: The 2/1 family of periodic orbits in the model of NGC 3992. The darker circle represents corotation at 5.5 kpc.

6

4

C)

0

-2

-4

6

-4

-2

0
kpc

4

6

Figure 3-7: The 4/1 family of periodic orbits in the model of NGC 3992.

Ko . p

Ico 5.5 kpc I

66

10

U.

0

-5

-10

Figure 3-8:

-10

-5

0

5

10

kpc

The -2/1 family of periodic orbits in the model of NGC 3992.

The darker curves represent the minima of the bar and spiral potentials.

The density

response

map was generated

according

to the

procedure outlined

in Chapter 2. calculation.

Table

3-6 gives the parameters used

for the

surface density

response

Table 3-5:

Resonance locations of the best model of NGC 3992.

Inner ResonancesI Outer Resonances

0.0 kpc 3.1 kpc 3.9 kpc 4.3 kpc

-8/1
-6/1
-41
-2/1

6.9 kpc 7.3 kpc 8.2 kpc 10.6 kpc

corotation = 5.5 kpc

I I I I I ~ I I I I I I I I I

2/1 4/1 6/1 8/1

67

Table 3-6:

Parameters used to calculate the surface

density response for the best model of NGC 3992.

Parameter

Minimum grid radius, RMIN (kpc) Maximum grid radius, RMAX (kpc) Radial cell width, DRS (kpc) Number of radial cells Azimuthal cell width (0) Number of azimuthal cells Number of radial start positions per Number of azimuthal start positions

grid cell, NOR per orbit, NOA

Length of (quasi)periodic orbit integration, PN Central velocity dispersion, uO (km s-1)

( ~period

Slope of the velocity dispersion dependence, or (km s-1 kpc-1)

Time of "circular" orbit integration (109 yr)

Grayscale representations of the resulting surface density

response are shown in

Figures 3-9 (unprojected) and 3-10 (projected to the actual orientation of NGC 3992, for comparison with Figure 3-1). These grayscale images were produced by first rebinning

the polar grid of density

values into a Cartesian grid.

The resulting image was then

smoothed using a "beam" five pixels square such that the smoothed pixel derived onethird of its value from the unsmoothed pixel, another one-third from the eight directly

and the

final third from the

sixteen outer pixels (giving a

standard

deviation for an equivalent Gaussian beam of 1.2 pixels). There are a couple of features

to note in these images.

Most obviously we see that the overall barred spiral appearance

of NGC 3992 is well-reproduced by the model. This point is perhaps best appreciated by directly comparing the projected figure (Figure 3-10) with the photograph of NGC 3992

(Figure 3- 1).

Also, we see that model spiral arm makes a slight bend at approximately

Value

0.1

12.1 0.2 60
1.4 128

2
6

12

100

-7

0.98

68

the position of the -4/1 resonance (8.2 kpc).

This feature seems also to be present

in NGC 3992 itself, particularly for the spiral arm in the northeast quadrant.

''elbows' 1

These

were also noted by Patsis et al. (1991) in their models of unbarred spirals.

After the surface response density was calculated, the self-consistency of the model was tested by comparing the response to the imposed surface density. Figures 3-11 and 3-12 give R* and AO, the two measures of self-consistency. These measures, the ratio of the relative amplitude of the response 20 component to the relative amplitude of the imposed 20 component and the phase difference between these two components, respectively, are described in detail in Chapter 2.

We note that the value of R* is almost constant at one (varying between 1.4 and

0.8) between 2.1 and 8.5 kpc. There is an upward divergence in R*

shown in Figure 3-11) due to the fact that the relative azimuthal variation of the imposed density drops rather rapidly inside the length of the bar minor axis, greatly decreasing the relative amplitude of the imposed 20 component, while the shape, and hence the density response, of the dominant 2/1 family remains rather elongated, thereby maintaining the

relative amplitude of the response 20 component.

These combined effects, together

with the possible existence of a nuclear bulge, indicate that, in general, we cannot meaningfully speak of a measureable bar component existing all the way to the center. Here, therefore, and in the cases of NGC 1073 and NGC 1398 also, we take the bar semiminor axis length as a practical inner limit for R*. The divergence near the OLR is due to several factors. First, the orbits here retain their somewhat elongated shape despite the fact that the strength of the imposed spiral has considerably diminished. Secondly, there exists a congestion of matter resulting from the response of the regular

69

-10

-10

-5

0

5

kpc

Figure 3-9: Grayscale image of the unprojected surface

density response of the best model of NGC 3992.

10

C)

0

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-10

10

71

trapped

(with only

a small dispersion

of velocities)

around

the -2/1 family.

Thirdly, and perhaps most importantly, the effect on R* of an azimuthal density variation

a given

absolute

size will be greatly

magnified

in the outer regions

where

axisymmetric component of the density becomes small.

The angle deviation AO is absolutely less than 0.12 radian

between 2.1

and 7.7

kpc. Beyond 7.7 kpc we see that the response spiral systematically lags the imposed up to the -4/1 resonance. Here we see a short sharp increase in AO and then a much steeper decline out to the OLR. This same systematic lag has been observed in similar models

of normal spirals with strong spiral arms

(Contopoulos and Grosbol 1988; Patsis et al.

1991), and seems to indicate that logarithmic spirals may not be the best mathematical description of real spirals.

Ir -- - T I

I I I I

S
0 ** es

I I I I

0

0 a 0 0.

CR

I I I I

I I I I

4

I I

S

I I I I I I I I I I I I I I I I I I I I I I I

S..
6@
0

-4/1 I I I I

6
kpc

8

Figure 3-11: The response -to -imposed ratio R* for the best model of NGC 3992.

20 component amplitude The positions of the major

outer resonances (corotation, -4/1, and outer Lindblad) are noted.

orbits

of

the

I I- -

' I

5

4

3

a

2

0

a

0* 0 009

1

0

0

0

2

OLR

12

10

I I

i I

I

I

I

I I I I

I I I

72

I I I I

I I I I

I I I I

I I I I

I I I i

0

*6986S. *e9

* S S S
S

0

a

0

0

0

0

I I I I

I I I I

2

4

CR

-4/1 I I I I i

6
kpc

a

OLR

II I

10

Figure 3-12: The phase difference AO (in radians) between the response
and imposed 20 components of the best model of NGC 3992. The positions of the major outer resonances are noted as in Figure 3-11.

NGC 1073

Observations

Neutral Hydrogen

In a continuation

of the

effort

initiated

by

GBHH

to study

the properties

neutral hydrogen in a carefully selected sample of barred

was observed by

spiral galaxies, NGC

1073

England, Gottesman, and Hunter (1990, hereafter EGH) at the VLA

between June, 1983 and June, 1984. The effective resolution obtained was 20./13 x 19.117,

with the major axis of the Gaussian beam at a position angle of 62-18.

For comparison

purposes, the diameter of the HI disk is 396" and the position angle of the line of nodes

is -15.'4 0.02.

The FWHP velocity

resolution obtained was

12.63 km s-1

. EGH

givs fll etalsof the VLA data reduction.

I I I

.5

0

-.5

-1

S
0

-1.5

0

0

12

of

I I

I

I

I

gives full details

73

The angle-averaged rotation curve, along with the following orientation parameters of the HI disk, was derived via a least-squares fit to the velocity field: major axis at a position angle of -15.'4 0.02, inclination of 18.'5 2.'5, and a systemic velocity

of 1208.9 0.2 km s-1

.The rotation curve data is given in Table 3-7. Here again,

the quoted uncertainties are simply the formal errors of the least-squares solution. In converting the measured angular radii to a linear scale we have assumed a distance to

NGC 1073

of 13.6 Mpc (de Vaucouleurs 1979).

The rotation curve data in Table 3-7

are plotted

as filled circles with error bars in Figure 3-13.

Table 3-7: HI rotation velocity data for NGC 1073.

R(arcmin) R (kpc) V (km s-1) R (arcmin) R (kpc) V (km s-)

(1) (1) (2)) (1) (2)

0.10 0.30 0.50
0.70 0.90 1.10 1.30 1.50
1.70 1.90
2.10 2.30 2.50

0.39 1.18 1.97 2.76 3.55
4.34 5.12 5.91 6.70
7.49 8.28 9.07 9.85

38.8 7.6 83.2 4.4 96.0 3.2 93.6 2.4 98.8 3.2 99.6 3.2 99.6 2.2 100.0 2.8 105.6 3.2 106.0 3.2 103.2 2.8
104.0 4.4 105.6 6.0

(1) Radius measured from the center of the

2.70 2.90 3.10 3.30 3.50 3.70 3.90
4.10 4.30 4.50 4.70 4.90

10.6

11.4 12.2 13.0 13.8
14.6 15.4 16.2 17.0 17.7 18.5 19.3

100.8 5.6 95.2 6.6
92.0 7.7 100.8 8.4 91.2 12.0 69.6 26.4 74.4 28.3 61.6 31.6 24.0 33.0 72.0 44.4 120.0 52.4 168.0 53.6

galaxy

(2) Rotation velocity

74

EGH find that the distribution of the neutral hydrogen in NGC 1073 is that of an almost circular disk, with a nearly complete ring surrounding a central depression;, however, this central depression is not as deep relative to the outer disk as in the case of NGC 3992. They also note the presence of a gas bar in NGC 1073 aligned with the

optical bar.

This feature shows up at around 30% of the peak HI density, compared

to the general disk gas density which is approximately 20% of peak. EGH find little evidence of spiral structure in the gas, however, in sharp contrast to the optical image in which there are two prominent spiral arms starting about 300 in azimuth from the ends of the bar. While the regions of high HI density (>50% of the peak density) show a correlation with the bright optical regions of the spiral arms, a broad gaseous ring

emerges when lower HII densities (-40% of peak) are considered.

This ring extends

from about r = 0.'7 to r = 2.'0, compared to the optical spiral arms which extend from about r = 0.'75 to r = 1.'1. Finally, EGH note a steep gradient in the gas density in the

While they postulate that this feature might be due

to an interaction, a search for 21 cm emission in a 1.05 x 1L'5 area of sky surrounding NGC 1073 yielded no evidence of such an interacting object.

The gas kinematics of NGC 1073 is dominated by circular motion. Still, irregularities in the velocity field due to gas streaming are observed where the optical spiral arms cross the isovelocity contours (EGH 1990). Like NGC 3992, NGC 1073 may possess a truncated disk. EGH cite as evidence for this the dropoff in the rotation curve starting at approximately r = 10 kpc, the corresponding drop in the angle-averaged, deprojected HI surface density, and the success achieved by truncated mass models in reproducing the rotation curve of NGC 1073 (cf. Casertano 1983; Hunter et al. 1984).

75

8 10

Figure 3-13:

Table 3-8:

Comparison of observed and theoretical rotation curves for NGC 1073.

Selected global and disk parameter values adopted for NGC 1073. Here again, the

errors listed are simply the formal errors of a least-squares analysis of the velocity field.

Parameter

Value

Systemic velocity Distance (Mpc)b

(heliocentric, km s-~

1208.9 0.2

13.6

Inclination angle (0')'

Position angle, line of nodes (')'

'England, Gottesman, and Hunter (1990)

bde Vaucouleurs

(1979)

Surface Photometry

A s for NG C

3992,

Elmegreen

and Elmegreen

(1985)

have provided B

andlI

passband surface photometry of NGC

1073.

They

determined the near-infrared disk

scale length of NGC 1073 to be 3.18 0.36 kpc assuming a distance of 13.2 Mpc.

For

our assumed distance of 13.6 Mpc, the scale length becomes 3.28 0.37 kpc, yielding

120 100

I I j I I I l i i I I I I I

dis

U
U2
S

-J
U
C
U

B0 60

40

20

0

0

2

4

6

12

14

16

18

18.5 2.5
-15.4 0.2

76

IThe initial value of the inverse scale

length of the spiral, derived from the slope of the spiral arm-interarm surface brightness difference given by BE, is 1.4 kpc- .

EGH have analyzed the bar figure of NGC 1073 in the infrared image taken by BE. Following the same procedure described for the case of NGC 3992, they find that the angle 40 between the major axis of the projected bar and the line of nodes is 75'. Also, they find that the bar has a semimajor axis = 43" and an apparent axial ratio /3o

= 7.19.

The actual linear dimensions of the bar derived from the Stark method are a

= 2.95 kpc, b = 0.39 kpc, and c = 0.295 kpc. The initial value Of the spiral arm Pitch angle, determined by a visual fit to the optical photograph of NGC 1073 reproduced in EGH from ATp and Sulentic (1979), is -100. Table 3-9 summarizes the quantities derived from the surface photometry.

Table 3-9: Photometrically derived parameter values for NGC 1073.

Parameter

Value

Inverse disk scale length (kpc1 )a Inverse spiral scale length (kpc1 )a Angle between projected bar major axis and disk line of nodes (,)b

Apparent bar semimajor axis (arcsec)b Apparent bar axial ratio b Long bar axis length (kpc)c Intermediate bar axis length (kpc)c Short bar axis length (kpc)c

Spiral arm pitch angle (~

0.305 +0.039 1.4 0.031

75

43

7.19 2.95 0.39
0.295

-10

aElmegreen and Elmegreen (1985) b England, Gottesman, and Hunter (1990) 'derived using method of Stark (1977) d visual fit

an inverse disk scale length of 0.305 +0*039 kpc-1
-0.031

77

Best Model

The best model for NGC 1073 was determined following the general procedure outlined for the case of NGC 3992. Table 3-10 summarizes the parameters of this model.

Table 3-10: Parameter values for the best self-consistent model of NGC 1073.

Ferrers B ar

MB= 1.8 x 109M0D

a = 2.95 kpc b = 0.39 kpc c = 0.295 kpc

Qp= 32.2 km s-1 kpc-1

Spiral

A= 9000 km 2 s 2 kpc-1 A=0 km 2 S2 kpc-1 ,s,=1.2 kpc-1, i0= -10' r, 2.95 kpc, r2 = 5.63 kpc IK1=K2= 0.5, A =0. 1kpc

Exponential Disk

co -- 250MO PC-2

6o= 0.305 kpc-1

The rotation curve of this

Plummer Sphere Halo

MA"H = 1.-0 X 10'0 l

bH =9 kpc

model is given along with the observations in Figure

3-13. The characteristics of the orbit families which comprise this model are shown in Figure 3-14. The most dynamically important periodic orbits, that is, those that are most effective in trapping quasi-periodic orbits around them, are the two stable branches of the 2/1 resonant orbits in the bar and the -2/1 resonant orbits in the outer disk (Figure

3-15).

Two sets of "circular" orbits are also included, a sizeable one extending from

r= 1.25 kpc to r, = 4.75 kpc and a smaller second one extending from r, = 5.25 kpc to r, = 5.6 kpc. As in the case of NGC 3992, the inclusion of these "circular" orbits does not imply that no periodic orbits exist in this region. We shall explore further the phase space structure of this region in Chapter 5. Table 3-11 gives the positions of the

main resonances in this case.

78

6

U

af

21

11

0

I

2

:3
rc (kpc)

4

5

6

Figure 3-14: Characteristics of the orbit families included in the best model of NGC 1073.

-6

-4

-2

0
kpc

2

4

6

Figure 3-15: Representative orbits of the three main periodic families in the best model of NGC 1073.

-3 /1 circular circular 2/11

0

I I I j I I 111111111 ~ I I I I

6

4

2

C.)

0

-2

-4

-6

79
Table 3-11: Resonance locations of the best model of NGC 1073.

Inner Resonances Outer Resonances

2/1 0.0 kpc -8/1 3.6 kpc
411 1. 8 kpc -6/1 3.8 kpc
6/1 2.2 kpc -4/1 4.3 kpc
8/1 2.4 kpc -2/1 5.6 kpc
corotation= 2.95 kpc

Table 3-12 gives the parameters of the surface density response calculation.

Table 3-12: Parameters used to calculate the surface density response for the best model of NGC 1073.

Parameter Value

0.02 6.02

Minimum grid radius, RMIN (kpc) Maximum grid radius, RMAX (kpc) Radial cell width, DRS (kpc) Number of radial cells Azimuthal cell width (0) Number of azimuthal cells Number of radial start positions per grid cell, NOR Number of azimuthal start positions per orbit, NOA

Length of (quasi)periodic orbit integration, PN (j-lo) Central velocity dispersion, o-o (km s-1) Slope of the velocity dispersion dependence, a, (km s-1 kpc-1

Time of circulara" orbit integration (109

yr)

0.1 60
1.4 128

2
6

12 40

-5

)

0.98

An unprojected grayscale representation of the surface density response is given

in Figure 3-16.

Since the inclination of the galaxy is only 18.'5 (i.e.

it is nearly face

on), a projected image is not given here. overall barred spiral structure is well-reprc

As in the case of NGC 3992, we see that the )duced. Also, in contrast to NGC 3992 and its

/ - - l ,

N

80

-4

-2

0

2

kpc

Figure 3-16: Grayscale image of the unprojected surface

density response of the best model of NGC 1073.

6

4

2

C)

0

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-2

-4

-6

-6

4

6

81

best model,

the spiral

arms of this

model

do

not exhibit

any

noticeable ''elbows."

Furthermore, it is observed that the model arms tend to wrap back onto themselves after

winding approximately 270' from their origin at the ends of the bar.

This "'wrapping"~

occurs at about the radius of the OLR. One note of discrepancy between the model and observations exists, though. While the actual spiral arms of NGC 1073 are observed to begin some 300 in azimuth from the ends of the bar (EGH 1990), the arms of the best

model come directly off the ends of the

bar.

I I I I I I I I

I I I I I I I I I I I I I I I I I I

4

0

S
S

0

0

0S* *.

S 0.

0

-4/1 I I I

I I I

I I I I I I -L

I

09 00 000 00 00006

CR

2

0
egos.

I I I

3
kpc

I I I

4

0

OLR

5

6

Figure 3-17: The response -to -imposed ratio R* for the best model of NGC 1073.

20 component amplitude The positions of the major

outer resonances (corotation, -4/1, and outer Lindblad) are noted.

Figures 3-17 and 3-18 give R* and AO for the best model of NGC 1073.

Again

we see the same general features here as in the case of NGC 3992. The value of R* is

almost constant (varying between 0.8 and 1.3) from 0.6 kpc to 4.1 kpc.

There is also,

for the same reasons as in the model of NGC 3992, an upward divergence in R * at large

5

4

3

9

2

0

I

0

7--r-

I

82

radii. The angle deviation AO remains absolutely less than 0.12 radian from 0.39 kpc

out to 3.9 kpc.

Again we

see a gradual systematic lag in the response 20 component

relative to the imposed from the end of the bar at corotation out to approximately the

-4/1resonance.

as sharply

Beyond this point AO drops more rapidly out to the OLR, albeit not

as in the case of NGC

3992.

I I I I j I I - I I

I I I I I I I

I I I I I 1I - I 1 1

1

I I- I

.0
0
0.

* 0see

S
0
S

0

so.

I I I I

I I I I

1

CR

I I I L

2

I I I I

3
kpc

-4/1 OLR

4

5

Figure 3-18: The phase response and imposed 20

difference A\O (in radians) between the components of the best model of NGC

1073. Again, the positions of the major outer resonances are noted.

NGC 1398

Observations

Neutral Hydrogen

Moore and Gottesman (1993) made high resolution HI observations of NGC 1398 at the VLA between February, 1991 and February, 1992. The effective linear resolution

9000400000000000 0000 ee* 0

.5

0

*0 0 .S

0

6

I I I I I

T--T

I I

I

Full Text

PAGE 1

SELF-CONSISTENT MODELS OF BARRED SPIRAL GALAXIES DAVID EUGENE KAUFMANN 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 1993

PAGE 2

To Mom, Dad, Greg, Eric, Elizabeth, and Carol

PAGE 3

ACKNOWLEDGMENTS With this dissertation my education has, in two very different senses, both drawn to a close and started anew. Although there will be no more classes, exams, or theses, I have just begun to learn. Many special people have helped to bring me this far. Dr. George Contopoulos, the de facto advisor of my thesis, not only has taught me essentially all that I know about the fascinating topics of nonlinear dynamics and chaos, he has also provided an unmatched role model for this young scientist. I personally know of no other person who strives harder to find truth. To this goal I also dedicate myself. Drs. James Hunter and Stephen Gottesman helped me greatly during the course of my research, especially during the periods of Dr. ContopoulosÂ’ absence. Dr. Hunter, who served as chairman for this dissertation, instilled in me the value of balancing purely computational results with the underlying theory. He taught me that one is of limited value without the other. Dr. Gottesman, who acted as cochairman for the dissertation, impressed upon me the need for theoretical results to address the observations and kept me from getting absorbed in the purely abstract. After all, there is the real world. Dr. Neil Sullivan, the final member of my committee, and Dr. James Fry, a member until the Fall of 1992, allowed me the freedom to conduct this research as I 111 deemed fit. For that I thank them.

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PAGE 5

Finally and most importantly, I would like to thank my mother Betty, my father John, my brothers Greg and Eric, and their respective wives Elizabeth and Carol. They have always stood behind me with unwavering support and encouragement. To them I dedicate this dissertation.

PAGE 6

TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTERS 1. INTRODUCTION 2. MODELING TECHNIQUES The Method of Contopoulos and Grosb0l (CG Method) The CG Method Modified for the Case of Barred Spiral Galaxies Gas Response Using Smoothed Particle Hydrodynamics (SPH) . 3. PROGRAM GALAXIES NGC 3992 Observations Best Model NGC 1073 Observations Best Model NGC 1398 Observations Best Model 4. VARIATION OF PARAMETERS Variation of A Variation of Ar Variation of io Variation of es Variation of Hp Variation of A Â’ Variation of ^2 Variation of ri, r2, ki, k 2 , and A4 Variation of the Bar Semiaxes a, b, and c Variation of the Bar Mass Mg Variation of cq and eo Ill Vlll . X xvii . . . 1 . . 11 . . 11 . . 31 . . 43 . . 50 . . 57 . . 57 . . 61 . . 72 . . 72 . . 77 . . 82 . . 82 . . 86 . . 94 . . 94 . . 96 . . 97 . . 97 . . 98 . 102 . 103 . 105 . 108 . 108 . 109 VI

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Variation of
PAGE 8

LIST OF TABLES 2-1 Initial conditions of the periodic orbit and the eight nonperiodic orbits used to calculate the effect of the velocity dispersion 24 2-2 The values of the Xj and wvi of Table 2-1 24 23 Adjustable parameters in the axisymmetric rotation curve fitting procedure. . 39 31 HI rotation velocity data for NGC 3992 58 3-2 Selected global and disk parameter values adopted for NGC 3992. The errors given are simply the formal errors of a least-squares analysis of the observed velocity field and do not imply that these quantities have been determined to this level of precision 60 3-3 Photometrically derived parameter values for NGC 3992 62 3-4 Parameter values for the best self-consistent model of NGC 3992 63 3-5 Resonance locations of the best model of NGC 3992 66 3-6 Parameters used to calculate the surface density response for the best model of NGC 3992 67 3-7 HI rotation velocity data for NGC 1073 73 3-8 Selected global and disk parameter values adopted for NGC 1073. Here again, the errors listed are simply the formal errors of a least-squares analysis of the velocity field 75 3-9 Photometrically derived parameter values for NGC 1073 76 3-10 Parameter values for the best self-consistent model of NGC 1073 77 3-11 Resonance locations of the best model of NGC 1073 79

PAGE 9

3-12 Parameters used to calculate the surface density response for the best model of NGC 1073 79 3-13 HI rotation velocity data for NGC 1398 84 3-14 Selected global and disk parameter values adopted for NGC 1398 85 3-15 Photometrically derived parameter values for NGC 1398 87 3-16 Parameter values for the best self-consistent model of NGC 1398 87 3-17 Resonance locations of the best model of NGC 1398 89 3-18 Parameters used to calculate the surface density response for the best model of NGC 1398 90 5-1 Summary of basic stochastic orbit types and their behaviors 137 5-2 Estimated breakdown of the mass-weighted orbit populations comprising the models of Chapter 3 according to type (trapped versus stochastic) and location (bar, disk, or both). The errors in the cited figures are somewhat uncertain, but are estimated to be of the order of 5% 143 IX

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LIST OF FIGURES 2-1 The effect of varying Sb on the shape of the model rotation curve given by Eq. (2-1): (1) Â£b = (2) Â£b = (3) Â£b = 2Â£di (4) Â£b = Â£dIn all cases fj, = 1. 13 2-2 The effect of varying fj, on the shape of the model rotation curve given by Eq. (2-1): (1) ff, = 0; (2) f^ = 0.5; (3) fb = 1; (4) f,, = 2. In all cases Â£b = 5ed14 2-3 Rotation curves of the exponential disk (solid curve) and a point with the same total mass (dotted curve) 33 2-4 Rotation curves of the Plummer sphere (solid curve) and a point with the same total mass (dotted curve) 35 25 Three normalized Â“axisymmetricÂ” bar rotation curves for cases where a = 1 and c = 0.1: (1) b = 0.15, (2) b = 0.3, and (3) b = 0.6. The dotted line represents the Keplerian rotation curve for a point mass equal to Mg 37 31 NGC 3992 (NASA Atlas of Galaxies Useful for Measuring the Cosmological Distance Scale 1988) 52 3-2 NGC 1073 (NASA Atlas of Galaxies Useful for Measuring the Cosmological Distance Scale 1988) 54 3-3 NGC 1398 (Sandage 1961, The Hubble Atlas of Galaxies) 56 3-4 Comparison of observed and theoretical rotation curves for NGC 3992. The theoretical curve is derived from our most successful model of NGC 3992. Also shown are the contributions of the separate components of this model to the total theoretical rotation curve 59 3-5 Characteristics of the orbit families included in the model. Each characteristic plots X, where a given orbit crosses the minor bar axis b, as a function of Jacobi constant, as parameterized by rc 63 3-6 The 2/1 family of periodic orbits in the model of NGC 3992. The darker circle represents corotation at 5.5 kpc 65

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3-7 The 4/1 family of periodic orbits in the model of NGC 3992 65 3-8 The -2/1 family of periodic orbits in the model of NGC 3992. The darker curves represent the minima of the bar and spiral potentials 66 3-9 Grayscale image of the unprojected surface density response of the best model of NGC 3992 69 3-10 Grayscale image of the surface density response of the best model of NGC 3992 projected to the galaxyÂ’s actual orientation 70 3-1 1 The response-to-imposed 20 component amplitude ratio R* for the best model of NGC 3992. The positions of the major outer resonances (corotation, -4/1, and outer Lindblad) are noted 71 3-12 The phase difference A0 (in radians) between the response and imposed 26 components of the best model of NGC 3992. The positions of the major outer resonances are noted as in Figure 3-11 72 3-13 Comparison of observed and theoretical rotation curves for NGC 1073. ... 75 3-14 Characteristics of the orbit families included in the best model of NGC 1073. . 78 3-15 Representative orbits of the three main periodic families in the best model of NGC 1073 78 3-16 Grayscale image of the unprojected surface density response of the best model of NGC 1073 80 3-17 The response-to-imposed 20 component amplitude ratio R for the best model of NGC 1073. The positions of the major outer resonances (corotation, -4/1 , and outer Lindblad) are noted 81 3-18 The phase difference A6 (in radians) between the response and imposed 29 components of the best model of NGC 1073. Again, the positions of the major outer resonances are noted 82 3-19 Comparison of observed and theoretical rotation curves for NGC 1398. The contributions of the separate model components are also shown 84 3-20 Characteristics of the orbit families included in the best model of NGC 1398. . 88 XI

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3-21 Representative orbits of the three main periodic families in the best model of NGC 1398 89 3-22 Grayscale image of the unprojected surface density response of the best model of NGC 1398 91 3-23 The response-to-imposed 29 component amplitude ratio R* for the best model of NGC 1398. The positions of the major outer resonances (corotation, -4/1, and outer Lindblad) are noted 92 324 The phase difference A.9 (in radians) between the response and imposed 29 components of the best model of NGC 1398. Again, the positions of the major outer resonances are noted 93 4Â— 1 The ratio R* and phase difference A9 in models of NGC 3992 where A = 2000 km^sÂ“^kpcÂ“' (Â“bestÂ” model, denoted by filled circles), A = 1000 km^s"^kpc~^ (open circles), and A = 4000 km^s"^kpcÂ“^ (plusses) 95 4-2 The ratio R* in models of NGC 1073 where the residual amplitude Ar = 0 km^sÂ”^kpcÂ“^ (Â“bestÂ” model, filled circles), 1000 km^sÂ“^kpc~^ (open circles), and 2000 km^sÂ“^kpc~^ (plusses). For comparison, the primary amplitude A = 9000 km^sÂ“^kpcÂ“^ 96 4Â—3 The ratio R* and phase difference A9 in models of NGC 3992 where io = -10Â° (Â“bestÂ” model, denoted by filled circles), io = -5Â° (open circles), and io = -15Â° (plusses) 99 4Â—4 The ratio R* and phase difference in models of NGC 3992 where Â£s = 0-2 kpcÂ“^ (open circles, A = 1213 km^sÂ“^kpc~^), 0.4 kpcÂ”^ (filled circles, Â“bestÂ” model), and 0.8 kpcÂ“^ (plusses, A = 5437 km^sÂ“^kpcÂ“^) 100 4Â—5 Amplitude ratio R* and phase difference A9 in models of NGC 3992 where flp = 34.7 km sÂ“*kpcÂ“^ (corotation = 4a/3, open circles), flp = 43.6 km sÂ”^kpc~^ (Â“bestÂ” model, corotation = a, filled circles), and Op = 55.7 km s"^kpcÂ“^ (corotation = 3a/4, plusses) 101 4Â—6 The ratio R* in models of NGC 3992 where A = 0.01 kpc (open circles), 0.1 kpc (filled circles, Â“bestÂ” model), and 1.0 kpc (plusses) 102 4Â—7 The ratio R* and phase difference A9 for models of NGC 3992 in which 92 = 0Â° (Â“bestÂ” model, filled circles), 15Â° (open circles), and 30Â° (plusses) 104 Xll

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4-8 The ratio R* in models of NGC 3992 where ri = 1.5 kpc (Â“bestÂ” model, filled circles), 0.5 kpc (open circles), and 2.5 kpc (plusses) 105 4-9 The ratio R* in models of NGC 1073 where ri = 2.95 (Â“bestÂ” model, filled circles), 2.0 kpc (open circles), and 3.5 kpc (plusses) 106 4-10 The ratio R* and phase difference for models of NGC 3992 in which Mg = 1.5 X lO^^MÂ© (Â“bestÂ” model, filled circles), 1.5 x 10Â®MÂ© (open circles), and 7.5 x 10^ MÂ© (plusses) 110 4-11 The ratio R* in models of NGC 3992 where Cq = 750MÂ©/pc^ (Â“bestÂ” model, filled circles), 500 A/Â© /pc^ (open circles), and 1000A/Â©/pc^ (plusses). . . Ill 4-12 The ratio R* in models of NGC 3992 where cq = 0.235 kpc~^ (Â“bestÂ” model, filled circles), 0.157 kpcÂ“* (open circles), and 0.353 kpcÂ“^ (plusses) 112 4Â—13 The ratio R* and phase difference in models where ao = 100 km s~\ err = -7.0 km sÂ“^kpcÂ“^ (Â“bestÂ” model, filled circles), ao = 30 km s~^
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5-8 More surfaces of section in the model of NGC 1398. Top: Jacobi constant equals -299940 km^ sÂ“^. Bottom: Jacobi constant equals -294526 km^ sÂ“^. . 131 5-9 More surfaces of section in the model of NGC 1398. Top: Jacobi constant equals -290724 km^ sÂ”^. Bottom: Jacobi constant equals -288987 km^ sÂ“^. . 132 5-10 Schematic drawing showing how loops in the orbits outside of corotation generate consequents on the surface of section. A portion of an orbit containing such a loop is shown, together with arrows indicating the direction of motion along the orbit. The drawn circle represents corotation, and the directions of the general motion are also indicated with arrows. . . 134 5-1 1 Stochastic orbit trapped within the bar. The value of its Jacobi constant (Ej = -191307 km^ sÂ“^) is slightly less than that of the Lagrange points Li and L 2 (Ej = -191188 km^ sÂ“^). The darker circle represents corotation at 5.5 kpc. 137 5-12 Stochastic orbit confined to the outer disk. The value of its Jacobi constant (Ej = -194202 km^ s~^) is also less than that of Li and L 2 . The bar and spiral potential minima are drawn for reference 138 5-13 Stochastic orbit which traverses both the bar and the outer disk. The value of its Jacobi constant (Ej = -189692 km^ sÂ“^) is greater than that of Li and L 2 , but less than that of L 4 and L 5 (Ej = -187300 km^ sÂ“^) 139 5-14 Stochastic orbit, whose Jacobi constant (Ej = -189476 km sÂ“^) is approximately the same as the orbit of Figure 5-13, calculated in a spiral potential of ten times greater amplitude than that of Figure 5-13 140 515 Stochastic orbit whose Jacobi constant (Ej = -1830(X) km^ sÂ“^) exceeds that of L 4 and L 5 and is energetically unconstrained 141 61 The gas response in the best model of NGC 3992 with the bar/spiral imposed. Top panel: 1.5 pattern rotations. Bottom panel: 2.4 pattern rotations 149 6-2 The gas response in the best model of NGC 3992 with the bar/spiral imposed (continued). Top panel: 3.3 pattern rotations. Bottom panel: 5.6 pattern rotations 150 6-3 The gas response in the best model of NGC 3992 with the bar/spiral imposed (continued). Top panel: 8.1 pattern rotations. Bottom panel: 10.4 pattern rotations 151 XIV

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6-4 The gas response in the best model of NGC 3992 with only the Ferrers bar present. Top panel: 1.5 pattern rotations. Bottom panel: 2.5 pattern rotations 152 6-5 The gas response in the best model of NGC 3992 with only the Ferrers bar present (continued). Top panel: 3.5 pattern rotations. Bottom panel: 5.3 pattern rotations 153 6-6 The gas response in the best model of NGC 3992 with only the Ferrers bar present (continued). Top panel: 7.6 pattern rotations. Bottom panel: 10.6 pattern rotations 154 6-7 The gas response in the best model of NGC 1073 with the bar/spiral imposed. Top panel: 1.1 pattern rotations. Bottom panel: 2.8 pattern rotations. . . . . 155 6-8 The gas response in the best model of NGC 1073 with the bar/spiral imposed (continued). Top panel: 4.5 pattern rotations. Bottom panel: 5.4 pattern rotations 156 6-9 The gas response in the best model of NGC 1073 with the bar/spiral imposed (continued). Top panel: 6.8 pattern rotations. Bottom panel: 8.3 pattern rotations 157 6-10 The gas response in the best model of NGC 1073 with only the Ferrers bar present. Top panel: 1.1 pattern rotations. Bottom panel: 2.6 pattern rotations 158 6-11 The gas response in the best model of NGC 1073 with only the Ferrers bar present (continued). Top panel: 4.5 pattern rotations. Bottom panel: 5.5 pattern rotations 159 6-12 The gas response in the best model of NGC 1073 with only the Ferrers bar present (continued). Top panel: 7.3 pattern rotations. Bottom panel: 8.3 pattern rotations 160 6-13 The gas response in the best model of NGC 1398 with the bar/spiral imposed. Top panel: 1.4 pattern rotations. Bottom panel: 3.3 pattern rotations. ... 161 6-14 The gas response in the best model of NGC 1398 with the bar/spiral imposed (continued). Top panel: 4.1 pattern rotations. Bottom panel: 5.0 pattern rotations 162 XV

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6-15 The gas response in the best model of NGC 1398 with the bar/spiral imposed (continued). Top panel: 7.2 pattern rotations. Bottom panel: 11.1 pattern rotations 163 6-16 The gas response in the best model of NGC 1398 with only the Ferrers bar present. Top panel: 1.9 pattern rotations. Bottom panel: 3.1 pattern rotations 164 6-17 The gas response in the best model of NGC 1398 with only the Ferrers bar present (continued). Top panel: 4.2 pattern rotations. Bottom panel: 5.3 pattern rotations 165 6-18 The gas response in the best model of NGC 1398 with only the Ferrers bar present (continued). Top panel: 7.5 pattern rotations. Bottom panel: 11.1 pattern rotations 166 XVI

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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 SELF-CONSISTENT MODELS OF BARRED SPIRAL GALAXIES By David Eugene Kaufmann August, 1993 Chairman: James H. Hunter, Jr. Cochairman: Stephen T. Gottesman Major Department: Astronomy Self-consistent models of barred spiral galaxies based on the observed properties of NGC 3992, NGC 1073, and NGC 1398 are constructed and analyzed. The method of model construction is a slight modification of the technique developed by Contopoulos and Grosbpl for the case of unbarred spirals. The main factors which influence selfconsistency are the amplitude, pitch angle, scale length and z-thickness of the spirals, the mass of the bar, the angular velocity of the bar/spiral pattern, the central surface density and scale length of the disk, and the central value and slope of the velocity dispersion. Stochastic orbits whose Jacobi constants lie between the values at the Lagrange points L\ and L 4 are found to play a significant role in supporting self-consistent spiral structure, especially in the regions just beyond the ends of the bar. Stochastic orbits whose Jacobi constants lie below this interval tend to fill more or less uniformly either rings in the outer disk or ovals in the bar region, depending on the regions to which they are confined. Stochastic orbits whose Jacobi constants lie above that of L 4 also tend not to support any imposed structure. The model bars are predominantly comprised of elongated orbits trapped around the X\ family and terminate close to corotation. XVll

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The response of gas to the forces of the most successful models is calculated using a two-dimensional smoothed particle hydrodynamics code. The results confirm that a bar alone is not sufficient to drive a strong spiral response in the gas of the outer disk. An underlying spiral structure in the more massive stellar component appears to be required. If stellar spirals are present, strong gas spirals may persist for long times. xviii

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CHAPTER 1 INTRODUCTION Spiral structure in galaxies was first observed by William Parsons, Third Earl of Rosse, in several nebulae that had been catalogued earlier by William Herschel. Because this discovery, made in the mid-nineteenth century, preceded the establishment of these nebulae as separate galaxies external to the Milky Way, they became known simply as spiral nebulae. In addition, their spiral shapes seemed to suggest that the systems rotate about an axis perpendicular to the plane of the spiral. Confirmation of this hypothesis was provided in 1914 by Vesto Slipher, whose spectroscopic observations of a number of these spiral nebulae revealed the Doppler shifts produced by rotation. The nature of the spiral nebulae, one of the subjects of the so-called Â“great debateÂ” between Huber D. Curtis and Harlow Shapley at the National Academy of Sciences in April 1920, was firmly established in the early 1920s by Edwin Hubble. In 1923, Hubble, using the 100-inch telescope at Mount Wilson, was able to resolve the outer parts of the spiral nebulae M3 1 and M33 into multitudes of apparent point sources. Although at first unable to determine whether these point sources were individual stars, he soon found that some were indeed Cepheid variables. Using the Cepheid period-luminosity relation, originally discovered in 1912 by Henrietta Leavitt, Hubble was able to show conclusively that M31 and M33 are at very large distances, and therefore galaxies external to the Milky Way. 1

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2 In 1926, Bertil Lindblad, working within ShapleyÂ’s concept of the Milky Way (i.e. that the SunÂ’s position is generally in the galactic plane but relatively far from the galactic center), developed a mathematical model for its rotation. This model depicted the galaxy as consisting of a number of subsystems, each rotating with its own characteristic angular velocity and degree of flattening about a common axis. Jan Oort, in 1927 and 1928, demonstrated the basic correctness of LindbladÂ’s model and showed that the galactic disk is in a state of dijferential rotation. That is, stars closer to the galactic center orbit around it with higher angular velocities than those farther out in the disk. At about the same time Lindblad began to try to understand the nature of spiral structure, a problem he worked on until his death in 1965. Perhaps the simplest interpretation of galactic spiral structure is that it represents a pattern of material arms that rotate as a rigid body about the galactic center. Differential rotation, however, presents this interpretation with a serious difficulty called the winding dilemma. Consider, for example, a galactic spiral arm of uniform pitch angle I'o = -30Â° at time r = 0. Also, suppose that the galaxyÂ’s angular rotation rate fl depends on radius r (i.e. the galactic disk rotates differentially). The azimuth of the arm at radius r and time t is given by ( 1 1 ) The pitch angle i at radius r and time t is defined by ( 1 2 ) For a typical galaxy with a flat rotation curve, Q,{r)r ^ 200 km s \ r w 10 kpc, and t PS 10^Â® yr. In this time the pitch angle i would have decreased from the original -30Â°

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3 to a value of -0.Â°28. Therefore, any material arms would have wound up and rendered the spiral structure unrecognizable on timescales comparable to the age of the galaxy. Binney and Tremaine (1987) summarize four possible solutions to the winding dilemma. Firstly, spiral structure may represent a statistical equilibrium in which the age of any given arm is quite young. The idea here is that clumpy features are continuously produced in the galactic disk and are sheared off into spiral segments by differential rotation. A spiral arm, then, would simply represent an aggregation of these shorter spiral segments. While this chaotic spiral arm hypothesis is believed to be applicable to flocculent galaxies (e.g. Toomre 1989), it has difficulty accounting for the striking global coherence exhibited by the spiral patterns of grand design galaxies. Secondly, spiral structure may be the temporary result of a recent tidal interaction with a companion galaxy. In this case the companion excites a transient global spiral density wave in the primary galaxy. The nature of density waves is discussed in more detail below. Models of this type are able to reproduce many features observed in grand design galaxies, such as dust lane location and strength as well as the main properties of the neutral hydrogen distribution and velocity field. According to Binney and Tremaine (1987, p. 394), however, Â“they cannot account for all spirals because encounters with massive companion galaxies in favorable orbits are not common enough.Â” Thirdly, spiral structure may represent a detonation wave of star formation, driven by supernovae explosions or expanding HII regions, that propagates around the disk. The wave is sheared into a trailing spiral by differential rotation and ultimately settles down with a fixed shape and pattern speed. This self-propagating star formation hypothesis, though, requires finely tuned star formation rates and does not adequately address the problems

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4 of the broad arms of the older disk components, the very regular patterns seen in some grand design galaxies, and the streaming observed along spiral arms. Finally, spiral structure may represent underlying waves in the density and gravitational potential of the disk itself. The orbiting stars and gas adjust their motions such that they tend to linger near the minima of the potential; moreover, the gas in these regions is compressed, thereby inducing rapid star formation that generates the bright young stars and HII regions which delineate the visible spiral pattern. While the material that constitutes the density waves is constantly changing, the waves themselves are neutrally stable and represent a quasi-stationary spiral pattern that rotates as a rigid body with angular velocity Q.p. Bertil Lindblad (1961, 1963) originally proposed such a scenario, but his approach, which emphasized the role of individual stellar orbits, was not well-suited for quantitative global analysis. C. C. Lin and Frank Shu (1964) first provided the mathematical formalism of the spiral density-wave theory. Therefore, the idea that spiral structure represents a quasi-stationary density-wave has become known as the Lin-Shu hypothesis. Since the present thesis concerns the structure of barred spiral galaxies, we must also consider those phenomena associated with the presence of a bar. Bars are found in a large fraction of disk galaxies. In fact, depending upon the reference catalog cited, between 25% and 35% of all disk galaxies are classified as strongly barred (Sandage and Tammann 1981; de Vaucouleurs et al. 1976; Nilson 1973). An additional fraction, about whose size there is greater disagreement, is classified as possessing weak bars or oval distortions. The class of barred galaxies is very heterogeneous. In it we find galaxies that span the entire range of Hubble types. In addition, Â“other properties, such as the size of the bar relative to the host galaxy, the

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5 degree of overall symmetry, the existence of rings and numbers (and position relative to the bar) of spiral arms in the outer disc, the gas and dust content, etc., vary considerably from galaxy to galaxyÂ” (Sellwood and Wilkinson 1992, p. 1). Observations of bars generally concern either the light distribution or the velocity field. Analyses of the light distributions reveal that bar radii (i.e. semimajor axes) are always less than the disk radii (defined, for example, by the radius, R25, at which a galaxyÂ’s surface brightness falls to 25 mag arcsec"^) of the host galaxies. Also, bar radii are typically shorter relative to disk radii in late-type galaxies than in early-type galaxies. In the rather diverse sample of barred spirals considered by Elmegreen and Elmegreen (1985), bars provide anywhere from -20% to -40% of the total luminosity within the radius of the bar, and from a few to -20% of the total luminosity within R25. Moreover, bars in early-type galaxies are generally stronger, more rectangular, and possess flatter major axis luminosity profiles than bars in late-type galaxies. The latter tend to be more elliptical in shape and possess light profiles that are centrally peaked and fall off exponentially along the bar major axis. Observations of the velocity fields of barred galaxies indicate that there exist significant noncircular streaming motions along the bar major axis (e.g. Kormendy 1983; Bettoni et al. 1988; Jarvis et al. 1988). This result is consistent with the idea, discussed in more detail below, that bar structure is dominated by orbits elongated along the bar. Comparison of bar radii with the azimuthally averaged rotation curves of their host galaxies indicates that early-type galaxies tend to have bars that extend beyond the rising parts of the rotation curves, while bars in late-type systems tend to end at or before the rotation curves peak or level out (Elmegreen and Elmegreen 1985). Typical

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6 values of the central stellar velocity dispersion are from 150 to 200 km sÂ“^, with radial profiles of the velocity dispersion vtuÂ’ying from flat to sharply falling (Sellwood and Wilkinson 1992). Finally, estimates of the angular rotation rates of bars can be made by matching the observed gas flow patterns with hydrodynamical models in which the bar tumble rate is varied (e.g. Athanassoula 1992). Although much work remains to be done concerning the structure and evolution of barred galaxies, our understanding of the origin of bars seems to be quite advanced. Some of the earliest A-body simulations of collisionless, rotationally supported stellar disks dramatically exemplified the bar instability (e.g. Miller and Prendergast 1968; Hockney and Hohl 1969; Hohl 1971). Shu (1970) and Kalnajs (1971) formulated the problem of the stability of axisymmetric stellar disks in terms of normal modes. The complete solution, however, is rather lengthy, and the shapes and eigenffequencies of the unstable modes have been calculated only in a small number of cases; nevertheless, A-body simulations of a few of these cases have yielded dominant unstable modes in close agreement with the theory (e.g. Sellwood and Athanassoula 1986). Toomre (1981) proposed that the bar instability is driven by the positive feedback of swingamplified waves. The idea here is that any initial leading disturbance will propagate out to corotation, where it will unwind and become a trailing disturbance. As it unwinds, it is swing-amplified by the coordination of the local epicyclic frequency with the rate of unwinding of the leading wave. The amplified trailing wave then propagates inward, and if it is able to reach the center, it will be reflected as an outgoing leading wave, thereby starting the amplification/reflection process over again. Wave action is conserved at corotation by the generation of a trailing wave which propagates outward

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7 toward the outer Lindblad resonance (OLR). Frequencies which cause the phases of the waves to match are associated with standing waves. The interference of equal-amplitude swing-amplified leading and trailing standing waves leads to bar formation. There are several ways to control the bar instability. One, noted by Hohl (1971), is simply to increase the velocity dispersion. This tends not only to stabilize a stellar disk against local axisymmetric instabilities by raising the value of ToomreÂ’s Q parameter,^ but also to inhibit the bar instability by reducing the effectiveness of the swing amplification mechanism. Another way is to immerse the disk in a massive halo. This idea stems from the work of Ostriker and Peebles (1973), who found empirically that their A-body models were stable against bar formation if the ratio of the rotational kinetic energy of the disk to the total absolute potential energy was less than a critical value of 0.14 Â± 0.02. A third way, suggested by Toomre (1981), is to interrupt the feedback loop. This will occur if the ingoing waves are prevented from reaching the center. The existence of an inner Lindblad resonance (ILR), which damps ingoing waves via wave-particle interactions, provides such a mechanism. Toomre proposed that strong spiral perturbations generated in an unstable galaxy shift material inward and raise the value of fl Â— f until an ILR appears and shuts off the feedback loop. Much work has been done concerning the structure and dynamics of bars. A major portion of it deals with the periodic orbit families that exist in model bar potentials. Contopoulos and Grosbpl (1989) give a comprehensive review of this topic. Sparke and Sellwood (1987) and Pfenniger and Friedli (1991) have examined the orbital 1. Toomre (1964) found that the condition for stability of a differentially rotating stellar disk to local axisymmetric instabilities is (? = wZqQv > 1Â» where Or is the local radial velocity dispersion, k is the local epicyclic frequency, G is the Newtonian gravitation constant, and S is the local surface density.

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8 structures of numerically generated two-dimensional and three-dimensional A^-body bars, respectively. While the work of Pfenniger and Friedli presents some exciting new results concerning the behavior of orbits and the structure of bars perpendicular to the primary disk plane, we restrict our attention in the present thesis to the two-dimensional case. The main results of all this work are (1) the overwhelming majority of particles that make up the bar are on orbits trapped around the Â“centralÂ” or, in the notation of Contopoulos (1975), Xi family (i.e. the family that reduces to circles in the axisymmetric case) and (2) bars end at, or slightly inside, corotation. In their study Sparke and Sellwood found a significant population of particles that had sufficient energy to be able to move freely between the bar and outer disk (i.e. their Jacobi constants exceeded the value at the Lagrange points L\ and L 2 ). Pfenniger and Friedli also noted such a population (which they termed Â“hotÂ”) in their study. The existence of a bar in a galaxy provides at least three additional possible mechanisms for the generation of spiral structure (cf. Elmegreen and Elmegreen 1985): (1) a rotating, growing bar may excite transient stellar spiral arms (a phenomenon observed by James and Sellwood [1978] and Sellwood [1981] in their numerical simulations), (2) a rotating, static bar may excite spiral waves in the gas of the outer disk (a phenomenon observed, for example, in the hydrodynamical model of Sanders and Huntley [1976]), and (3) a rotating, static bar may drive stellar spirals in a strongly self-gravitating disk (Julian and Toomre 1966; Feldman and Lin 1973; Lin and Lau 1975; Goldreich and Tremaine 1978). The blue and near-infrared surface photometry of fifteen barred spiral galaxies provided by Elmegreen and Elmegreen (1985) allows us to begin to discriminate

PAGE 27

9 between these structure-generating mechanisms. They find that, in general, the spiral arms of their sample galaxies contain star formation regions superposed on density enhancements in the underlying stellar disk. Also, they find that the amplitudes of the spiral arms, as measured by the near-infrared brightness difference between the arm and interarm regions, decreases outward for early-type galaxies (typically with flat bars, as mentioned before) and increases outward for late-type galaxies (which generally have exponential bars); moreover, they find that galaxies with strong bars also tend to have strong spirals. Regarding spiral arm morphology, they note that all of their sample galaxies that have irregular spiral arms also have short bars of exponential or indeterminate type. On the other hand, all galaxies with flat bars have long, continuous arms or a smooth ring. The Elmegreens suggest that this apparent bimodal distribution of barred spiral types indicates fundamentally different origins and natures. According to this suggestion, early-type barred spirals have bars that extend close to their corotation radii and continuously excite spiral structure as they slowly grow. Late-types, on the other hand, have rather weak bars that extend only to the ILR and excite correspondingly weak, often multiple, asymmetric and patchy spiral arms. These interpretations have received some recent support from the numerical simulations of Combes and Elmegreen (1992). Given the numerous possibilities concerning the nature of the spiral arms and the positions of the major resonances in barred spiral galaxies, it is of interest to ascertain whether or not self-consistent models of these galaxies can be constructed. Since realistic models of galaxies are, in general, non-integrable, we must resort to numerical techniques in order to produce self-consistent models. The first effort of this type was

PAGE 28

10 made by Martin Schwarzschild (1979), who succeeded in constructing a self-consistent model of a nonrotating, triaxial stellar ellipsoid via the method of linear programming. His approach consisted basically of computing a library of orbits in the potential derived from an assumed density distribution. The time-averaged density in each cell of a grid that spanned the volume of the model was calculated for each orbit. The technique of linear programming was then employed to derive the nonnegative population weights required in order for the ensemble of orbits to give back the imposed model density. Pfenniger (1984b) has used a nonnegative least squares technique to find the population weights for a self-consistent model of a rapidly rotating, two-dimensional Ferrers bar. The most recent approach to the problem of self-consistent galactic models has been developed by Contopoulos and Grosbpl (1986, 1988). Patsis (1990) and Patsis et al. (1991) have successfully applied this method to a sample of thirteen unbarred spiral galaxies. We extend that work in this thesis by applying the method of Contopoulos and Grosbpl to a sample of three barred spirals: NGC 1398, NGC 3992, and NGC 1073. In Chapter 2 we describe in detail the method of Contopoulos and Grosbpl, our modifications of this method in order to apply it to the case of barred spirals, and the smoothed particle hydrodynamics (SPH) method used to calculate the gas response. In Chapter 3 we present the relevant observations together with our most successful models of the program galaxies. In Chapter 4 we present the results of varying the parameters of the most successful models. In Chapter 5 we detail the role of stochastic orbits in these models, and we give the results of the gas dynamical calculations in Chapter 6. Finally, we summarize the main results in Chapter 7.

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CHAPTER 2 MODELING TECHNIQUES The Method of Contopoulos and Grosbdl (CG Method) Before discussing the method of Contopoulos and Grosbpl as applied to barred spiral galaxies, it is useful to describe the technique as it was originally applied to unbarred spiral galaxies (Contopoulos and Grosbpl 1986, 1988; Patsis 1990; Patsis et al. 1991). The starting point of any modeling attempt is the observational data. Here the relevant data are the axisymmetric rotation curve and the surface photometry, preferably in the red or near infrared. The former gives a rough estimate of the gravitational mass distribution in the galaxy, at least insofar as the dynamics of the disk are concerned. The latter gives important information not only about the axisymmetric distribution of the stellar component of the disk, but also about the shape and strength of any bar and/or spiral perturbation which may be present. One kinematic result from observations of spiral galaxies is that the overwhelming majority of them exhibit rotation curves which, after steep initial rises, are flat or slowly rising as far out as they can be determined (e.g. Rubin et al. 1978; Bosma 1981). Any reasonable model rotation curve must exhibit this behavior as well. Secondly, it is observed that some spiral galaxies have more prominent nuclear bulges than others. In fact, this feature has been used as a classification criterion for these galaxies (Sandage 1961). Hence it is desirable to have a model rotation curve which can represent the bulge and disk contributions separately. Lastly, the underlying axisymmetric potential. 11

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12 Vo(r), derived from the rotation curve should be expressible using standard analytic functions. The two model rotation curves employed by Contopoulos and Grosbpl both have these three features. The one that has been used most extensively to date is v(r) = Umax exp (-Â£fcr) + [1 exp(-Â£rfr)], ( 2 1 ) where Sb~^ and ed~^ are the scale lengths for the bulge and disk components, respectively. The quantity Umax is the asymptotic value of the circular rotation velocity for large r, and the bulge fraction ft gives the importance of the bulge relative to the disk. Figures 2-1 and 2-2 show the effects of varying et and fb on the shape of the rotation curve. The axisymmetric potential Vo(r), derived from the relation is given by = -v{^^^{fbexp{-ebr) [Inr + Ei{Â€dr)]), ( 2 2 ) where Â£i(jc) is the first exponential integral. In this case the angular velocity fl(r) = goes to infinity for r = 0, thereby giving exactly one inner Lindblad resonance (ILR). For this reason a similar, but slightly modified, rotation curve was developed which allows either zero or two ILRs. The circular rotation velocity for this model is given by u(r) Â— Umax -Sbv) -f[1 (1 -IÂ£(/r) exp (-Â£(tr)]5 (2-3) while the corresponding axisymmetric potential is Vo{r) = -vl^^^{fb{l + Sbr)exp{-Sbr) [Inr + E\{edr) + exp(-Â£rfr)]}. (2-4)

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13 Figure 2-1: The effect of varying Â£b on the shape of the model rotation curve given by Eq. (2-1): (1) Â£b = 10e
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16 that extends beyond its cutoff radius r 2 Â„Â„ and K2m defines the steepness of the cutoff. By analogy with equation (5), esm~^ is the scale length and /omthe pitch angle of the mO component. Also, differences in the phases of the various components are allowed with the corresponding values of OmAnalysis of the surface photometry can yield (at least rough) estimates of most or all of the parameters needed to specify the spiral perturbation. It is not necessary at this stage to find the specific values of the parameters which yield the most selfconsistent model. Variations of and corrections to the parameters are made after the results of subsequent modeling attempts are known. In fact, this is the essence of the CG method. The observations do, however, place limits on the ranges of acceptable parameter values, as well as provide good initial guesses. After the parameters of the spiral perturbation are determined, at least roughly, from the surface photometry, the next step is to calculate the periodic orbits. For this the equations of motion are needed. Since these models are restricted to two dimensions (i.e. the disk plane), the equations of motion can be derived from the two-dimensional Hamiltonian // = i nÂ„Jo + v^r) + Viir, 0) = h, (2-8) where r is the radial component of the velocity, Jq the angular momentum in the inertial frame of reference, Q,p the angular velocity of the reference coordinate system (i.e. the frame corotating with the spiral pattern) with respect to the inertial frame, and h the numerical value of H. Also, Vo(r) and V\{r,0) are the axisymmetric and perturbation potentials, respectively. From Eq. (2-8) the necessary equations of motion can be

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17 derived. They are, in polar coordinates in the rotating frame. M dt j],), dr dt (2-9) These equations can be solved numerically by any of the standard integration techniques. The one that has been used by Contopoulos and Grosbpl (1986, 1988) is a fourth-order Runge-Kutta method with a variable stepsize, which allows the relative error in each variable to be maintained below a preset level (typically one part in 10^). The error controls allow for energy conservation along the integrated orbit to the order of one part in 10^ for a typical orbit integration. The periodic orbits of particular interest here are the ones comprising the Â“central,Â” or in the notation of Contopoulos (1975), Xi family, which reduce to circular orbits in the purely axisymmetric case. The introduction of a bar or spiral perturbation breaks this central family into an infinity of families by gaps at all even resonances between the epicyclic frequency Â«:(r), defined by kÂ“{v) = -I1 = 4fl'^(r) + 2rfl(r)^^^, and the orbital angular frequency measured in the frame corotating with the perturbation Q(r) flp (Contopoulos 1983). Here H(r) = That is, where fl(r) Qj, ( 2 10 ) Also, regions of instability are produced in the central family near the odd resonances, where /v(r) H(?') Â— flp ( 2 11 ) From these unstable regions bifurcate the odd resonant families.

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18 Since the available dynamical evidence (Contopoulos and Grosb 0 l 1986, 1988; Patsis 1990; Patsis et al. 1991) indicates that the spiral structure of normal spiral galaxies extends from the ILR (2/1 resonance) to either the 4/1 resonance (for strong spirals) or the corotation resonance (for weak spirals), one can get a reasonable estimate of the pattern speed Qp. This parameter is notoriously difficult to determine observationally, and therefore must be left adjustable in the fitting procedure. Nevertheless, observations of the extent of the spiral structure, together with the axisymmetric rotation curve, can produce reasonable initial values. The significant branches of the central family, then, that exist in the fully perturbed model between the inner Lindblad resonance and corotation are found. In this region there is a one-to-one correspondence between the radius rc of the circular periodic orbit of the axisymmetric case and its value of the Hamiltonian h. This allows for a rather simple algorithm to find the periodic families of the fully perturbed case. For example, consider the branch of the central family which exists between the ILR and the 4/1 resonance. The radius of the periodic orbit halfway between these resonances in the axisymmetric case will have the value ro = rc = 7 (riLR + ^ 4 / 1)5 where riLR and r 4 /i are the positions of the inner Lindblad and 4/1 resonances, respectively. The orbit will cross the 0 = 0 axis with a radial velocity kq = 0. Now let the Â“amplitudeÂ” of the perturbation potential be increased from zero by some small fraction of its maximum value. Also, let the value of the Hamiltonian for the new orbit remain h = h{rc). If this new orbit is started at radius ro with zero radial velocity, however, it will not, after being integrated for one revolution, cross the = 0 axis at radius ro with zero radial velocity. That is, it will not be periodic due to the imposed perturbation. But since the

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19 applied perturbation is small, the orbit will be close to a periodic orbit, thus the actual periodic orbit can be readily determined using a two-dimensional Newton method. Consider an arbitrary orbit of fixed Â“energyÂ” h, close to a periodic orbit, starting with initial conditions (ro, ro) along the 0 = 0 axis. After one revolution it will cross the 9 = 0 axis again but with different radius and radial velocity (ri, ri). Now vary separately ro and ro by Ar and Ar, respectively, and integrate these new orbits for one revolution each, producing the following mappings of initial to final conditions: (ro -iAr, ro) Â— ^ (r 2 , K 2 ) and (ro, ro + Ar) (r^, rs). These quantities are now used to compute the partial derivatives of the final radius and radial velocity, Vf and r/, with respect to the initial radius and radial velocity, r/ and r,-. Specifically, dvf V2 Â— ri drf T3 Â— r\ dri Ar Â’ dri Af Â’ drf r2 Â— ri drf rs Â— r\ ( 2 12 ) dvi Ar Â’ dri Ar These numerically determined partial derivatives are used to find the first-order corrections to ro and ro necessary to adjust ro and ro more closely to the initial conditions of the periodic orbit. The corrections, 8 r and 8 r, are determined by solving the following equations: dvf drf n + -::;Â— 8 r -f -Trr-or = ?Â’o + 8 r on on (2-13) drf dvf fi -f 7 ^ 8 r -f ~^ 8 r ?'o -f 8 r, UT i UT I These corrections are added to ro and ro. ro Â— > ro -I8 r ro Â— ^ ro + 8 r, (2-14)

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20 thereby improving the estimates of the initial conditions of the periodic orbit. This procedure is then repeated until |^r| and l^fj fall below some predefined tolerance level. There is usually no problem with convergence of this method if the starting initial conditions are not too Â“farÂ” from the periodic orbit. In this manner it is possible to Â“step upÂ” the perturbation amplitude from zero to its full value in small increments, finding accurately (typically to one part in 10^Â®) the periodic orbit at each step. After the first step, better starting guesses at the initial conditions of the periodic orbit of the next step are provided by an extrapolation procedure. This whole process is done for only one orbit between any given pair of even resonances inside corotation. The orbit Â“energyÂ” h, parameterized by rc, is chosen to be that of the circular orbit (in the axisymmetric case) halfway between the positions of the resonances in question, as in the example of the ILR and 4/1 resonance above. Once this periodic orbit is found in the fully perturbed case, the whole family of orbits between the resonances is calculated by varying the energy rc both positively and negatively. Again, after the first step each way, an extrapolation procedure is used to provide better first guesses at the initial conditions of the periodic orbit. In models of unbarred spiral galaxies, the resonance ranges probed for branches of the x\ family are usually limited to: 2/1 (ILR) to 4/1, 4/1 to 6/1, and 6/1 to 8/1. This is because in a realistic model of a spiral galaxy, the positions of the resonances bunch up near corotation and most of the galaxy inside corotation is also inside the 8/1 resonance. For example, in the standard model for NGC 5247 considered by Contopoulos and Grosbpl (1986), corotation is roughly at 23 kpc. In comparison, the ILR is around 1.5 kpc, the 4/1 resonance around 12 kpc, the 6/1 resonance around 16 kpc, and the 8/1

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21 resonance around 17.5 kpc. And, according to Contopoulos and Grosbpl (1986, p.l55), Â“the congestion of infinite resonances n/1 near corotation produces a large degree of stochasticity there.Â” Still, in cases where significant stable periodic families exist beyond the 8/1 and/or 10/1 resonances, orbits in these regions are generally included. Now that the families of periodic orbits to be included have been determined, the next step, undoubtedly the most important step in the method of Contopoulos and Grosbpl, is to generate the density response map. For this a two-dimensional polar grid is employed; moreover, since the entire problem is symmetric, only half of the grid is actually used in order to save computational time and storage space. Use of a polar grid permits both a simple way of analyzing the self-consistency of the density response, and higher resolution near the center where the density and density gradient become large. The radial range of the grid, as well as the radial and azimuthal widths of the grid cells, can be varied in the analysis. The only restriction is that the number of azimuthal cells should be a power of 2. The reason for this will be outlined later in the discussion of the analysis of the density response. The density response map, then, is constructed by computing a representative sample of the periodic orbits, and a set of orbits dispersed in radial velocity around them, and incrementing the value of the density of a given cell by the product of the weight of the orbit, to be discussed in detail shortly, and the amount of time spent by the orbit in the cell. Each periodic orbit family included in the analysis is sampled at fixed intervals of Â“energy,Â” as parameterized by the radius of the circular orbit of equivalent energy in the axisymmetric case. The sampling step in the Â“energyÂ” so parameterized is taken to be times the radial width of a grid cell, where NOR =

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22 1, 2, 3, ... is an input parameter. In all of the models considered in this thesis we have taken the value of NOR to be 2. Every periodic orbit chosen by the sampling process to be included in the density map is then integrated for one half period (since only half of the polar grid is used) in order to have its contribution to the density map added and to generate initial conditions for the dispersed orbits. The periodic orbits, as originally determined, all have their initial conditions (r, r) specified along the 0 = 0 axis. It is desirable, however, in order to estimate more accurately, and with reduced integration times per orbit, the actual density response of orbits dispersed around the periodic orbit, to start orbits for the density response calculation away from this axis but along the trajectory of the periodic orbit. Therefore, during the integration of the periodic orbit, its radius r, radial velocity r, azimuth 6, and angular momentum Jq are stored every fraction of its half period, where NOA = 1, 2, 3, ... is also an input parameter. After the density contribution of each periodic orbit is added to the density map, the orbits dispersed about the periodic orbit are computed. Whereas the periodic orbit is calculated for its full half period, it is not really necessary to calculate the dispersed orbits for this length of time due to the increased numbers of dispersed orbits calculated per periodic orbit. The actual time of integration for the dispersed orbits is controlled by a parameter PN such that integration time = to = PN x 1 PZ NOA ^ (2-15) where PZ is the period of the base periodic orbit. The value of PN is typically chosen to be of order NOA, empirically determined to be a good compromise between the need to generate an adequate representative density response for the orbit and the need to

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23 minimize computational time per orbit. The dispersed orbits are started at the NOA stored points along the periodic orbit. The distribution of dispersed orbits is assumed to be Gaussian with velocity dispersion a, thus the orbits must be dispersed and weighted appropriately relative to the central periodic orbit in order to estimate best, using a finite number of orbits, a continuous Gaussian distribution. A quadrature scheme is indicated. Consider a periodic orbit, characterized by (r, 9, r, Jq), a point along the trajectory of the orbit. The problem is to compute the surface response density of a continuum of orbits all starting at this point, but distributed in radial velocity about r in a Gaussian fashion with dispersion a. Let the surface response density of the orbit started at (r, 6, r, Jo) and calculated for time ro be given by s{r,6,r, Jo^to). The problem, then, can be stated mathematically as OO S{r,6,Jo,to)Â— J exp Â— OO Jo, to) dr , (2-16) where S{r, 6 , Jo, to) represents the total surface response density of all orbits distributed about the period orbit and calculated for time toIn order to estimate the above integral, a quadrature scheme employing the Hermite polynomial Hn(x) is used (in all cases considered in this thesis we have taken n = 9). Specifically (Abramowitz and Stegun 1965, p. 890), J n / exp (-x^)f{x)dx = ^ wvif(xi) + Rn, -OO where xi is the zero of Hnix), the weights wvi are given by 'n?[Hn-i{xi)fÂ‘'Â‘ (2-17) (2-18) and Rn is a small remainder.

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24 Since we use a nine point (n = 9) Hermite integration to calculate the effect of velocity dispersion on the density response, a total of nine orbits, the one periodic orbit and eight dispersed about it, four on either side in r space, are integrated, with the OO initial conditions given in Table 2Â—1. Since f exp (^Â—x^)dx = y/w, the weights wvi Â— OO must be normalized by dividing by a/x before the integrations are done. Table 2-1: Initial conditions of the periodic orbit and the eight nonperiodic orbits used to calculate the effect of the velocity dispersion. Orbit Number Initial Conditions Orbit Weight 1 (periodic orbit) r, 6>,r + (jxi, Jo WVi 2,3 r, 0, r Â± ax 2 , Jo WV2 4,5 r,0,rÂ± ax3,Jo wvs 6,7 r, 0, r Â± (7X4, Jo WV 4 8,9 r,0,fÂ± (7X5, Jo WVs Table 2-2: The values of the xi and wv{ of Table 2-1. Â• 1 Xi WVi 1 0.00000 7.20235... X 10-1 2 0.72355... 4.32651... X 10-1 3 1.46855... 8.84745... X 10-2 4 2.26658... 4.94362... X 10-2 5 3.19099... 3.96069... X 10-2 In addition to the Hermitian weight mentioned above, we must also include some further weighting factors to measure properly the contribution of a particular orbit to the response density map. These weights are introduced to account for (1) the relative abilities of different periodic families coexisting at Â“energyÂ” rc to trap matter around them, (2) the population of a particular orbit of Â“energyÂ” Vc according to the imposed

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25 axisymmetric surface density at radius Vc, (3) the Â“gridÂ” effects of the parameters NOA and NOR, and (4) the time of orbit integration. Sometimes, in the course of calculating the periodic orbit families to include in the model, we find more than one stable family at the same Â“energyÂ” Therefore the question arises as to the relative importance of each orbit family in the energy intervals where there is overlap. Contopoulos (1979) has given an estimate of the relative amount of trapping done by a periodic orbit of a given energy r^. This estimate is measured, for the m* orbit family, by the weight KÂ“{rc)x 2 < f2 > (2-19) where n{rc) is the epicyclic frequency at radius Vc, Xm is the maximum deviation of the periodic orbit from r<;, and is the square of the RMS radial velocity dispersion. Since the velocity dispersion a mentioned earlier is assumed to be isotropic, = Therefore the relative weight Wm of the m* family at Â“energyÂ” rc is given by ( 2 20 ) where n is the total number of stable orbit families coexisting at energy rc. In energy intervals where only a single orbit family exists, this weighting factor is obviously equal to one. The next weighting factor to be considered is the most obvious. Orbits of lower energy rc, which exist nearer to the center of the galaxy, should be weighted more than orbits of higher energy, which exist in the outer regions of the galaxy. This simply reflects the fact that the matter density in the disk decreases outward. Therefore a

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26 weight wv, of the form Â• wi: = S(rc), (2-21) is used. Here S(rc) is the surface density of the unperturbed disk at radius rc. This quantity cannot simply be derived from the axisymmetric rotation curve, because it is not clear that the majority of the gravitational mass in spiral galaxies is concentrated in or near the observed disk. In fact, observed disk luminosity profiles typically decrease exponentially outward (Freeman 1970). Unless the actual mass-to-light ratio varies wildly across the disk, it is plausible to infer that the matter density in the disk also exhibits an exponential decrease with radius, and that the excess gravitational mass responsible for the observed rotation curve is distributed in a spherical or oblate spheriodal distribution, but away from the disk in any case. Given these facts, the surface density of the unperturbed disk is taken to be exponential, given by S(r) = coexp(-Â£or), (2-22) where cq is the central surface density and the scale length of the disk. Thus the weight factor applied for an orbit of energy is wj: = Co exp (-Â£o^c)(2-23) The final weighting factors are introduced in order to eliminate the effects of the particular choices of NOR, the number of sampling steps in Â“energyÂ” Kc per grid cell, NOA, the number of different starting positions for dispersed orbits along the periodic orbit, and to, the time of integration. The grid effects are taken out by a weight of

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27 the form 1 1 tOgj'id X NOR NOA (2-24) Also, since the contribution of a particular orbit to any one grid cell is the previous weight of the orbit multiplied by the amount of time spent by the orbit in the cell, the orbit weight must be divided by the total time of orbit integration. As noted before, the integration time is ^PZ for periodic orbits and ^PZ x PN x j;;j^ for orbits dispersed about the periodic orbit. Again, PZ is the period of the base periodic orbit and PN the parameter which controls the length of integration for nonperiodic orbits through Eq. (2-15). Therefore the corresponding weights applied are simply the inverses of these times. ^time 2 PZÂ’ 2xNOA periodic orbits (2-25) M X P Z Â’ nonperiodic orbits The total weight applied to a given orbit, then, can be written as W^tot = WV X Wm X Wj: X tCgrid X tCtime (2-26) The complete density response map, then, is produced by (1) specifying the minimum (RMIN) and maximum (RMAX) radial extent of the grid, as well as the radial (DRS) and azimuthal cell widths, (2) specifying the parameters NOR, NOA, and PN described above, (3) specifying the central value, ao, and slope, ar, of the velocity dispersion (the form of the velocity dispersion is assumed to be a(r) = ao + rur, a linear relation being the simplest functional form possible which allows a{r) to vary across the disk), and (4) stepping in radius Vc from RMIN to RMAX in increments of

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28 and integrating all orbits existing at each value of rc, weighted as described above. The results from this raw map are then preprocessed slightly before being stored for later analysis. Each semiannulus of the grid is processed in turn. Thus, for a fixed semiannulus, the density value of a each azimuthal cell is Â“smoothedÂ” with the values of the two adjacent cells according to the smoothing formula DEN(i) = iDEN(i) + i(DEN(i 1) + DEN(i+l)), (2-27) 2 d where i is the azimuthal cell index. At the same time, the total Â“density,Â” RHO, in the semiannulus is computed. n RHO = J]DEN(i), (2-28) where n is the number of azimuthal cells. The value of each cell is then normalized by the Â“averageÂ” cell value of the semiannulus and finally decreased by one, such that nxDEN(0 , (2-29) represents directly the fractional deviation of the azimuthal cell of the semiannulus from the average. The final step of the method of Contopoulos and Grosbpl is to analyze the density response for consistency with the imposed density. In other words, the selfconsistency of the model is measured. Here again, the analysis is done on the semiannuli separately. Two basic quantities are used to estimate the self-consistency of the model. The first compares the amplitude of the 29 component of the response density to the corresponding amplitude of the imposed density. The second measures the difference between the phases of the response and imposed density maxima in the semiannulus.

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29 First, a Fast Fourier Transform (FFT) with respect to azimuth is performed on the equally spaced data in array DEN. The reason for requiring the number of azimuthal cells to be an integer power of 2 now becomes clear. While the FFT can be made to accomodate data sets of other lengths, the algorithm is applied most easily to a set of length n = 2*, yt a nonnegative integer. The power and phase of each positive frequency component fi = ^, / = 0, 1, 2, ..., nil, are computed. For the i = 1 frequency component (i.e. the 26 component), the difference A0 between the response density maximum and imposed potential minimum is then calculated. This angle difference is measured with respect to the imposed potential minimum rather than the imposed density maximum because these two coincide almost exactly and because the angle of the the imposed potential minimum is more readily ascertained. In a fully self-consistent model, should be zero for all values of the radius. Therefore A^ constitutes one measure of the self-consistency of a given model. The amplitude of the / = 1 frequency component corresponds to the quantity ^2,resp ^0,resp (2-30) where a2,resp is the actual amplitude of the 26 component of the response density in the semiannulus under consideration and ao^resp is the mean surface response density in the semiannulus, or RHO/n. It is of interest to compare this with the corresponding quantity of the imposed density, or ^2, imp ^0,imp (2-31) The mean imposed density at the radius of the semiannulus is given by the formula for the exponential disk [Eq. (2-23)]. The amplitude of the 26 component of the imposed

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30 density is derived from the imposed potential via PoissonÂ’s equation. To this end Vandervoort (1970; see also Contopoulos and Grosbpl 1988) has given the following formulae, valid in the case of a tightly wrapped spiral density wave, relating the full three-dimensional potential Vi and density p\ of the spiral: = yiosechl^'^l(^//A), (2-32) + 0(i), where Vio is the spiral potential formula in the disk plane (z = 0), A is the z thickness of the spiral and k is the wave number. The parameter A, then, allows the vertical thickness of the spiral to be controlled and must be included in the adjustment of parameters. For the form of spiral potential used by Contopoulos and Grosbpl, k = Â— (2-33) r tan ^o where m is the number of spiral arms. From the above relations (J2,imp can be computed. Specifically, OO OO <72, imp ~ J Pi{z)dz = 2 J pi{z)dz. (2-34) Â— OO 0 Here A is taken to be constant over the entire disk of the galaxy, quantity
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31 Â“imposedÂ” disk. The maximum value of cq is limited only by the total axisymmetric potential. Therefore, practical self-consistency requires that R simply be constant with radius. This is the second measure of the self-consistency of a given model. The whole procedure outlined above is then repeated for different sets of model parameters, and a Â“bestÂ” model achieved by a sort of manual iteration. It would be desireable to automate the whole procedure, but at this point too much human decisionmaking is required at intermediate steps for this to be feasible. The CG Method Modified for the Case of Barred Spiral Galaxies In this section are described the modifications made to the CG method in order to apply it to the case of barred spiral galaxies. Perhaps the most significant change is the nature of the assumed model. In contrast to the original CG method where a simple function is fit to the observed axisymmetric rotation curve, thereby allowing the total axisymmetric potential to be derived in a correspondingly simple form, model components are used. This is necessitated by the fact that the full complement of nonaxisymmetric structure cannot, in this case, be adequately modeled as a simple perturbation of the axisymmetric disk. This approach was tried initially, by replacing the quantity ,1^ Â— m0-\-m9m with md in the formula for V\{r,9) for values of radius less than the length of the bar, with somewhat limited success. It was found that even with the inclusion of significant higher harmonics the imposed bar density distribution could not adequately simulate realistic bars, even qualitatively. Therefore a three component model is employed, with components introduced to represent separately a bar, a disk. and a halo.

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32 The axisymmetric component is modeled as the superposition of disk and halo components. In view of the fact, mentioned above, that the observed brightness profiles of disk galaxies are typically exponential, an exponential disk is used. The corresponding potential in the disk plane is readily derived (cf. Binney and Tremaine 1987, p.77) and can be written Voir) = -7cGcQr\lQ{y)Ki{y) Ii{y)Ko{y)], (2-36) where G is the Newtonian gravitation constant, Co is the central surface density of the disk, y = \ear, and the /Â„ and Kn are modified Bessel functions of the first and second kind, respectively. Again, eo~^ is the scale length of the exponential disk. The circular rotation velocity VÂ£>{r) corresponding to this potential is readily derived from the relation dVp{r) dr (2-37) using the fact (Abramowitz and Stegun 1965, p.376) that dlp{x) dx = h(x) dKo(x) dx = -Ki{x), dh{x) dKi = Io[x) , and dx dx 1 Â—Ko{x) Ki{x). X (2-38) The result is Vpir) = yj i:GcQeQrÂ‘^[h{y)KQ{y) h{y)K\{y)]. (2-39) The general form of this rotation curve, along with the Keplerian curve for a point mass equal to the mass Md = of the exponential disk is shown in Figure 2-3 (essentially Figure 2-17 of Binney and Tremaine 1987, p. 78).

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33 Figure 2-3: Rotation curves of the exponential disk (solid curve) and a point with the same total mass (dotted curve). As can be readily seen from Figure 2-3, the rotation curve of an exponential disk declines after approximately two disk scale lengths and rather quickly approaches the Keplerian value. This, as described above, is in contradiction to the observed rotation curves of most disk galaxies. Hence a separate halo component is included in order to Â“hold upÂ” the rotation curve at large radii. The density distribution chosen for this component is that of a Plummer sphere: pjj{r) = (2-40) Here M// is the total halo mass and b is a parameter which controls the central condensation. The potential corresponding to this density distribution is particularly simple: GMh \/r^ + 6Â“ (2-41)

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34 From the potential the circular rotation velocity is derived: (2-42) Figure 2-4 shows the rotation curve of the Plummer sphere and the corresponding Keplerian rotation curve for a point mass equal to MhAt first glance it seems as though nothing has been gained by including the halo component. The rotation curve of the Plummer sphere declines and approaches the Keplerian curve also at large radii. There is a crucial difference, however, between the halo and the disk. Since the disk is luminous, its exponential scale length can be observed. The halo, though, is made up of dark matter which is not accessible to direct observation. The totality of knowledge about the matter distribution in the halo must be inferred by its gravitational effects. The end result is that the parameter b, unlike the exponential disk scale length eo, is free, constrained only by the observed rotation curve. Hence the value of b can be near or beyond the observed disk (optical or HI), thereby placing the disk completely within the rising part of the halo rotation curve. For modeling the bar, the triaxial homeoidal density distribution given by Â™<; (2-43) to 772 > 1, O 7/^ 2 2 where = fr + fr + TT and a > & > c > 0 are the long, intermediate, and short bar CtÂ« o axes, respectively, is used. This is the n = 2 Ferrers potential (Ferrers 1877; see also Binney and Tremaine 1987, p.61). This particular bar component was chosen not only because it represents many features of observed bars rather well Â— Hunter et al. (1988) and Ball (1984, 1992) found that bar components with a Gaussian brightness profile, similar to that of the n 2 Ferrers bar, best fit the infrared observations of the bars

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35 of NGC 3992 and NGC 3359, respectively Â— but also because its dynamics has been the subject of a number of previous investigations (e.g. de Vaucouleurs and Freeman 1972; Papayannopoulos and Petrou 1983; Athanassoula et al. 1983; Pfenniger 1984a, 1985). The strict inequalities in the bar axesÂ’ lengths are only required for mathematical reasons in the derivation of the associated potential and forces. They do not represent a real restriction from essentially spherical, oblate or prolate figures. Exact solutions for these special cases already exist. Figure 2^: Rotation curves of the Plummer sphere (solid curve) and a point with the same total mass (dotted curve). The central density pc of the bar can be expressed in terms of the total mass Mp and its axesÂ’ lengths as 105 Mb pc 32tt ahc (2-44)

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36 The projected surface density of this figure is obtained by integrating p(x,y,z) with respect to z over the range where p is nonzero: Zrnax p{x,y,z)dz, Â— Z max where Zmax Â— / 1 jf. The result is expressed by ^B{x,y) = ^cpc 5 2 2 \ 2 IP' U4b ( 27rab V 1-^ 2 X 5 2 \ 2 62 (2-45) (2-46) The potential in the disk plane can be written as VB{x,y) = -^mWnoxV WnoxS/ W2wxV D kkiooy +lTo2oa:^ + 1^2002/^ Â— kkoio^:^) (2-47) +Vfooo Â— Â— VV3002/^}? where C = 27rGpcabc = ^GMb, and the Vkp are coefficients dependent on both the axesÂ’ lengths a, b, and c and the position {x,y). Complete derivations of the general three-dimensional form of the potential, the Cartesian components of the force, and the coefficients ITj/yt are given by Pfenniger (1984a, appendix). A summary of all the bar quantities needed for this study is included in the appendix of this thesis. Despite the fact that there is no Â“axisymmetricÂ” rotation velocity associated with this bar potential, an estimate of this quantity can be obtained. By analogy with the axisymmetric disk and halo rotation velocities, the Â“circularÂ” velocity VB{r,0) due solely to the bar potential is given by VB(r,0) JVB{r,6) dr (2-48) Above, the velocity and potential have been taken in terms of the polar coordinates (r,9) instead of the Cartesian ones (x,y). The Â“axisymmetricÂ” rotation velocity u^(r), then, is 7 T 2 / VB{r) = 2 7T or (2-49) 0

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37 In a similar manner, all other Â“axisymmetricÂ” quantities of the bar can be estimated. Figure 2-5 gives representative Â“axisymmetricÂ” rotation curves for three sets of bar dimensions. Figure 2-5; Three nomralized Â“axisymmetricÂ” bar rotation curves for cases where a = 1 and c = 0.1: (1) b = 0.15, (2) b = 0.3, and (3) b = 0.6. The dotted line represents the Keplerian rotation curve for a point mass equal to Mg. As in the original CG method, the observed angle-averaged rotation curve is fit by the model rotation curve. The model curve in this case, though, is derived from the component rotation curve by the relation v'^{r) = v]){r)^rv]J{r) + v%{r), (2-50) where vj'{r) is the total rotation velocity. The method of curve fitting is somewhat more crude in our modified method than in the original CG method. In particular, the use of a least squares fitting routine is forgone in lieu of simple visual fitting. There are a couple of reasons for this. First, in barred spiral galaxies the noncircular motions can be

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38 much more pronounced than in normal spirals (e.g. Duval and Athanassoula 1983). The uncertainty in the Â“correctÂ” axisymmetric rotation curve, then, grows in proportion to the magnitude of the noncircular motions. Large uncertainties, in themselves, however, do not preclude the use of a mathematical fitting scheme. A second and more serious problem was that least square fits in which all input parameters were allowed to adjust freely resulted, in some cases, in physical implausibilities, such as nearzero bar masses. For these reasons the following procedure for fitting rotation curves was implemented. First, since the component contributions to the rotation curve are somewhat independent of each other (the bar component dominates the inner region of the observed rotation curve, the disk component the central region, and the halo component the outer region), it was decided to fit visually one component at a time, starting from the center outward. The first major contributor to the rotation curve is the bar. The bar that was used was the one with the maximum allowable mass under the constraints of the rotation curve. This component typically fit the steep initial rise of the rotation curve. The next component included was the exponential disk. Again, the maximum disk allowable by the rotation curve was used. The addition of the disk component permitted good fits to the observed rotation curve over most of the radial range. The Plummer sphere halo component was added last in order to Â“hold upÂ” the rotation curves at large radii. The parameter values obtained this way were adjusted slightly in order to achieve the best visual fit. The parameters that were adjusted in order to provide the fits to the rotation curve are given in Table 2-3. While the bar contribution to the rotation curve certainly depends on the axesÂ’ lengths a, b, and c, these parameters, as we will see, are fixed by the surface photometry.

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39 Table 2-3: Adjustable parameters in the axisymmetric rotation curve fitting procedure. Component Adjustable Parameter(s) Ferrers Bar Mb Exponential Disk cq and eo Plummer Sphere Halo M// and b The method of Stark (1977) is used to derive the lengths of the barÂ’s axes. A complete explanation of the method can be found in Stark (1977) and in Ball (1984), thus the derivation will not be included here. In short, StarkÂ’s method assumes that the volume brightness of the bar is constant on similar, nested, triaxial ellipsoidal surfaces. Contopoulos (1956) has shown that the projected isophotes of such a bar, from any viewing angle, are similar, concentric ellipses. Assuming that the triaxial bar figure has one of its principle axes normal to the disk plane, one can combine the observed axial ratio of the bar isophotes, /?o, the inclination angle of the galaxy, i, and the angle between the line of nodes of the disk and the major axis of the bar isophotes, tp, to give a one parameter family of triaxial figures as the solution. The parameter , which controls the family, is in some sense a measure of the true azimuth of the triaxial figure in the disk plane referred to some fiducial direction. In the present analysis p is taken to be the angle measured in the disk plane between the long axis of the bar and the line of nodes of the disk. A useful property of this family of solutions is that there is a one-to-one correspondence between p and the axial ratios of the bar. Specifying, say, the axial ratio da is equivalent to specifying (f> directly, and that is the actual procedure used here. The value of da is taken to be 1/10 after Pfenniger (1984a [see also references therein, de Vaucouleurs and Freeman 1972; Kormendy 1982], 1985). In all of the cases considered in this thesis the long and intermediate bar axes a and b are

PAGE 58

assumed to lie in the principle disk plane of the galaxy, and the short axis c is assumed to be perpendicular to the disk plane. After the determination of the lengths of the bar axes from the Stark analysis, and the visual fitting of the axisymmetric rotation curve, we derive starting values for Qp and for the parameters which define the spiral perturbation. There is ample theoretical and numerical evidence (Contopoulos 1981; Schwarz 1981; Teuben and Sanders 1985; Sellwood 1980, 1981; Thielheim and Wolff 1981, 1984; Sparke and Sellwood 1987) that bars end at or slightly inside corotation, the distance at which the angular rotation rate of matter in the galaxy equals flp. For a first estimate of the pattern speed, then, the bar is assumed to end at corotation. As before, the parameters characterizing the spiral pattern beyond the bar are derived from the available surface photometry. A new feature is now added, however, to the form of the perturbation. Since the spiral is assumed to begin at or near the end of the bar, an inner cutoff is added to the formula for the spiral amplitude Amir): As we shall see later, the optimum positions for these inner cutoffs yield important information concerning the nature of the bars in the most successful models. Also, some changes to the argument of the cosine in Eq. (2-6) are required. Namely, (2-51) argument = < At this point, the entire barred spiral model is specified. r > a (2-52) r < a. As in the original CG method, the next step is to calculate the main families of periodic orbits. In the barred spiral models, periodic orbit families were searched

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41 for between the ILR and the 8/1 resonance inside corotation, and between the -8/1 and -2/1 (outer Lindblad) resonance outside corotation. For bars of moderate to high strength, however, there are substantial regions of Â“energyÂ” r^, both inside and outside of corotation, where there are no stable, dynamically significant periodic families. In order to represent matter existing in these regions for the surface response density calculation, we included orbits started here on Â“circularÂ” trajectories. The initial radius of a Â“circularÂ” orbit of energy rc was taken to be r^, and the initial radial velocity was taken to be zero (hence the designation Â“circularÂ”). Since the majority of orbits in these regions turn out to be stochastic, the surface density response they generate is not particularly sensitive to the particular choice of initial conditions. Thus we have given them the most simple initial conditions possible, the Â“circularÂ” orbit initial conditions described above. Despite the fact that most of these orbits turn out to be stochastic, they yield interesting density responses nonetheless, as we will see later in Chapters 3, 4, and 5. Next, the surface response density map is created. The procedure is exactly the same as in the original CG method, except for the handling of the Â“circularÂ” orbits described above. The differences are (1) in the formula for the weight Wm, which gauges the relative amount of trapping done by the m* family, Xm, the deviation of the orbit from r^, is taken to be 1 kpc, a typical value for this type of orbit, (2) the starting points for dispersed orbits along the Â“circularÂ” orbits are taken to be every interval 2 xNOA azimuth, (3) in cases where the given Â“energyÂ” and initial conditions of a Â“circularÂ” orbit result in an imaginary angular momentum solution (i.e. where the initial conditions lie outside the limiting curve) the Â“energyÂ” is artificially increased to the point

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42 where the angular momentum becomes real (this procedure is justified by noting that both observations of real barred galaxies and N-body simulations of barred galaxies show that these regions are populated with stars), and (4) all Â“circularÂ” orbits and orbits dispersed around them are integrated for a fixed time T, chosen such that good estimates of the average density response are derived from these orbits and computation times axe held to reasonable lengths. Typically T is chosen to be approximately 1 billion years. These changes require some changes in the precise form of the weighting factors; however, the meaning of each factor remains the same. Specifically, for all orbits, 77 2 ms = Co exp (-eoxc) + Â“ / ^^(rc, 6)dd, (2-53) 0 and for the Â“circularÂ” orbits, 1 ^time Â— (2Â—54) The resulting density map is preprocessed exactly as before prior to the analysis for self-consistency. The essence of the analysis of the density response map remains unchanged in the modified method (i.e. the application of the CG method to barred spiral galaxies); however, one detail concerning the computation of the amplitudes of the Fourier components of the imposed density has been altered. Since the bar potential is not put into the model explicitly in the form of its Fourier components, the amplitude of the 29 component of the imposed density is derived in exactly the same way as the 29 component of the response density; specifically, the imposed density is calculated as a function of azimuth, for a fixed semiannulus radius, at exactly the same azimuthal resolution as the response density map. The imposed density at a given position is

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43 derived from (1) the surface density of the exponential disk and the projected bar, and (2) the spiral perturbation according to the formula of Vandervoort. A Fast Fourier Transform is then performed on this array of imposed densities at the same time as on the array of response densities. At each radius, then, the amplitudes of the 20 components are compared by the ratio of response to imposed, as in the original method. The phase difference calculation is unchanged. Gas Response Using Smoothed Particle Hydrodynamics (SPH) To complement the self-consistent stellar models achieved by the modified method, the gas response of the Â“bestÂ” model in each case has been investigated via the use of a two-dimensional smoothed particle hydrodynamics (SPH) code generously provided to the author by Dr. Nikos Hiotelis. The SPH method was introduced independently by Lucy (1977) and Gingold and Monaghan (1977) and uses a Monte Carlo method to solve the equations of hydrodynamics. The complete details of the SPH method are given in several review articles (Hemquist and Katz 1989; Benz 1990; Monaghan 1985; Steinmetz and Muller 1992) and references therein, thus only a brief overview of the general method (in two dimensions) will be given here, along with the relevant details of the specific SPH implementation used for the present calculations. SPH is a Lagrangian method which does not require the use of a computational grid; therefore, unlike finite-difference schemes, which require a grid, no computational resources are wasted simulating large voids that may arise (Steinmetz and Muller 1992). Since the SPH method is not very complicated, it does not require a lengthy code; furthermore, it is quite robust. Steinmetz and Muller (1992) mention also three new developments which have greatly enhanced the potential of SPH: (1) the achievable

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44 resolution has been improved dramatically with the advent of the variable smoothing length, (2) global time step restrictions have been alleviated via the use of a separate time step for each particle, and (3) the introduction of the hierarchical tree method to calculate the gravitational forces has reduced computation times for SPH with self-gravity from the order of to MogA^ without altering its original Lagrangian formulation. Finally, the SPH scheme and hierarchical tree method can be vectorized to take advantage of highly parallelized computing architectures. In the SPH method a continuous fluid medium is modeled as an ensemble of N fluid elements (in this thesis N is approximately 12,000 in all cases), each taken to be a Â“particleÂ” with position r,and mass mi, smoothed out according to some chosen smoothing kernel W{r r'. A). The kernel is a function strongly peaked around r Â— r = 0, and the quantity A is called the smoothing length. The kernel must fulfull certain requirements. First, it must guarantee conservation of momentum, both linear and angular. An easy way to do this is to make the kernel spherically symmetric. Secondly, it must have the properties J W{r,\)dÂ‘^r = 1 and lim mr,A) = (^(r). A-^O (2-55) The ensemble of particles together with the smoothing kernel can be used to estimate the mean value of any spatially varying physical variable at any arbitrary position r. The actual mean value at r of the spatially varying function A(r) is given by (A(r)) = J A(r')W (r Â— r' , X)(Pr' . (2-56) The estimate using the Â“smoothedÂ” particles is given by the sum N (^(r)) = ^mj-^lT"(rrj,A), PAj) j=i (2-57)

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45 where, for the density p(Xj), the similarly estimated value N 2 = 1 (2-58) is used. The time derivative and gradient, respectively, of the quantity (A(r)) can be written (Steinmetz and Muller 1992) as d and (2-59) V(A(r)) = j A{v')\7W{r-v',X)d^r'. These results can be used to derive the averaged hydrodynamic equations of motion. Specifically, they are obtained by multiplying each term of the exact equations by the smoothing kernel and summing over all particles (e.g. Monaghan and Gingold 1983; Monaghan 1985; Benz 1990). Including also the artificial viscosity terms, the following set of equations (Hiotelis et al. 1991) are obtained: dvi dt dui dt 1 2 N j=i >Â«Â• , p/ [p1 p)_ (1 + Uij)ViW dvi _ V,', dt f 7 'P^ , Pi /i !>) J (1 + n,y)(vi (2-60) Above, Vf, Pi, pi, Ui, Vi, and Vj-l-T are the velocity, pressure, density, thermal energy per unit mass, gravitational potential, and gradient of the kernel W, respectively, at the position T; of the particle. The term II, y includes the effect of artificial viscosity. The form of II, y is taken from Monaghan and Gingold (1983) and is the same as that used by Hiotelis et al. (1991): n r\ Oipij + PPij(2-61)

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46 Here, a and (3 are constant coefficients and j.iij is defined by l^ij = i cij(k.-rfi W) 0, otherwise. (2-62) Above, Ay and cij are the average smoothing length and sound speed, respectively. y between positions r,and r^'. The parameter = 0.01 A^ is introduced to avoid possible numerical divergences in mj. In the expression for Ily, the term linear in /ty produces a shear and bulk viscosity (Monaghan 1985), while the quadratic term is included to handle high Mach number shocks and is roughly equivalent to the Von NeumannRichtmyer viscosity used in finite-difference methods. In all calculations done here, a and /? are taken to be unity. According to Hernquist and Katz (1989), these values represent a fair compromise for the typical range of Mach numbers encountered. In order to integrate the hydrodynamic equations an equation of state must be specified. The gas in this case is assumed to behave like an ideal gas. Therefore, the pressure, density and thermal energy of the gas is related by Pj = (7 l)piUi, (2-63) where 7 is defined to be the ratio of specific heats. Since the calculations are done in two dimensions instead of three, the values of 7 corresponding to the adiabatic and isothermal cases are 2 and 1 , respectively. The actual behavior of interstellar gas most probably is somewhere between these two extremes, therefore a compromise value of 7 = 5/3 is used. The wide variation present in the temperature of the interstellar medium presents a problem concerning the appropriate choice of the thermal energy ui. An initial value of w/ corresponding to an average temperature of around 3500 K is used. We consider this a fair compromise between the larger expanses of very hot, low density

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47 interstellar gas (T ft; 10Â® K, n ft; 10Â“^ cm~^) and the smaller expanses of relatively cooler and denser HI clouds (10^ K < T < 6 x 10^ K, 20 cmÂ“^ > 0.3 cm~^) and HII regions (T ft^ 8 x 10^ K, n > 0.5 cmÂ”^). It still remains to specify the exact smoothing kernel used. The two-dimensional version of the spherically symmetric three-dimensional spline-based kernel proposed by Monaghan and Lattanzio (1985) and employed by Hiotelis et al. (1991) is used. It is defined by 1 -h 0 < u < 1 i(2-^^)^ l 2, (2-64) where u = I . Also, the smoothing length is assumed to be variable in space and time, hence A = A(r, t). This is done to increase the resolution possible with the SPH method, since A directly limits the resolution (all structure on scales smaller than A are strongly smoothed out). Whereas the introduction of a time-dependent A is straightforward, there exist two possibilities to introduce a spatially variable A (Steinmetz and Muller 1992). These are the Â“gatherÂ” and Â“scatterÂ” approaches (cf. Hernquist and Katz 1989). In the Â“gatherÂ” method, the smoothed average value of the arbitrary physical quantity A(r) is calculated by summing the contributions from the whole of space weighted by the smoothing kernel centered on r using the local value A(r) of the smoothing length: {A{r)) gather = / A(r') fP (r r, A(r)) dV' . (2-65) Hence the contributions to A at r are Â“gatheredÂ” in from the surrounding elements, weighted appropriately by the kernel at r. The Â“scatterÂ” method, on the other hand, derives its estimate for the same quantity by using the kernel local to the contributing

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48 element and Â“scatteringÂ” its appropriate contribution to position r: {A{r)) scatter = A{v')W { v V , \{r'))(fr' . (2-66) Exclusive use of one or the other of these approaches does not guarantee the conservation of energy, linear momentum, or angular momentum. A combination is usually used. Benz (1990) has symmetrized the smoothing length by using Wij = W(ri rj, (Aj + Xj) / 2), while Hernquist and Katz (1989) have symmetrized the kernel itself Wij = 0.5(Vk(r/ rj. A;) + W{Vi Tj, Xj)). In the present work, the first method of symmetrizing the smoothing length is used. Two further questions remain concerning A(r, t): (1) How should A be defined? and (2) how should it evolve during the integration? It seems natural to associate the smoothing length with the local density such that denser regions have smaller smoothing lengths in order to provide better resolution. One way to do this (Steinmetz and Muller 1992) is to let Vr.) = c(^)', (2-67) where C is a parameter of order unity. The problem with this definition is that A(r,) is needed in order to compute p(ri) in the first place. One way around this problem is to use the value of the density at the previous time step to calculate the smoothing length (Miyama et al. 1984). Another way, proposed by Benz (1990), is to take the time derivative of both sides above and eliminate the density altogether using the equation of continuity to yield dX[v) dt 1 -AV Â• V. 2 ( 2 68 ) In this way the smoothing length can be considered to be just another hydrodynamic variable to be evolved in the course of the integration. Yet another method (Hernquist

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49 and Katz 1989) is to link the value of A to the number of neighboring particles (within a radius of 2A), and let A change in space and time so as to keep this number fixed. The method used here is the second one where A is given an initial value for each particle and is subsequently evolved according to Eq. (2-68) above. The algorithm used to integrate the hydrodynamic equations of motion is a predictor-corrector time stepping rule. At any given point in the integration a single time step St is used to advance all the Â“particles.Â” The stepsize varies from one step to another and is given by the relation St Â— min (2-69) is the estimate of the circular orbit period at position r,-, and (the Courant number) is an input parameter of order 0.5 utilized to suppress numerical instabilities. In all cases we take c/v to be 0.4. Also, F,and v,are the acceleration and velocity of the particle. The terms in the expression for St are included to allow, respectively, fast-moving Â“particles,Â” highly accelerated Â“particles,Â” and Â“particlesÂ” very close to the center to be followed accurately. where T Â— 2Tr

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CHAPTER 3 PROGRAM GALAXIES The galaxies chosen for this study are NGC 3992, NGC 1073, and NGC 1398, and are shown in Figures 3-1, 3-2, and 3-3. These galaxies were chosen for several reasons. First, the neutral hydrogen emission from all three galaxies has been observed at the Very Large Array (VLA) of the National Radio Astronomy Observatory (NRAO) at high resolutions and with high signal-to-noise ratios. These observations reveal information about the kinematics, and therefore the total gravitational potential, of the disk. Secondly, detailed surface photometry exists for each of the galaxies. Elmegreen and Elmegreen (1985) have provided both blue and near-infrared surface photometry of NGC 3992 and NGC 1073. Ohta et al. (1990) have observed NGC 1398 in the B band. In addition, Grosbpl (1985) has scanned and analyzed the images of all three galaxies on the red Palomar Sky Survey copy plates. The surface photometry yields information concerning the distribution of luminous matter in the disk. Finally, these galaxies span a moderately wide range of barred spiral morphological types, de Vaucouleurs et al. (1976) classify NGC 1398 as (R')SB(r)ab, NGC 3992 as SB(rs)bc, and NGC 1073 as SB(rs)c. This latitude allows us to look for possible correlations between the model parameters of the most successful models and morphological type. Furthermore, since these galaxies show no gross peculiarities, they can be considered to be representative examples of their respective types. 50

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Vh a Â• o oo c/5 oo ON C/5 (D o X 13 ct3 o 00 o o o 4h o Â•4Â— > c/5 c/5 Oj Q < 13 < o 00 bX) < o O g (N On ON c/5 O U cn o U 4 Â— > O bX) Z G 'C Â• Â• G 1 c/5 1 m c3
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NGC 3992 52 r f SBb(rs) |. V=1134

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Ui <2 a Â• o 0? c/3 oo D Os c/3 o o X n cd o 00 o (D O o ^ c/3 cd Â• ^ T3 Q < 13 < o 00 bX) < o 2 O s cn o o U U :3 W) Â• iÂ»4 Uh

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NGC 1073 54 4 SBc(rs) II V=1318

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c/3
PAGE 74

56 / K * Â• J \ \ Â» Â« > n \ Â• f < 4 c

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57 NGC 3992 Observations Neutral Hydrogen High-resolution neutral hydrogen observations of NGC 3992 were obtained by Gottesman, Ball, Hunter, and Huntley (1984, hereafter GBHH) at the VLA between September, 1980 and January, 1983. The effective resolution obtained was 26." 1 x 20. "0, with the major axis of the Gaussian beam at a position angle of-22.Â°l. In comparison, the diameter of the HI disk is 504" and the position angle of the line of nodes is -1 1 1.Â°5 Â± 0.Â°6. The fullwidth half-power (FWHP) velocity resolution was 25.2 km s~^ The reader is referred to GBHH for further details of the data reduction. The angle-averaged rotation curve was derived, along with the following orientation parameters for the HI disk, via a least-squares fit to the velocity field: major axis at a position angle of -lll.Â°5 + 0.Â°6, inclination of 53.Â°4 + 0.Â°9, and a systemic velocity of 1045.8 + 0.6 km sÂ“^. The data points so derived are given in Table 3-1. The quoted errors are the formal errors of the least-squares solution. The angular radii of columns 1 and 4 in Table 3-1 have been converted to linear dimensions in columns 2 and 5 using an assumed distance to NGC 3992 of 14.2 Mpc (de Vaucouleurs 1979). The data in Table 3-1 are plotted as filled circles with error bars in Figure 3-4. GBHH find that the spiral structure of NGC 3992 is only weakly discernible in the neutral hydrogen distribution. The optical image, nevertheless (Figure 3-2), is seen clearly to reveal spiral structure which extends from an incomplete ring around the bar.

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58 Table 3-1: HI rotation velocity data for NGC 3992. R (arcmin) (1) R (kpc) (1) V (km s (2) R (arcmin) (1) R (kpc) (1) V (km s ^) (2) 0.14 0.58 141.8 + 37.8 2.89 11.9 269.1 + 5.5 0.37 1.53 123.6 Â± 52.7 3.13 12.9 270.9 + 5.5 0.64 2.64 170.9 Â± 32.7 3.36 13.9 267.3 Â± 7.1 0.84 3.47 195.6 Â± 16.5 3.63 15.0 265.5 Â± 7.5 1.14 4.71 222.5 Â± 10.9 3.86 15.9 258.2 + 10.9 1.37 5.66 240.0 Â± 9.1 4.09 16.9 249.1 Â± 11.1 1.64 6.77 254.2 + 7.3 4.36 18.0 247.3 Â± 12.5 1.90 7.85 258.2 + 5.8 4.63 19.1 240.0 Â± 10.9 2.14 8.84 262.2 Â± 4.7 4.89 20.2 240.0 + 20.0 2.36 9.75 265.5 Â± 4.2 5.10 21.1 238.2 Â± 18.2 2.63 10.9 267.3 + 4.4 (1) Radius measured from the center of the galaxy (2) Rotation velocity They also find a deficiency of HI inside this optical ring. Two possibilities exist. Either the total gas surface density suffers a real depression in the central region of the galaxy, or most of the hydrogen in this region has been converted to molecular form and is not being seen in the HI observations. GBHH cite observations of NGC 3992 by Young, which set an upper limit to the H 2 surface density of only a few times larger than the neutral hydrogen surface density, and conclude that the total gas surface density near the center appears to be less than in the disk. They add that Â“it is possible that dynamical effects associated with the bar may have exaggerated the phenomenonÂ” (GBHH, pp. 477-478). As we shall see from the results of our gas calculations (Chapter 6), bars can indeed Â“sweep outÂ” gas from large regions inside the bar radius.

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59 Figure 3-4: Comparison of observed and theoretical rotation curves for NGC 3992. The theoretical curve is derived from our most successful model of NGC 3992. Also shown are the contributions of the separate components of this model to the total theoretical rotation curve. As far as the kinematics of NGC 3992 are concerned, GBHH find that the motion of the gas is dominated by circular rotation, with only weak irregularities associated with the optical spiral arms. They interpret these irregularities as streaming of the gas along the spiral arms. Also noted is a discontinuity in the isovelocity contours of the velocity field across the major axis of the galaxy, which the authors tentatively interpret as an effect of the flow of gas around the bar of NGC 3992. The uncertainty in this interpretation stems from the limited resolution in this region due to low gas densities. Finally, GBHH note some peculiarities in the neutral hydrogen velocities along the major axis but at radial distances greater than about 3. '8 (15.7 kpc). Specifically, the velocities show a rather sudden decrease at this radius, followed by a more gradual decline thereafter. GBHH identify this kinematical feature with a possible truncation of the disk.

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60 Table 3-2: Selected global and disk parameter values adopted for NGC 3992. The errors given are simply the formal errors of a least-squares analysis of the observed velocity field and do not imply that these quantities have been determined to this level of precision. Parameter Value Systemic velocity (heliocentric, km sÂ“^)^ 1045.8 + 0.6 Distance (Mpc)'Â’ 14.2 Inclination angle (Â°)Â® 53.4 + 0.9 Position angle, line of nodes (Â°)Â® -111.5 + 0.6 ^Gottesman, Ball, Hunter, and Huntley (1984) 'Â’de Vaucouleurs (1979) Surface Photometry Elmegreen and Elmegreen (1985, hereafter EE) have provided blue {B passband) and near-infrared (/ passband) surface photometry of fifteen barred spiral galaxies, including NGC 3992. They photographed these galaxies between 1979 and 1981 using the 1.2 meter Palomar Schmidt telescope. The blue images were taken using baked 103a-O emulsions with a GG 385 filter, yielding an effective wavelength of 4350 A. The near infrared images were taken using hypersensitized IV-N emulsion with Wr 88 A filter, yielding an effective wavelength of 8250 A. Here again, the reader is referred to EE for further details concerning the observations and data reduction. Primary emphasis has been placed on the / passband images because they better describe the distribution of the older disk stars, the gravitationally dominant component of the disk. EE determine the near-infrared disk scale length of NGC 3992 to be 3.38 Â± 0.52 kpc, assuming a distance of 11.3 Mpc to the galaxy. Converting this scale length to our assumed distance of 14.2 Mpc yields a value of 4.25 + 0.65 kpc, or an inverse disk scale length of 0.235 kpc ^ EE also determined the slope of the

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61 radial dependence of the spiral arm-interarm surface brightness difference, from which we derive the inverse scale length of the spiral perturbation of 0.4 kpcÂ“ . Hunter et al. (1988) have analyzed the bar figure of NGC 3992 in the infrared image taken by EE. They determined the orientation and apparent extent of the bar from the positions of the extreme ends of its apparent major axis. Further, the apparent axial ratio /5o was estimated by fitting ellipses to the outermost isophotes of the bar, after subtraction of the bulge component. They found that the major axis of the projected bar figure makes an angle = 35Â° with respect to the line of nodes of the disk, that the semimajor axis of the bar is 62", and that the apparent axial ratio /?o = 2.6. These three quantities were used along with the inclination angle i and the adopted distance to derive, via the method of Stark (1977, also see Chapter 2), the actual linear dimensions of the triaxial bar figure (again, assuming da = 0.1). The resulting long bar axis a is 5.5 kpc, the intermediate axis 6 is 2.1 kpc, and the short axis c is 0.55 kpc. Again, it is assumed that the long and intermediate bar axes lie in the disk plane. Finally, Kennicutt (1981) gives the spiral arm pitch angle of NGC 3992 as -11Â° Â± 2Â°. Table 3-3 summarizes the quantities derived from the surface photometry that are used in this work. Best Model The first step in determining the most successful model of NGC 3992 was to fit visually the observed rotation curve with the theoretical curve comprised of contributions from the bar, disk, and halo. Next, the pattern speed ftp was fixed by the assumption that the long bar axis is equal to the corotation radius. Finally, initial values of the parameters characterizing the bar/spiral perturbation were set by the surface photometry.

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62 Table 3-3: Photometrically derived parameter values for NGC 3992. Parameter Value Inverse disk scale length (kpcÂ“')Â® n OQK +0-043 Inverse spiral scale length (kpcÂ“')^ 0.4 Angle between projected bar major axis and disk line of nodes ('Â’)'Â’ 35 Apparent bar semimajor axis (arcsec)'Â’ 62 Apparent bar axial ratio'Â’ 2.6 Lxjng bar axis length (kpc)'Â’ 5.5 Intermediate bar axis length (kpc)'Â’ 2.1 Short bar axis length (kpc)*Â’ 0.55 Spiral arm pitch angle (Â°)Â‘' -11 Â± 2 Â“Elmegreen and Elmegreen (1985) 'Â’Hunter, Ball, Huntley, England, and Gottesman (1988) Â‘Â’derived using the method of Stark (1977) Â‘'Kennicutt (1981) The periodic orbits of the model were then determined, and the surface response density was calculated and measured for self-consistency with the imposed surface density. The free model parameters, primarily the ones characterizing the spiral perturbation, were adjusted and the model differentially corrected until the most successful model was found. Table 3-4 summarizes the parameters of the most successful model of NGC 3992. The rotation curve of this model is shown in Figure 3-4. Also shown are the contributions of the separate model components. Figure 3-5 shows the characteristics of the different orbit families which are included in the model. The most important periodic families in the bar are the 2/1 (Figure 3-6) and 4/1 (Figure 3-7) resonant families. The 3/1 resonant family exists over a shorter range of energy, and these orbits

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63 are found not to be very critical to the overall structure of the bar. Also included inside the bar radius <3 is a set of Â“circularÂ” orbits extending from Vc = 3.5 kpc to = 5.5 kpc. Table 3-A: Parameter values for the best self-consistent model of NGC 3992. Ferrers Bar Spiral Mb = 1.5 X lO^'^MÂ© A = 2000 km^ s"^ kpcÂ“^ a = 5.5 kpc Ar = 0 km^ sÂ“^ kpcÂ“^ b = 2.1 kpc Ss = 0.4 kpc~^, iQ = -10Â° c = 0.55 kpc ri = 1.5 kpc, K 2 = 10.6 kpc Vlp = 43.6 km sÂ“^ kpc~^ = K 2 = 1, A = 0.1 kpc Exponential Disk Plummer Sphere Halo CO = 750MÂ© pc-2 Mh = 2.75 x 10Â“MÂ© Â£0 = 0.235 kpc-i bH = 12 kpc Figure 3-5: Characteristics of the orbit families included in the model. Each characteristic plots x, where a given orbit crosses the minor bar axis b, as a function of Jacobi constant, as parameterized by

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64 Outside the bar the main family is the -2/1 resonant family (Figure 3-8). These orbits support the imposed spiral in the region between the -4/1 resonance (outside of corotation) and the outer Lindblad resonance (OLR). The -4/1 and -6/1 resonant families are also included, but they play a much less significant role than does the -2/1 family in supporting the imposed spiral structure. Two more sets of Â“circularÂ” orbits are included outside of corotation. The first, a continuation of the set inside of corotation, extends from = 5.5 kpc to 7.5 kpc. The second, included to sample the response of matter near and slightly beyond the OLR, extends from rc = 10 kpc to 12 kpc. The use of the Â“circularÂ” orbits near corotation does not imply that there are no stable periodic orbits in this region. However, as we shall see in Chapter 5, the available phase space in this interval of energy is dominated by stochasticity and the trapping done by periodic orbits is small. Therefore, given the magnitude of the imposed velocity dispersion, it makes essentially no difference to the resulting density response map whether these orbits have the initial conditions of periodic orbits or not. Furthermore, despite the fact that the majority of these orbits are stochastic, they do provide significant enhancement of the imposed spiral structure and are instrumental in achieving self-consistency. The behavior of these stochastic orbits will be considered in more detail in Chapter 5. Table 3-5 summarizes the positions of the main resonances in the model. The epicyclic frequency Â«(r) and angular rotation rate 0(r), necessary for determining the resonance positions, are obtained by averaging and in azimuth, where Vsir, 9) is the potential of the bar, and adding them to the corresponding quantities of the disk and halo. k{v), then, is simply 0(r) is i , where Vr(r) is the total axisymmetric potential.

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65 Figure 3-6: The 2/1 family of periodic orbits in the model of NGC 3992. The darker circle represents corotation at 5.5 kpc. Figure 3-7: The 4/1 family of periodic orbits in the model of NGC 3992.

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66 Figure 3-8: The -2/1 family of periodic orbits in the model of NGC 3992. The darker curves represent the minima of the bar and spiral potentials. The density response map was generated according to the procedure outlined in Chapter 2. Table 3-6 gives the parameters used for the surface density response calculation. Table 3-5: Resonance locations of the best model of NGC 3992. Inner Resonances Outer Resonances Type Location Type Location 2/1 0.0 kpc -8/1 6.9 kpc 4/1 3.1 kpc -6/1 7.3 kpc 6/1 3.9 kpc -4/1 8.2 kpc 8/1 4.3 kpc -2/1 10.6 kpc corotation = 5.5 kpc

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67 Table 3-6: Parameters used to calculate the surface density response for the best model of NGC 3992. Parameter Value Minimum grid radius, RMIN (kpc) Maximum grid radius, RMAX (kpc) Radial cell width, DRS (kpc) Number of radial cells Azimuthal cell width (Â°) Number of azimuthal cells Number of radial start positions per grid cell, NOR Number of azimuthal start positions per orbit, NOA Length of (quasi)periodic orbit integration, PN Central velocity dispersion, ao (km sÂ“^) 1 1 Slope of the velocity dispersion dependence, Cr (km sÂ“ kpcÂ“ ) Time of "circular" orbit integration (10^ yr) 0.1 12.1 0.2 60 1.4 128 2 6 12 100 -7 0.98 Grayscale representations of the resulting surface density response are shown in Figures 3-9 (unprojected) and 3-10 (projected to the actual orientation of NGC 3992, for comparison with Figure 3-1). These grayscale images were produced by first rebinning the polar grid of density values into a Cartesian grid. The resulting image was then smoothed using a Â“beamÂ” five pixels square such that the smoothed pixel derived onethird of its value from the unsmoothed pixel, another one-third from the eight directly adjacent pixels, and the final third from the sixteen outer pixels (giving a standard deviation for an equivalent Gaussian beam of 1.2 pixels). There are a couple of features to note in these images. Most obviously we see that the overall barred spiral appearance of NGC 3992 is well-reproduced by the model. This point is perhaps best appreciated by directly comparing the projected figure (Figure 3-10) with the photograph of NGC 3992 (Figure 3-1). Also, we see that model spiral arm makes a slight bend at approximately

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68 the position of the -4/1 resonance (8.2 kpc). This feature seems also to be present in NGC 3992 itself, particularly for the spiral arm in the northeast quadrant. These Â“elbowsÂ” were also noted by Patsis et al. (1991) in their models of unbarred spirals. After the surface response density was calculated, the self-consistency of the model was tested by comparing the response to the imposed surface density. Figures 3-11 and 3-12 give R and A^, the two measures of self-consistency. These measures, the ratio of the relative amplitude of the response 29 component to the relative amplitude of the imposed 29 component and the phase difference between these two components, respectively, are described in detail in Chapter 2. We note that the value of R is almost constant at one (varying between 1.4 and 0.8) between 2.1 and 8.5 kpc. There is an upward divergence in R* at small radii (not shown in Figure 3-1 1) due to the fact that the relative azimuthal variation of the imposed density drops rather rapidly inside the length of the bar minor axis, greatly decreasing the relative amplitude of the imposed 29 component, while the shape, and hence the density response, of the dominant 2/1 family remains rather elongated, thereby maintaining the relative amplitude of the response 29 component. These combined effects, together with the possible existence of a nuclear bulge, indicate that, in general, we cannot meaningfully speak of a measureable bar component existing all the way to the center. Here, therefore, and in the cases of NGC 1073 and NGC 1398 also, we take the bar semiminor axis length as a practical inner limit for R . The divergence near the OLR is due to several factors. First, the orbits here retain their somewhat elongated shape despite the fact that the strength of the imposed spiral has considerably diminished. Secondly, there exists a congestion of matter resulting from the response of the regular

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69 o 10 -5 0 5 10 kpc Figure 3-9: Grayscale image of the unprojected surface density response of the best model of NGC 3992.

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70 -10 -5 0 5 kpc Figure 3-10: Grayscale image of the surface density response of the best model of NGC 3992 projected to the galaxyÂ’s actual orientation. 10

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71 orbits trapped (with only a small dispersion of velocities) around the -2/1 family. Thirdly, and perhaps most importantly, the effect on R of an azimuthal density variation of a given absolute size will be greatly magnified in the outer regions where the axisymmetric component of the density becomes small. The angle deviation A9 is absolutely less than 0.12 radian between 2.1 and 7.7 kpc. Beyond 7.7 kpc we see that the response spiral systematically lags the imposed up to the -4/1 resonance. Here we see a short sharp increase in Ad and then a much steeper decline out to the OLR. This same systematic lag has been observed in similar models of normal spirals with strong spiral arms (Contopoulos and Grosbpl 1988; Patsis et al. 1991), and seems to indicate that logarithmic spirals may not be the best mathematical description of real spirals. Figure 3-11: The response-to-imposed 29 component amplitude ratio R* for the best model of NGC 3992. The positions of the major outer resonances (corotation, -A !\ , and outer Lindblad) are noted.

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72 kpc Figure 3-12: The phase difference A9 (in radians) between the response and imposed 20 components of the best model of NGC 3992. The positions of the major outer resonances are noted as in Figure 3-11. NGC 1073 Observations Neutral Hydrogen In a continuation of the effort initiated by GBHH to study the properties of neutral hydrogen in a carefully selected sample of barred spiral galaxies, NGC 1073 was observed by England, Gottesman, and Hunter (1990, hereafter EGH) at the VLA between June, 1983 and June, 1984. The effective resolution obtained was 20. "3 x 19. "7, with the major axis of the Gaussian beam at a position angle of 62.Â° 8. Eor comparison purposes, the diameter of the HI disk is 396" and the position angle of the line of nodes is -15. Â°4 Â± 0.Â°2. The FWHP velocity resolution obtained was 12.63 km s~^. EGH gives full details of the VLA data reduction.

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73 The angle-averaged rotation curve, along with the following orientation parameters of the HI disk, was derived via a least-squares fit to the velocity field: major axis at a position angle of-15.Â°4 + 0.Â°2, inclination of 18.Â°5 Â± 2.Â°5, and a systemic velocity of 1208.9 + 0.2 km s"^. The rotation curve data is given in Table 3-7. Here again, the quoted uncertainties are simply the formal errors of the least-squares solution. In converting the measured angular radii to a linear scale we have assumed a distance to NGC 1073 of 13.6 Mpc (de Vaucouleurs 1979). The rotation curve data in Table 3-7 are plotted as filled circles with error bars in Figure 3-13. Table 3-7: HI rotation velocity data for NGC 1073. R (arcmin) (1) R (kpc) (1) V (km s-^) (2) R (arcmin) (1) R (kpc) (1) V (km s ^) (2) 0.10 0.39 38.8 Â± 7.6 2.70 10.6 100.8 + 5.6 0.30 1.18 83.2 Â± 4.4 2.90 11.4 95.2 Â± 6.6 0.50 1.97 96.0 + 3.2 3.10 12.2 92.0 + 7.7 0.70 2.76 93.6 Â± 2.4 3.30 13.0 100.8 + 8.4 0.90 3.55 98.8 + 3.2 3.50 13.8 91.2 + 12.0 1.10 4.34 99.6 Â± 3.2 3.70 14.6 69.6 Â± 26.4 1.30 5.12 99.6 Â± 2.2 3.90 15.4 74.4 + 28.3 1.50 5.91 100.0 Â± 2.8 4.10 16.2 61.6 + 31.6 1.70 6.70 105.6 Â± 3.2 4.30 17.0 24.0 + 33.0 1.90 7.49 106.0 + 3.2 4.50 17.7 72.0 + 44.4 2.10 8.28 103.2 Â± 2.8 4.70 18.5 120.0 Â± 52.4 2.30 9.07 104.0 + 4.4 4.90 19.3 168.0 Â± 53.6 2.50 9.85 105.6 Â± 6.0 % (1) Radius measured from the center of the galaxy (2) Rotation velocity

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74 EGH find that the distribution of the neutral hydrogen in NGC 1073 is that of an almost circular disk, with a nearly complete ring surrounding a central depression; however, this central depression is not as deep relative to the outer disk as in the case of NGC 3992. They also note the presence of a gas bar in NGC 1073 aligned with the optical bar. This feature shows up at around 30% of the peak HI density, compared to the general disk gas density which is approximately 20% of peak. EGH find little evidence of spiral structure in the gas, however, in sharp contrast to the optical image in which there are two prominent spiral arms starting about 30Â° in azimuth from the ends of the bar. While the regions of high HI density (>50% of the peak density) show a correlation with the bright optical regions of the spiral arms, a broad gaseous ring emerges when lower HI densities (-40% of peak) are considered. This ring extends from about r = 0.'7 to r = 2.'0, compared to the optical spiral arms which extend from about r = 0.'75 to r = l.'l. Finally, EGH note a steep gradient in the gas density in the northwest quadrant of the galaxy. While they postulate that this feature might be due to an interaction, a search for 21 cm emission in a l.Â°5 x l.Â°5 area of sky surrounding NGC 1073 yielded no evidence of such an interacting object. The gas kinematics of NGC 1073 is dominated by circular motion. Still, irregularities in the velocity field due to gas streaming are observed where the optical spiral arms cross the isovelocity contours (EGH 1990). Like NGC 3992, NGC 1073 may possess a truncated disk. EGH cite as evidence for this the dropoff in the rotation curve starting at approximately r = 10 kpc, the corresponding drop in the angle-averaged, deprojected HI surface density, and the success achieved by truncated mass models in reproducing the rotation curve of NGC 1073 (cf. Casertano 1983; Hunter et al. 1984).

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75 Figure 3-13; Comparison of observed and theoretical rotation curves for NGC 1073. Table 3-8: Selected global and disk parameter values adopted for NGC 1073. Here again, the errors listed are simply the formal errors of a least-squares analysis of the velocity field. Parameter Value Systemic velocity (heliocentric, km sÂ“^)^ 1208.9 + 0.2 Distance (Mpc)Â’Â’ 13.6 Inclination angle (Â°Y 18.5 + 2.5 Position angle, line of nodes (Â°)^ -15.4 + 0.2 ^England, Gottesman, and Hunter (1990) Â’Â’de Vaucouleurs (1979) Surface Photometry As for NGC 3992, Elmegreen and Elmegreen (1985) have provided B and / passband surface photometry of NGC 1073. They determined the near-infrared disk scale length of NGC 1073 to be 3.18 Â± 0.36 kpc assuming a distance of 13.2 Mpc. For our assumed distance of 13.6 Mpc, the scale length becomes 3.28 Â± 0.37 kpc, yielding

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76 an inverse disk scale length of 0.305 ^Q ggj kpcÂ“^ The initial value of the inverse scale length of the spiral, derived from the slope of the spiral arm-interarm surface brightness difference given by EE, is 1.4 kpcÂ“^ EGH have analyzed the bar figure of NGC 1073 in the infrared image taken by EE. Following the same procedure described for the case of NGC 3992, they find that the angle ^ between the major axis of the projected bar and the line of nodes is 75Â°. Also, they find that the bar has a semimajor axis = 43" and an apparent axial ratio j3o = 7.19. The actual linear dimensions of the bar derived from the Stark method are a = 2.95 kpc, b = 0.39 kpc, and c = 0.295 kpc. The initial value of the spiral arm pitch angle, determined by a visual fit to the optical photograph of NGC 1073 reproduced in EGH from Arp and Sulentic (1979), is -10Â°. Table 3-9 summarizes the quantities derived from the surface photometry. Table 3-9: Photometrically derived parameter values for NGC 1073. Parameter Value Inverse disk scale length (kpc~')Â“ n QnK+O-039 U.OUO_Q Q3^ Inverse spiral scale length (kpcÂ“')^ 1.4 Angle between projected bar major axis and disk line of nodes (Â°)'Â’ 75 Apparent bar semimajor axis (arcsec)'Â’ 43 Apparent bar axial ratioÂ’Â’ 7.19 Long bar axis length (kpc)Â° 2.95 Intermediate bar axis length (kpc)Â° 0.39 Short bar axis length (kpc)Â° 0.295 Spiral arm pitch angle (Â°)*' -10 ^Elmegreen and Elmegreen (1985) 'Â’England, Gottesman, and Hunter (1990) ^derived using method of Stark (1977) Â‘'visual fit

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77 Best Model The best model for NGC 1073 was determined following the general procedure outlined for the case of NGC 3992. Table 3-10 summarizes the parameters of this model. Table 3Â—10: Parameter values for the best self-consistent model of NGC 1073. Ferrers Bar Ms = 1.8 X lO^MÂ© a = 2.95 kpc b = 0.39 kpc c = 0.295 kpc ftp = 32.2 km sÂ“^ kpcÂ“^ Spiral A = 9000 km^ sÂ”^ kpc"^ Ar = 0 km^ sÂ“^ kpc"^ = 1.2 kpcÂ“^ io = -10Â° ri = 2.95 kpc, V 2 = 5.63 kpc Ki = K 2 = 0.5, A = 0.1 kpc Exponential Disk Plummer Sphere Halo CO = 250MÂ© pc"2 Mh = 1.0 X lO^^MÂ© Â£0 = 0.305 kpc-i bn = 9 kpc The rotation curve of this model is given along with the observations in Figure 3-13. The characteristics of the orbit families which comprise this model are shown in Figure 3-14. The most dynamically important periodic orbits, that is, those that are most effective in trapping quasi-periodic orbits around them, are the two stable branches of the 2/1 resonant orbits in the bar and the -2/1 resonant orbits in the outer disk (Figure 3-15). Two sets of Â“circularÂ” orbits are also included, a sizeable one extending from Kc = 1.25 kpc to Tc = 4.75 kpc and a smaller second one extending from Vc = 5.25 kpc to Vc = 5.6 kpc. As in the case of NGC 3992, the inclusion of these Â“circularÂ” orbits does not imply that no periodic orbits exist in this region. We shall explore further the phase space structure of this region in Chapter 5. Table 3-11 gives the positions of the main resonances in this case.

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78 Figure 3-14: Characteristics of the orbit families included in the best model of NGC 1073. kpc Figure 3-15: Representative orbits of the three main periodic families in the best model of NGC 1073.

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79 Table 3-11: Resonance locations of the best model of NGC 1073. Inner Resonances Outer Resonances Type Location Type Location 2/1 0.0 kpc -8/1 3.6 kpc 4/1 1.8 kpc -6/1 3.8 kpc 6/1 2.2 kpc -4/1 4.3 kpc 8/1 2.4 kpc -2/1 5.6 kpc corotation = 2.95 kpc Table 3-12 gives the parameters of the surface density response calculation. Table 3-12: Parameters used to calculate the surface density response for the best model of NGC 1073. Parameter Minimum grid radius, RMIN (kpc) Maximum grid radius, RMAX (kpc) Radial cell width, DRS (kpc) Number of radial cells Azimuthal cell width (Â°) Number of azimuthal cells Number of radial start positions per grid cell, NOR Number of azimuthal start positions per orbit, NOA Length of (quasi)periodic orbit integration, PN f 2 * x N oA ) Central velocity dispersion, ao (km sÂ“^) Slope of the velocity dispersion dependence,
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80 Figure 3-16: Grayscale image of the unprojected surface density response of the best model of NGC 1073.

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81 best model, the spiral arms of this model do not exhibit any noticeable Â“elbows.Â” Furthermore, it is observed that the model arms tend to wrap back onto themselves after winding approximately 270Â° from their origin at the ends of the bar. This Â“wrappingÂ” occurs at about the radius of the OLR. One note of discrepancy between the model and observations exists, though. While the actual spiral arms of NGC 1073 are observed to begin some 30Â° in azimuth from the ends of the bar (EGH 1990), the arms of the best model come directly off the ends of the bar. Figure 3-17: The response -to-imposed 20 component amplitude ratio R* for the best model of NGC 1073. The positions of the major outer resonances (corotation, -4/1 , and outer Lindblad) are noted. Figures 3-17 and 3-18 give R* and AO for the best model of NGC 1073. Again we see the same general features here as in the case of NGC 3992. The value of R* is almost constant (varying between 0.8 and 1.3) from 0.6 kpc to 4.1 kpc. There is also, for the same reasons as in the model of NGC 3992, an upward divergence in R* at large

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82 radii. The angle deviation /S.6 remains absolutely less than 0.12 radian from 0.39 kpc out to 3.9 kpc. Again we see a gradual systematic lag in the response 26 component relative to the imposed from the end of the bar at corotation out to approximately the -4/1 resonance. Beyond this point A0 drops more rapidly out to the OLR, albeit not as sharply as in the case of NGC 3992. kpc Figure 3-18: The phase difference (in radians) between the response and imposed 20 components of the best model of NGC 1073. Again, the positions of the major outer resonances are noted. NGC 1398 Observations Neutral Hydrogen Moore and Gottesman (1993) made high resolution HI observations of NGC 1398 at the VLA between February, 1991 and February, 1992. The effective linear resolution

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83 obtained was 22."3 x 20. "2, with the major axis of the Gaussian beam at a position angle of -35.Â°9. The diameter of the HI disk was found to be approximately 720" and the position angle of the line of nodes is -84.Â° 5 + 0.Â°5. The FWHP velocity resolution was 25.4 km sÂ“^. From the neutral hydrogen velocity maps an angle-averaged rotation curve for NGC 1398 was derived in the same way as for NGC 3992 and NGC 1073. In the fitting procedure the HI disk was determined to have the following properties: major axis at a position angle of -84.Â°5 Â± 0.Â°5, inclination of 46.Â°3 Â± 0.Â°7, and a systemic velocity of 1397.0 Â±1.0 km sÂ“^. As in the cases of NGC 3992 and NGC 1073, the HI surface density distribution of NGC 1398 shows evidence of a depletion in the central region. Moore (private communication) notes that this HI Â“holeÂ” is rather small (at a lower spatial resolution Â— 32" x 60" beam) extending about 30" to 40" from center, a distance only slightly less than the deprojected semimajor axis of the bar. Also, in the high resolution HI density map, a rather clumpy gas ring extending over the region of the outer optical spiral arms is observed. Analysis of the high resolution data yielded a rotation curve which contained no information about the central rising part of the curve. Therefore, another rotation curve was then derived using the lower spatial resolution in order to increase the signal-to-noise ratio in the inner region. The rotation curve finally adopted for NGC 1398 is, then, a composite of the two. In the inner region the rotation data come from analysis of the low resolution map only. For the remainder of the rotation curve an average of the two data sets is used. Near the edge of the HI disk the two curves diverge for reasons unknown, but this occurs beyond the end of our adopted rotation curve and does not affect our results. The rotation curve data are

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84 given in Table 3-13. The distance to NGC 1398 is taken to be 18.7 Mpc (Buta and de Vaucouleurs 1983). The data are plotted as filled circles with error bars in Figure 3-19. Table 3-13: HI rotation velocity data for NGC 1398. R (arcmin) (1) R (kpc) (1) V (km s-^) (2) R (arcmin) (1) R (kpc) (1) V (km s ^) (2) 0.26 1.41 128.2 + 65.7 2.97 16.2 301.5 + 5.0 0.54 2.94 250.6 + 19.8 3.32 18.1 295.4 Â± 5.4 0.89 4.84 293.3 + 5.8 3.67 20.0 291.2 + 6.4 1.24 6.75 299.1 Â± 4.9 4.03 21.9 289.2 Â± 4.1 1.59 8.65 300.7 Â± 4.3 4.37 23.8 289.6 Â± 1.3 1.93 10.5 307.7 Â± 4.6 4.72 25.7 285.9 Â± 2.5 2.28 12.4 307.7 + 2.9 5.06 27.5 279.8 + 6.4 2.64 14.4 305.2 + 3.8 (1) Radius measured from the center of the galaxy (2) Rotation velocity Figure 3-19: Comparison of observed and theoretical rotation curves for NGC 1398. The contributions of the separate model components are also shown.

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85 Table 3-14; Selected global and disk parameter values adopted for NGC 1398. Parameter Value Systemic velocity (heliocentric, km sÂ“^)^ 1397 + 1 Distance (Mpc)*Â’ 18.7 Inclination angle (Â°)'^Â’Â‘^ 43 Position angle, line of nodes (Â°)*^^ 99.8 ^Moore and Gottesman (1993) 'Â’Buta and de Vaucouleurs (1983) ^Ohta, Hamabe, and Wakamatsu (1990); this source quotes no errors for these quantities. ^*These values are used since the corresponding values from Moore and Gottesman (1993) were not available at the time of the model runs. Surface Photometry Ohta et al. (1990) have provided B passband photometry of six early-type barred spiral galaxies, including NGC 1398. They have taken two photographs of NGC 1398 (one 6 minute exposure and one 30 minute exposure) with the 2.5 m du Pont telescope at the Las Campanas Observatory using hypersensitized 103a-O emulsions. The reader is referred to Ohta et al. (1990) for complete details on the observations and data reduction. They decomposed the azimuthally averaged luminosity profile of NGC 1398 into an exponential disk and an r^^'^-law bulge by an iterative scheme. They determine the scale length of the exponential disk to be 59."6. Converting this scale length to linear dimensions by using our assumed distance yields an inverse scale length of 0.185 kpcÂ“^. This is the value we have adopted here. We were unable to find published data regarding the scale length of the spirals, thus we left the inverse spiral scale length parameter to be fit in the modeling procedure.

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86 They also analyzed the bar figure of NGC 1398 in their study. They give inclination-corrected values for the half-length and full width of the bar of 53" and 18", respectively. However, NGC 1398 possesses a significant bulge component and these bar dimensions are derived only from portions of the bar which extend beyond the bulge radius (i.e. where the light contributions from the two components are approximately equal), which they determine to be 22" for NGC 1398. This method overestimates the axial ratio of the bar in cases, such as this thesis, where only one component is used to represent the bar. Instead, the following method was used to derive the observed axial ratio !3q. Ohta et al. (1990) also give isophotes of the extracted bars (i.e. with the axisymmetric components subtracted, see Ohta et al. [1990] for details of the extraction process). The published isophotes for the bar of NGC 1398 derived in this way were measured for axial ratio, and an average value of = 2.15 was obtained. The given deprojected half-length of the bar was then reprojected, yielding a projected bar semimajor axis of 39". Also, from the isophotes of the extracted bar we measured the angle between the projected bar and the line of nodes to be i/Â’ = 91Â°. These three quantities, along with the adopted distance to NGC 1398 and the assumption that da = 0.1, were then used in the Stark analysis to derive the actual linear dimensions of the triaxial model bar. Finally, Kennicutt (1981) gives the spiral arm pitch angle of NGC 1398 as -6Â° Â± 2Â°. Table 3-15 summarizes the parameters derived for NGC 1398 from the surface photometry. Best Model The best model of NGC 1398 was determined by the same method as the best models for NGC 3992 and NGC 1073. Table 3-16 summarizes its parameters.

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87 Table 3-15: Photometrically derived parameter values for NGC 1398. Parameter Value Inverse disk scale length (kpcÂ“^)^ 0.185 Angle between projected bar major axis and disk line of nodes (Â°)^ 91 Â± 1 Apparent bar semimajor axis (arcsec)Â® 39 Apparent bar axial ratio^ 2.15 Long bar axis length (kpc)'Â’ 4.8 Intermediate bar axis length (kpc)'Â’ 1.6 Short bar axis length (kpc)*Â’ 0.48 Spiral arm pitch angle {Â°y Â— 6 Â± 2 Â®Ohta, Hamabe, and Wakamatsu (1990) ^derived using method of Stark (1977) Â‘^Kennicutt (1981) Table 3-16: Parameter values for the best self-consistent model of NGC 1398. Ferrers Bar Spiral Mb = 1.2 X 10^Â°MÂ© a = 4.8 kpc b = 1.6 kpc c = 0.48 kpc Qp = 54.2 km sÂ“^ kpc"^ A = 3000 km^ sÂ“^ kpcÂ“^ Ar = 0 km^ sÂ“^ kpcÂ“^ Â€s = 0.4 kpc"\ iÂ‘o = -6Â° r\ = 1.0 kpc, f 2 Â» OLR K\ = K 2 = 0.5, A = 0.1 kpc Exponential Disk Plummer Sphere Halo Co -1475 Mq pcÂ“^ Â£o = 0.185 kpcÂ“^ Mh = 6.0 X 10Â“ MÂ© bn = 35 kpc

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88 Figure 3-20: Characteristics of the orbit families included in the best model of NGC 1398. The rotation curve of this model is given along with the observations in Figure 3-19. The characteristics of the orbit families which comprise this model are shown in Figure 3-20. Once again the most important periodic family in the bar is the 2/1 resonant family. There exists in the model a branch of 4/1 resonant orbits, but they are of less dynamical importance than the 2/1 family. This is because they exist over a much smaller range of energy and trap much less matter around them than do the 2/1 orbits. Also, while there is a very small family of -2/1 orbits at about Vc = 12.0 kpc, the dominant orbital behavior in the outer disk of this model is decidedly stochastic. Accordingly, we include a set Â“circularÂ” orbits extending all the way from = 3.3 kpc to Vc = 11.8 kpc. We shall examine this stochasticity further in Chapter 5. Figure 3-21 shows representative orbits from the three stable periodic families included in the model.

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89 Figure 3-21: Representative orbits of the three main periodic families in the best model of NGC 1398. Table 3-17 gives the positions of the major resonances of the model. Table 3-17: Resonance locations of the best model of NGC 1398. Inner Resonances Outer Resonances Type 1 Location Type Location 2/1 0.0 kpc -8/1 6.1 kpc 4/1 2.5 kpc -6/1 6.6 kpc 6/1 3.2 kpc -4/1 7.5 kpc 8/1 3.6 kpc -2/1 9.8 kpc corotation = 4.8 kpc Table 3-18 summarizes the parameters used in the generation of the surface density response of the model.

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90 Table 3-18: Parameters used to calculate the surface density response for the best model of NGC 1398. Parameter Minimum grid radius, RMIN (kpc) Maximum grid radius, RMAX (kpc) Radial cell width, DRS (kpc) Number of radial cells Azimuthal cell width (Â°) Number of azimuthal cells Number of radial start positions per grid cell, NOR Number of azimuthal start positions per orbit, NO A Length of (quasi)periodic orbit integration, PN j Central velocity dispersion, ao (km sÂ“^) Slope of the velocity dispersion dependence, ar (km sÂ“^ kpcÂ“^) Time of "circular" orbit integration (10^ yr) Value 0.01 12.01 0.2 60 1.4 128 2 6 12 80 -5 0.98 Figure 3-22 shows a grayscale representation of the surface density response for the best model of NGC 1398. Since the bar is almost perpendicular to the line of nodes and the inclination angle is not very large (thus the main effect of the galaxyÂ’s inclination is merely to foreshorten the bar), we do not present a projected grayscale image. As Figure 3-22 shows, the model exhibits a well-defined structure, with the outer spirals extending smoothly outwards with no Â“elbowsÂ” and no strong tendency to wrap back onto themselves. This feature is consistent with the observed optical arms (Figure 3-3) which are long and continuous and Â“can be traced for about one and a quarter revolutions from their origin ...Â” (Sandage 1961). One feature of NGC 1398 which is not well-reproduced by our model is the inner ring, composed of nearly circular spiral segments, which encircles the bar. It is not clear, however, that this is actually a significant feature in the distribution of the underlying disk stars.

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91 Figure 3-22: Grayscale image of the unprojected surface density response of the best model of NGC 1398. 10

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92 Figures 3-23 and 3-24 give R* and AO for the most successful model of NGC 1398. In this case we do not have such a pronounced divergence of R at large radii as we had in the cases of NGC 3992 and NGC 1073. Except for a small dip to 0.4 around 9.4 kpc, R remains fairly constant between 0.8 and 1.3 from 1.6 kpc to 11.2 kpc. The angle deviation AO remains absolutely less than 0.4 radian all the way from 1.6 kpc out to 10.6 kpc. While this value is somewhat larger than in the cases of NGC 3992 and NGC 1073, the agreement is still rather good because the angle deviation is magnified by the small absolute value of the pitch angle (i.e. given the same deviation perpendicular to the spiral, the angular difference will be greater at a specified radius the lower is the pitch angle). In this case again, we see the tendency for the response spiral progessively to lag the imposed. Figure 3-23: The response-to-imposed 20 component amplitude ratio R* for the best model of NGC 1398. The positions of the major outer resonances (corotation, -A/\, and outer Lindblad) are noted.

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93 Figure 3-24: The phase difference A9 (in radians) between the response and imposed 26 components of the best model of NGC 1398. Again, the positions of the major outer resonances are noted.

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CHAPTER 4 VARIATION OF PARAMETERS In this chapter we examine how systematically varying the parameters affects the self-consistency of the Â“bestÂ” models described in Chapter 3. The purpose of such an examination is twofold. Firstly, we would like to identify those parameters which exert the most influence on self-consistency. In this way we might hope to identify quantities which act as eigenvalues of the self-consistency problem. Secondly, we would like to detect, if possible, any systematic trends in this set of controlling parameters as we move from early to late Hubble types. Since varying the same parameter in different models often produces essentially the same effect, we present complete details for the case of NGC 3992, and include the results of NGC 1073 and NGC 1398 only where they differ significantly from those of NGC 3992. Variation of A Figure 4Â—1 shows how varying the amplitude A of the bar/spiral perturbation potential affects self-consistency. ^ Decreasing the amplitude leads to worse selfconsistency as regards both R* (at all radii) and AO (outside corotation). On the other hand, while increasing A produces somewhat smoother runs of R and AO with radius, R is seen to decrease systematically below unity outside of corotation. 1. In each of the figures in this chapter we have included, for the sake of completeness, the mns of R (or A9) versus radius for the entire radial range spanned by the computational grid. For reasons detailed in Chapter 3, though, we do not expect good agreement over this entire radial range. Specifically, the upward divergence noted at very small radii can be safely ignored. 94

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95 kpc kpc Figure 4-1: The ratio R* and phase difference A0 in models of NGC 3992 where A = 2000 km^sÂ“^kpc~^ (Â“bestÂ” model, denoted by filled circles), A = 1000 km^sÂ“^kpcÂ“^ (open circles), and A = 4000 km^sÂ“^kpcÂ“^ (plusses).

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96 Variation of i4^ The only galaxy whose Â“bestÂ” model shows any real sensitivity to variations in the residual amplitude Ar is NGC 1073. The reason is that in the models of NGC 3992 and NGC 1398 the inner and outer cutoff radii (ri and ^ 2 ) for the primary amplitude A are, respectively, close to the center and at or beyond the OLR. That is, in these cases the imposed outer spiral perturbation is continued well inside of corotation as a bar perturbation in addition to the Ferrers bar component. Therefore, as the primary amplitude extends over most of the computational grid in these models, the residual amplitude does not really come into play. The Â“bestÂ” model of NGC 1073, however, has the inner cutoff r\ located at the corotation resonance (i.e. the end of the bar), and variations in the residual amplitude do produce any noticeable effects in the bar region. Figure 4-2: The ratio R* in models of NGC 1073 where the residual amplitude Ar = 0 km^sÂ“^kpcÂ“^ (Â“bestÂ” model, filled circles), 1000 km^sÂ“^kpcÂ“' (open circles), and 2000 km^sÂ“^kpcÂ“^ (plusses). For comparison, the primary amplitude A = 9000 km^sÂ“^kpcÂ“^

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97 Figure 4Â—2 shows these effects. Here, the inclusion of any residual amplitude inside the bar leads to worse self-consistency. Variation of /q Figure 4-3 shows the results of varying the spiral pitch angle /q. Rather large systematic displacements of R*, similar to those due to variations in A (Figure 4-1), are seen. These effects, in fact, are not independent. Contopoulos and Grosbpl (1986) showed that, depending on the assumed geometry, the maximum perturbed density of the spiral is proportional to either tanÂ“^ io (cylindrical geometry) or Itan io\~^ (flat geometry). Decreasing the pitch angle, therefore, produces the same qualitative effect on R* as increasing A. The plots of phase difference A9 exhibit somewhat larger variations here than in the case of varying A, particularly for the case /Â’o = -5Â°. This is due to the fact that, given the same perpendicular offset between imposed and response spiral, azimuthal differences between the two are magnified more in the case of a smaller pitch angle. Variation of Es A decrease of Cs produces a relative increase of the strength of the spiral outwards. As can be seen in Figure 4-4, this causes a greater variation in the imposed spiral amplitude than in the response; therefore, R* displays a systematic decrease outwards when the value of Ss is too small. Conversely, when Ss is too large, the imposed spiral dies off too quickly and the ratio R quickly becomes large. These results hold despite the fact that the value of A was adjusted in order to keep the bar/spiral perturbation amplitude the same at r = 2.5 kpc in all three cases. The phase agreement AO is

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98 approximately the same for the two lower values of Es. This is because the imposed spiral remains strong enough to force the response spiral to remain almost in phase with it. When Es is large, though, the imposed spiral is not strong enough to produce good phase agreement beyond corotation. Variation of ftp One of the assumptions of our modeling technique is that the bar figure ends at corotation. This assumption is based on results of orbit calculations in realistic bar potentials (e.g. Contopoulos 1980; Teuben and Sanders 1985) and of A^-body simulations (e.g. Sellwood 1981). Still, there is some disagreement as to the exact placement of corotation, and many authors claim that bars end somewhat inside of corotation. In order to test whether either of these changes improve self-consistency, we varied the bar/spiral pattern speed Qp of the Â“bestÂ” model of NGC 3992. Figure 4Â—5 shows the results of this variation. The open circles represent the case where corotation is placed at |a, a being the bar semimajor axis. The filled circles represent, as usual, the results of the Â“bestÂ” model. The plusses represent the case where corotation is placed at |a. The resonance positions marked in the diagram are for the Â“bestÂ” case with corotation at r = a. We judge that the best results, as regards both R* and A9, occur when the bar is assumed to end at corotation. Still, the results of the case where the bar ends before corotation are not very much worse than those of the Â“bestÂ” model. Therefore, we concur with those who claim that bars end at or slightly inside corotation, and in any case do not extend beyond corotation.

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99 kpc kpc Figure 4-3: The ratio R* and phase difference A9 in models of NGC 3992 where io = -10Â° (Â“bestÂ” model, denoted by filled circles), io = -5Â° (open circles), and io = -15Â° (plusses).

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100 kpc kpc Figure 4^: The ratio R* and phase difference A9 in models of NGC 3992 where Ss = 0.2 kpc~^ (open circles, A = 1213 km^sÂ“^kpcÂ“^), 0.4 kpcÂ“^ (filled circles, Â“bestÂ” model), and 0.8 kpcÂ“^ (plusses, A = 5437 km^s"^kpc"^).

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101 Di 0 2 4 6 8 10 12 kpc kpc Figure 4-5: Amplitude ratio R* and phase difference in models of NGC 3992 where fip = 34.7 km sÂ“'kpcÂ“^ (corotation = 4a/3, open circles), flp = 43.6 km s"^kpc~* (Â“bestÂ” model, corotation = a, filled circles), and Dp = 55.7 km sÂ“'kpcÂ“* (corotation = 3a/4, plusses).

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102 Variation of A Figure 4Â—6 shows the results of varying A, the z-thickness of the bar/spiral perturbation. We see, as expected, a similar sort of effect on the ratio R as varying the amplitude A directly (Figure 4-1). This result is easily understood. Since the zthickness does not influence the potential, or therefore the orbital behavior, in the disk plane, the response density is independent of A. Decreasing A, then, is tantamount to decreasing the amplitude of the imposed density relative to the response (i.e. increasing the value of R*). We find that, for all three galaxies modeled, the optimum value of A is in the range from 0.05 kpc to 0.2 kpc, with 0.1 kpc being judged as yielding the best overall self-consistency in each case. Finally, since A does not affect the response density, the phase difference AO also remains unaffected. Figure 4-6: The ratio R* in models of NGC 3992 where A = 0.01 kpc (open circles), 0.1 kpc (filled circles, Â“bestÂ” model), and 1.0 kpc (plusses).

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103 Variation of 62 The phase angle 62 represents the angular offset between the innermost part of the spiral and the end of the bar. It is defined such that positive values displace the spiral in the direction of pattern rotation (i.e. the spirals Â“leadÂ” the bar). Negative values correspond to spirals that Â“trailÂ” the bar. Figure 4Â—7 compares the results of setting 62 = 15Â° (open circles) and 30Â° (plusses) to the Â“bestÂ” model (filled circles) of NGC 3992. We note large deviations in both R* and in the region around corotation, indicating that angle offsets between the ends of the bar and the starting points of the spiral arms are not preferred orientations for self-consistent stellar models which assume quasi-stationary structure that rotates at a single pattern speed. We emphasize positive values of O 2 here because the observations of our sample galaxies seem to preclude the existence of negative offsets. Nevertheless, one model of NGC 1398 was calculated with \$2 = -15Â°. The results of that run are as unsuccessful as those shown in Figure 4-7, but the deviations of R* and /S .6 are in the opposite sense. We had hoped for better results in this case of NGC 1073, which in fact displays an offset of some 30Â° between the bar and spirals (Sandage 1961), but they also were no better than those shown in Figure 4Â—7 for NGC 3992. The fact that NGC 1073 and some other barred spirals exhibit angle offsets between the bars and spirals may indicate either that the spirals in this region are primarily gaseous in nature or that the bar and spiral strucures are independent entities that possess separate pattern speeds. There is some numerical (e.g. Sellwood and Sparke 1988) and observational (e.g. NGC 1365) evidence for independent pattern speeds for bars and spirals. This possibility will be discussed again in Chapter 7.

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104 kpc kpc Figure 4-7: The ratio R* and phase difference A6 for models of NGC 3992 in which 62 (Â“bestÂ” model, filled circles), 15Â° (open circles), and 30Â° (plusses).

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105 Variation of r\, V 2 , k\, K 2 , and A 4 In this section we consider the effects of varying the remaining parameters of the bar/spiral perturbation. These are the cutoff radii (ri,r 2 ) and steepnesses (/ci, K 2 ) of the bar/spiral and the amplitude A 4 of the A9 component. Changes in all of these parameters were found to have much less effect on model self-consistency than those already discussed. Figure 4Â—8 shows the effect on the ratio R* of the Â“bestÂ” model of NGC 3992 of varying the position of the inner bar/spiral cutoff radius r\ . Not surprisingly, the effects are only noticeable in the vicinity of the cutoff itself {r\ = 1.5 kpc in the Â“bestÂ” model of NGC 3992). Positioning the cutoff closer to the center of the grid (ri = 0.5 kpc, the open circles in Figure 4-8) reduces somewhat the upward divergence of R in that region. Figure 4-8; The ratio R* in models of NGC 3992 where r\ = 1.5 kpc (Â“bestÂ” model, filled circles), 0.5 kpc (open circles), and 2.5 kpc (plusses).

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106 In fact, since the amplitude of the imposed 29 component does not decrease rapidly enough toward the center in this case, R eventually turns downward and reaches zero at the center. This case is arguably an improvement over the Â“bestÂ” model, but in our opinion the overall agreement (i.e. at all radii) is not significantly better. Figure 4Â—9 shows the results of varying r\ in the case of NGC 1073. Here we see much a much more pronounced effect on the ratio R than in the case of NGC 3992 (or NGC 1398). The difference is that the cutoff radius is relatively farther out in the disk in the model of NGC 1073 than it is in the models of either of the other two galaxies. Also, since the Â“bestÂ” model of NGC 1073 employs a rather shallow inner cutoff steepness Ki, the variation in r\ causes a systematic shift in R similar to the effect of increasing the bar/spiral amplitude A directly, as this is in essence what has happened. In the case of each galaxy, varying r\ did not affect the phase agreement A9 at all. Figure 4-9: The ratio R* in models of NGC 1073 where r\ = 2.95 (Â“bestÂ” model, filled circles), 2.0 kpc (open circles), and 3.5 kpc (plusses).

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107 Models of NGC 3992 and NGC 1073 were also calculated in which the position of the outer cutoff of the bar/spiral perturbation, r 2 , was varied (an outer spiral cutoff was not employed in the case of NGC 1398). These models showed only very minor differences with the Â“bestÂ” models of each case. There are several reasons for this. Since V 2 in the Â“bestÂ” models of NGC 3992 and NGC 1073 is located at the corresponding OLR, the spiral amplitude is rather small; moreover, the cutoff is not infinitely sharp but mediated by the cutoff steepness k 2 (1.0 and 0.5, respectively, for the Â“bestÂ” models of NGC 3992 and NGC 1073, denoting rather moderate cutoff steepnesses). Therefore, the exact position of the outer cutoff did not affect the self-consistencies of these models to a significant extent. The inner and outer cutoff steepnesses ki and k 2 were varied in each case where the corresponding cutoff was employed. Decreasing either value (i.e. making the cutoff too shallow) resulted in a Â“leakageÂ” of excess bar/spiral perturbing potential beyond the cutoff and a corresponding depression of R*, the same effect seen in Figure 4Â—1 when A is too large. The opposite effect occurred when the cutoff steepnesses were taken to be too large. In all cases the phase agreement A0 was essentially unaffected. We also calculated models in which we included A9 components with amplitudes 20% and 40%, respectively, of the 20 amplitude A of the Â“bestÂ” model. We find that the ratio R remains almost completely unaffected. The phase agreement A0 is altered somewhat, particularly near the -4/1 resonance, but always towards larger negative values (i.e. worse self-consistency). In contrast, Contopoulos and Grosbpl (1988) found that the inclusion of a moderate AO component improves self-consistency in models of unbarred spirals. This difference may be due in part to the relatively

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108 large fraction of stochastic orbits responsible for supporting the outer spiral structure in the barred models. We now consider variations of the parameters which characterize the Ferrers bar potential and the exponential disk. In all cases we restrict the variations to within the limits set by the surface photometry and the observed rotation curve, and we adjust the Plummer sphere halo mass M// and shape parameter bu in order to ensure a good fit to the observed HI rotation curve. Variation of the Bar Semiaxes a, b, and c Since varying the semimajor axis a of the Ferrers bar potential is essentially equivalent to varying the pattern speed Q,p (the main effect of both is to vary the length of the bar relative to the corotation resonance), it is not surprising that the effects on R and A9 in each case are qualitatively the same as those shown in Figure 4Â—5. The magnitude of the changes are less, though, since the surface photometry constrains the value of a (and therefore the difference between a and the corotation radius) more tightly than was allowed when ftp was varied. Furthermore, we find that varying b and c, the intermediate and minor bar axes, within the limits set by the surface photometry produces essentially no effect on R* and A9. Variation of the Bar Mass Mb Since the only observational constraints on the masses of the bars of our program galaxies are the (somewhat uncertain) inner parts of the corresponding HI rotation curves, we have considerable latitude to vary Mg. Figure 4Â—10 shows the effects of varying Mb on the measures of self-consistency in the case of NGC 3992. We see

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109 that R* is affected the most in the region of the bar, while Ad is affected most outside the bar in the region of the spiral. While a moderate decrease in bar mass improves the self-consistency as regards R* in the region of the bar (e.g. the case where Mq = 7.5 X IO^Mq, denoted by plusses in Figure 4Â—10), this improvement is more than offset by the deterioration of the phase agreement in the outer spiral. We consider the model with the highest value of Mb (i.e. 1.5 x 10^*^ MÂ©) to be the most self-consistent overall. This conclusion is strengthened by noting that Hunter et al. (1988) obtained a comparable mass of 2.5 x 10^^ Mq for the bar of NGC 3992 by an independent modeling technique. The results of varying Mb in the cases of NGC 1073 and NGC 1398 yielded results similar to those presented here for NGC 3992. Variation of cq and eo Next we consider the central surface density cq and inverse scale length eo of the exponential disk. Figure 4Â—1 1 shows the effect of varying cq on the ratio R in models of NGC 3992. We note systematic vertical shifts in R , with higher values of central surface density corresponding to higher values of response-to-imposed amplitude ratio. This is due to the fact that when cq is increased and the amplitude of the imposed bar and spiral perturbation are held fixed, the amplitude of the bar/spiral relative to the axisymmetric background is decreased. The response amplitude does not suffer such a decrease because the amplitude of the response bar and spiral is modulated by cq through the weighting of the orbits. The ratio R*, then, tends to go up when cq is increased and down when it is decreased.

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no Figure 4-10: The ratio R* and phase difference AO for models of NGC 3992 in which Mb = 1.5 X 1O^Â°M0 (Â“bestÂ” model, filled circles), 1.5 x IO^Mq (open circles), and 7.5 X 10^ M 0 (plusses).

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Ill Figure 4-11: The ratio R* in models of NGC 3992 where cq = 75OM0/pc^ (Â“bestÂ” model, filled circles), SOOMq/pc^ (open circles), and lOOOM0/pc^ (plusses). Figure 4-12 shows the effect of varying the inverse disk scale length eo on the ratio R* in models of NGC 3992. Here again we see systematic shifts of the responseto-imposed amplitude ratio and for the same reasons outlined above for the variation of central surface density cq. A lower value of eo means a less rapid dropoff in axisymmetric density away from the center. This, in turn, results in a progressive decrease with radius of the bar/spiral amplitude relative to underlying disk, and a corresponding increase in R*. The opposite is true for the case of higher sq. We can clearly see this divergence in R away from center for the different cases. We detect essentially no changes in the phase agreement A0 when cq and Â£q are varied. This result might seem surprising at first, but it really is not. The phase agreement between the response and imposed bar/spiral is basically determined by the shapes of the orbits, which, in turn, are determined by the total potential in the disk.

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112 We noted above that we altered the halo parameters at the same time as we changed the exponential disk parameters in order to maintain a good fit to the observed rotation curve. Therefore, we have compensated the gravitational potential in the disk for changes in the disk parameters and ensured that the orbits have maintained their essential shapes. This is why the is not affected by variations in the disk parameters. Figure 4-12: The ratio R* in models of NGC 3992 where eo = 0.235 kpc ^ (Â“bestÂ” model, filled circles), 0.157 kpcÂ“^ (open circles), and 0.353 kpcÂ“* (plusses). Variation of cto and <7r Finally, we consider the effects on model self-consistency of varying the central value (Jo and radial slope Cr of the velocity dispersion employed. Following the results of observations (e.g. Jarvis et al. 1988) and of V-body simulations (e.g. Sparke and Sellwood 1987; Pfenniger and Friedli 1991), we consider here only those models in which the velocity dispersion decreases with radius. We have calculated models in

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113 which the velocity dispersion is constant across the disk, but these, in general, yield results inferior to those in which the dispersion decreases. Figure 4Â—13 shows the effects of varying ao and ar on R* and A9 in models of NGC 3992. We see that a lower velocity dispersion (open circles in Fig. 4Â—13) leads to worse agreement as regards R* but good agreement as regards A9. On the other hand, a higher velocity dispersion (plusses in Fig. 4Â—13) yields slightly worse agreement in both R* and A^. These results are understood by realizing that lower velocity dispersions lead to a more sharply peaked response (and hence a greater 29 response component) which lies close to the imposed potential minima. Higher velocity dispersions, conversely, tend to smear out the response (and hence lower the 29 response component), as well as cause it to depart farther from the minima of the potential. Thus it seems that the appropriate velocity dispersion is a compromise between these two competing effects. The velocity dispersion which works best for our model of NGC 3992 has a central value of 100 km sÂ“^ and drops to around 25 km sÂ“^ near the OLR (10.6 kpc). Interpretation of Results In this final section we briefly summarize the conclusions that can be drawn from the results of varying the model parameters. The properties which seem to exert the most influence on self-consistency are (1) the bar/spiral amplitude A, (2) the spiral pitch angle io, (3) the inverse spiral scale length Ss, (4) the perturbation thickness A, (5) the bar/spiral pattern speed (6) the bar mass Mb, (7) the cental surface density Cq and inverse scale length eo of the exponential disk, and (8) the central value ao and radial slope ar of the velocity dispersion. Parameters which control the detailed structure of the bar/spiral amplitude such as the cutoff radii r\ and V 2 and steepnesses k\ and K 2 , as

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114 kpc Figure 4-13: The ratio R* and phase difference A0 in models where ao = 100 km sÂ“^ = -7.0 km sÂ“^kpcÂ“^ (Â“bestÂ” model, filled circles), ao = 30 km sÂ“^ a^ = -1.0 km sÂ”^kpc"^ (open circles), and ao = 150 km sÂ“^ a^ = -11.0 km s"^kpcÂ“^ (plusses).

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115 well as the residual amplitude Ar, do exert influence on model self-consistency, but their effects are secondary to the parameters which control its global shape and strength (i.e. A, io, Ss, A, and Mb). Also, while the bar length a affects self-consistency significantly, this simply reflects the influence of the pattern speed flp. Thus we may group the important parameters into four main categories: (i) those that control the global shape and amplitude of the bar/spiral potential, (ii) the pattern speed, (iii) those that control the magnitude and shape of the azimuthally averaged radial mass distribution, and (iv) those that control the magnitude and shape of the velocity dispersion. Finally, we again note the rather strong preference in the models of this thesis for the spiral arms to emanate directly from the ends of the bar. Since we only have three galaxies in our sample, it is difficult to discern possible correlations of the parameters of the Â“bestÂ” models with Hubble type. Still, some comments are warranted. Three basic classification criteria for spiral (and therefore barred spiral) galaxies are given by Sandage (1961): (1) the openness of the spiral arms, (2) the degree of resolution of the arms into stars, and (3) the relative size of the unresolved nuclear region. Although there are, of course, exceptions, earlier type galaxies (here early, intermediate, and late type barred spirals refer to SBa, SBb, and SBc, respectively, in HubbleÂ’s [1926] original classification scheme) tend to have both smoother, more tightly wound spiral arms that are less resolvable into individual stars and relatively larger unresolved nuclear regions (i.e. Â“bulgesÂ”). The distribution of spiral pitch angles in our Â“bestÂ” models is consistent with these expectations. The Â“bestÂ” model of our representative early-type galaxy NGC 1398 has a pitch angle of -6Â°, while the Â“bestÂ” models of the two later-type representatives (NGC 3992 and NGC

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116 1073) both have iÂ‘o = -10Â°. While we might have expected the Â“bestÂ” model of NGC 1073 to display a somewhat larger pitch angle, we note that the observed scatter in pitch angle as a function of Hubble type is rather large (Kennicutt 1981) and that the value of i'o for NGC 1073 is not unusual. We are unable to say anything based on our model results concerning the latter two criteria. This is because criterion (2) is uniquely observational in nature and criterion (3) refers to components (bulges) not included in the present models. One final point we would like to make concerns an observation of Elmegreen and Elmegreen (1985). In their paper they distinguish between two types of bars, and exponential, depending on the shape of the luminosity profile along the major axis. They find that early-type barred spirals are more likely by far to possess a flat bar, while exponential bars are preferentially found in late-type galaxies. The results of our modeling tend to corroborate this finding. In particular, the Â“bestÂ” models of NGC 3992 and NGC 1398 require bar/spiral potentials which extend significantly inwards of corotation, while the inner cutoff radius of the bar/spiral in the Â“bestÂ” model of NGC 1073 is located at corotation. At the same time we note that our assumed Ferrers bar potential more closely represents an exponential bar than a flat bar. We interpret the inward extensions of the bar/spiral perturbations in the models of NGC 3992 and NGC 1398 as compensating for the intrinsic Â“non-flatnessÂ” of the Ferrers bar potential. Combes and Elmegreen (1992) give further results concerning flat versus exponential bars. Their A-body simulations indicate that the flat bars of early-type galaxies are limited in extent by their corotation radii (i.e. by stochasticity), whereas the exponential bars of late-type galaxies are limited by the scale length of the disk in which they are

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117 located (i.e. the bar stops growing when the matter density in between corotation and the OLR is not sufficient to absorb the angular momentum transported outwards by the growing bar). They claim, therefore, that the ends of exponential bars do not necessarily correspond to any major resonance radius. We did not check this possibility here.

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CHAPTER 5 STOCHASTIC ORBITS In this chapter we examine the role of stochasticity in the models presented in Chapter 3. Again, since we are considering here only the surface density response, we restrict our attention to stochastic motion in the disk plane. While this restriction simplifies the present analysis, it completely misses potentially important phenomena such as vertical resonances, Arnold diffusion, complex instability, and collisions of bifurcations, all of which only appear in cases of at least three degrees of freedom (Contopoulos 1987). Indeed, analyses of the orbital structures of realistic bar potentials (e.g. Pfenniger 1984a, 1985; Pfenniger and Friedli 1991) and the results of 3-D A-body simulations (e.g. Combes and Sanders 1981; Pfenniger and Friedli 1991) indicate that vertical instability strips in the periodic orbit families, as well as global buckling (Â“fire hoseÂ”) instabilities of the bars themselves, can play significant roles in determining the final 3-D (often box or peanut) shape that a given bar assumes. These vertically thickened bars of the simulations, in fact, have been identified with the similarly shaped bulges seen in some edge-on galaxies (Combes and Sanders 1981). These results notwithstanding, we are, for several reasons, of the opinion that the 2-D problem should be addressed in more detail first. For one, the collective longterm surface density response of stochastic orbits in realistic two-dimensional barred spiral potentials has simply never been tested. Pfenniger (1984b), in his attempts to 118

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119 construct self-consistent 2-D bars, found that he usually had to include roughly 10% stochastic orbits in order to achieve success. Other authors (e.g. Teuben and Sanders 1985) make the claim that if an orbit is stochastic (i.e. it only respects one integral of motion, the energy), then its time-averaged density response fills more or less uniformly the region bounded by its zero-velocity curve. However, Contopoulos and Polymilis (1993) have shown that in near-integrable systems (such as galactic models), stochastic orbits may remain for long times in the phase-space neighborhood of unstable periodic orbits. Contopoulos and Kaufmann (1992) have emphasized this Â“stickinessÂ” property as the phenomenon that explains both the almost-regular behavior of stochastic orbits over long times and the fact that these orbits do not escape to infinity for long times, even though their energy exceeds the escape energy and their motion is not restricted by any other integral of motion. A second reason to study the density response of stochastic orbits in realistic twodimensional galactic potentials is indicated by the results of Sparke and SellwoodÂ’s (1987) dissection of a 2-D A-body bar. First, they found that the bar was comprised of orbits trapped and semi-trapped around the central x\ family of periodic orbits. The trapped orbits are true quasi-periodic orbits which lie on invariant tori in phase space. The semi-trapped orbits are actually stochastic orbits whose exploration of the available phase space is restricted for long times by cantori (invariant tori which have developed an infinity of Â“holesÂ” due to an increase in the perturbation, thereby allowing communication between formerly well-separated regions of stochasticity). We discuss these ideas in more detail below. Secondly, and perhaps more importantly for barred spiral models, they found that there existed a substantial Â“hotÂ” population of particles

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120 which had sufficient energy to traverse the bar and the outer disk (i.e. their Jacobi constants exceeded the value at the Lagrange points L\ and L 2 ). Pfenniger and Friedli (1991) also found such a Â“hotÂ” population in their similar study of a 3-D A^-body bar. As we shall see, it is this class of orbits which plays the crucial role in supporting the imposed spiral structure of our models from the ends of the bar (near corotation) to a point in the outer disk where regular orbits again can dominate the spiral structure. Finally, observations of edge-on spirals seem to indicate that at least some bars are as thin as the rest of the disk. This statistical argument says that if bars are as common in edge-on systems as in all disk galaxies (-25 35%), then we should see a substantial fraction of thick disks if bars are substantially thicker than the disks in which they reside. Because we do not, bars must be as thin as the disks of their host galaxies. While the observation of box and peanut-shaped bulges tends to undercut this argument, the proportion of box and peanut-shaped bulges in edge-on galaxies (-20%, e.g. Shaw 1987) does not match the proportion of bars in all disk galaxies. Also, Athanassoula (cf. Sell wood and Wilkinson 1992) raises the concern that the sizes of the observed box/peanut-shaped bulges relative to the extent of the disk are less than the sizes of strong bars to their host galaxies. These facts imply that a significant proportion of bars are thin as well. If this is true, the importance of phenomena which depend on a strong coupling between motion in the disk and motion in the third dimension will be lessened. A typical value quoted for the ratio of bar thickness to bar major axis is 1:10 (e.g. Kormendy 1982; Wakamatsu and Hamabe 1984). A number of studies have addressed the problem of stochasticity in two-dimensional galactic models (e.g. Contopoulos 1981, 1983, 1987; Contopoulos and Grosbpl 1989;

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121 Athanassoula et al. 1983; Teuben and Sanders 1985). The transition from the completely ordered motion of an integrable system (i.e. a system in which all particles obey the same number of integrals of motion as there are degrees of freedom) to the stochastic motion of a chaotic system is now fairly well-understood. The introduction to an integrable two-dimensional system of a small perturbation which couples the independent motions renders the system near-integrable. That is, the system contains an infinite number of stable and unstable periodic orbits, and regions of chaos are found in the vicinity of each unstable orbit. Because the perturbation is small, however, the regions of chaos are also small, and the system retains a large set of ordered motions. These motions lie, in phase space, on toroidal surfaces called KAM (Kolmogorov, Arnold and Moser, e.g. Moser 1973) tori, which separate the chaotic regions. The intersections of the KAM tori by a Poincare surface of section (cf. Poincare 1892) are closed invariant curves. As the perturbation increases, the relative sizes of the stochastic regions also increase. There is, however, no communication between different stochastic regions until the perturbation exceeds a critical value and the so-called last KAM curve separating them is destroyed. The evolution of the last KAM curve has been described by Greene (1979) and Shenker and Kadanoff (1982). When the last KAM curve is destroyed, it develops an infinity of holes which form a Cantor set; therefore, it is called a cantorus (Aubry 1978; Percival 1979; Mather 1982; Katok 1982; Aubry and Le Doeron 1983). The existence of these holes allows diffusion of orbits through the cantorus and establishes communication between previously well-separated regions of chaos. The rate of diffusion through the holes of the cantorus for perturbations slightly above the critical value has been estimated theoretically by Bensimon and Kadanoff

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122 (1984) and by MacKay, Meiss, and Percival (1984). This rate depends very much on the sizes of the holes in the cantorus, and therefore on the amplitude of the perturbation (Contopoulos and Barbanis 1989). This explains why stochastic orbits often do not cover the available phase space in a smooth way. Also, if the diffusion time scale is much greater than the average orbit period (as may be the case in galactic models), then we may consider, for practical purposes, a cantorus to be an effective barrier to the diffusion of stochastic orbits. In the next section we present several Poincare surfaces of section (each representing a different value of the Jacobi constant) for the most successful models of NGC 3992, NGC 1073, and NGC 1398 that were presented in Chapter 3. The surface of section is a very useful tool not only for visualizing the periodic orbits that are present at a given value of the Jacobi constant, but also for gauging the amount of trapping done by each periodic orbit and the amount of stochasticity present. We construct the surface of section by numerically integrating an orbit and plotting one position and and one velocity component whenever it crosses some reference axis in a specified direction. For example, in our surfaces of section we take the reference axis to be along the intermediate axis of the bar (i.e. y = 0), and plot x and x whenever the orbit crosses the reference axis with y > 0. Such plotted points are called consequents. We shall see how the orbital structure of each model, with emphasis on the stochastic regions, evolves as a function of the Jacobi constant. After presenting the surfaces of section, we shall examine briefly the qualitative behavior of the stochastic orbits as a function of Jacobi constant. We conclude the chapter with estimates of the percentage of stochastic orbits in each Â“bestÂ” model.

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123 Surfaces of Section In Figures 5-1 through 5-9 we present surfaces of section showing the evolution of orbital structure and the onset of stochasticity in the models of Chapter 3. Each surface of section is generated by plotting 100 consequents per orbit for a large number of orbits (about 100 per surface of section), all with the same value of the Jacobi constant. Figures 5-1 through 5-3 refer to the model of NGC 3992, Figures 5-4 through 5-6 the model of NGC 1073, and Figures 5-7 through 5-9 the model of NGC 1398. The limiting curves that delineate accessible regions are shown as dashed lines. For a given model the sequence of figures is arranged in order of increasing Jacobi constant. Figure 5-1 shows surfaces of section for the two lowest values of the Jacobi constant Ej considered in the model of NGC 3992. The most striking feature in the top panel of Figure 5-1 {Ej = -213438 km^ sÂ“^) is the predominant regularity of the orbits, both in the region of the bar and in the outer disk. (The consequents of orbits in the bar are found in the bounded region centered on x = 0, while those in the outer disk are found in the two accessible regions on the left and right extremes of the diagram. Also, since we focus on the behavior of orbits in our models, we include only consequents generated by direct orbits. That is why the consequents in the bar region generally have positive X values while those in the outer disk generally have negative x values. We have included, in the bar region of the top panel of Figure 5-1, one invariant curve centered near [x = -1 kpc, i: = 0 km s~^] generated by a retrograde orbit to show that there is indeed structure to these regions.) At this lowest value of Ej, which corresponds in the axisymmetric case to a circular radius Kc = 12.0 kpc (also Cc = 1.68 kpc), the orbital structure in the bar region and in the outer disk region is dominated by the x\ family.

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124 Figure 5-1: Surfaces of section in the model of NGC 3992. Top: Jacobi constant equals -213438 km^ sÂ”^. Bottom: Jacobi constant equals -202713 km^ sÂ“^.

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125 c/3 !Â«! > -15 10 5 X (kpc) 0 10 Figure 5-2: More surfaces of section in the model of NGC 3992. Top: Jacobi constant equals -198084 km^ s"^. Bottom: Jacobi constant equals -194202 km^ s~^.

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126 X (kpc) Figure 5-3: More surfaces of section in the model of NGC 3992. Top: Jacobi constant equals -191307 km^ s~^. Bottom: Jacobi constant equals -189692 km^ s~^.

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127 0 X (kpc) X (kpc) Figure 5^: Surfaces of section in the model of NGC 1073. Top: Jacobi constant equals -34143 km^ sÂ“^. Bottom: Jacobi constant equals -32393 km^ sÂ“^.

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128 Figure 5-5: More surfaces of section in the model of NGC 1073. Top: Jacobi constant equals -30871 km^ sÂ“^. Bottom: Jacobi constant equals -29746 km^ s~^.

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129 0 X (kpc) 0 X (kpc) Figure 5-6: More urfaces of section in the model of NGC 1073. Top\ Jacobi constant equals -29405 km^ sÂ“^. Bottom: Jacobi constant equals -29265 km^ sÂ“^.

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200 130 0 (kpc) X (kpc) Figure 5-7: Surfaces of section in the model of NGC 1398. Top: Jacobi constant equals -314204 km^ sÂ“^. Bottom: Jacobi constant equals -306600 km^ sÂ“^.

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131 -5 0 X (kpc) Figure 5-8: More surfaces of section in the model of NGC 1398. Top: Jacobi constant equals -299940 km^ sÂ“^. Bottom: Jacobi constant equals -294526 km^ sÂ“^.

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132 X (kpc) X (kpc) Figure 5-9: More surfaces of section in the model of NGC 1398. Top: Jacobi constant equals -290724 km^ sÂ“^. Bottom: Jacobi constant equals -288987 km^ sÂ“^.

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133 The xi orbits at this value of Â£7 are highly elongated along the bar in the bar region (see Figure 3-6) and essentially circular in the outer disk. Since this outer branch of the Xi family is located beyond the OLR where the matter density is low, it is of negligible importance in our models. The lower panel of Figure 5-1 represents the case where Ej = -202713 km^ sÂ“^ (rc = 10 kpc, 2.44 kpc). The orbital structure of the bar is still dominated by the Xi family, while outside we see two main stable orbit families and a significant amount of stochasticity. The two stable families represent -2/1 resonant orbits near the OLR. The family with deformed and elongated invariant curves centered near (x = 9.2 kpc, ;fc = 0 km sÂ“^) is the same orbit family as the X\ family in the top panel of Figure 5-1. Instead of being circular, though, at this value of Ej the orbits 0 of this family are elongated in the y direction and are out of phase with the imposed spiral. Orbits of the other 2/1 family, centered near (x = -11 kpc, jc = 10 km s~^), are elongated along the x direction and are in phase with the imposed spiral (note: the consequents outside the bar with x > 0 kpc, seen in the lower panel of Figure 5-1 and later plots, are formed by loops counter to the general motion of the outer orbits around the galaxy. Figure 5-10 shows schematically how such consequents are generated.). Figure 5-2 shows surfaces of section for Ej -198084 km^ sÂ“^ {Tc = 9 kpc, 2.94 kpc; top panel) and Ej = -194202 km^ sÂ“^ {Tc = 8 kpc, 3.52 kpc; bottom panel). At the lower value of Ej we continue to see almost complete regularity as regards the orbits in the bar. Outside the bar, except for a small stable -2/1 family near (x = -9.2 kpc, x = 0 km sÂ“^), stochasticity is dominant. While there are stable regions evident at large values of x and Ixl, their dynamical significance is small because the amount of matter existing in these regions of phase space is small (the matter density in these surfaces of

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134 section decreases exponentially with increasing x and Lxl). In the lower panel of Figure 5-2 we see a significant change in the orbital structure of the bar. At this value of Ej the xi family in the bar has been broken by a gap into two 4/1 resonant families, located near (x = 0.2 kpc, A: = 0 km sÂ“^) and (x = 2.2 kpc, x = 0 km s~^). The first 4/1 family, however, is dynamically insignificant because it is stable over only a very small range of energy. Outside the bar region there are two regions of stability. The one centered roughly at (x = -9 kpc, j: = 10 km sÂ“^) is an extension of the -2/1 family of the top panel of Figure 5-2. The other, located approximately at (x = -7.8 kpc, x = -20 km sÂ“^), is a -4/1 resonant family that is stable only over a small range of values. Figure 5-10: Schematic drawing showing how loops in the orbits outside of corotation generate consequents on the surface of section. A portion of an orbit containing such a loop is shown, together with arrows indicating the direction of motion along the orbit. The drawn circle represents corotation, and the directions of the general motion are also indicated with arrows. Finally, in Figure 5-3 we see surfaces of section for E/ = -191307 km^ s~^ (rc = 1 kpc, 4.21 kpc; top panel) and Ej = -189692 km^ sÂ“^ {Vc = 6 kpc, 5.04 kpc; bottom

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135 panel). These two surfaces of section are extremely similar. Inside the bar region the stochasticity is almost complete. Outside the bar there is also a large region of stochasticity. Regions of regular motion are confined to large x values (relative to r^) and large iJcl values where matter density is low, thus their dynamical significance in this range of Ej is small. In the bottom panel of Figure 5-3 the consequents around (x = 3.8 kpc, X = 0 km sÂ“^) are generated by stochastic orbits that traverse the Lagrange points Li and L 2 and travel between the bar and the outer disk. These orbits, which only exist for values of Ej > -191188 km^ sÂ“^ (i.e. the Jacobi constant at Li and L 2 , form the so-called Â“hotÂ” population. As we shall see shortly, they are instrumental in supporting the imposed spiral in the region near corotation. Figures 5-4 through 5-6 show surfaces of section derived from the model of NGC 1073 that was presented in Chapter 3. Figures 5-7 through 5-9 show analogous figures for the case of NGC 1398. In both progessions of plots we see an evolution similar to that in the case of NGC 3992. At relatively low values of Ej the orbital structure within the bar is very regular. As the Jacobi constant is increased, however, a large degree of stochasticity develops. The stochasticity increases with increasing Ej, and it dominates the motion in the bar region when Ej is sufficiently close to the Jacobi constant at L\ and L 2 . Stochasticity in the bar region of the model of NGC 1073 develops somewhat more rapidly and completely than in the model of NGC 1398. This is due primarily to the high value of the bar axial ratio alb in the model of NGC 1073. Another point of difference is the lack of any significant regular motion in the outer disk of NGC 1398 at low values of Ej. This is due to the fact that, at large distances, the outer spiral in the model of NGC 1398 is relatively stronger than in the model of NGC 1073,

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136 and it induces greater stochasticity at the lower values of Ej. The large regions of stochasticity and the few significant periodic orbit families that exist in these models, and in the model of NGC 3992 as well, indicate why we chose to include the Â“circularÂ” orbits mentioned in Chapters 2 and 3. Since we needed some means of estimating the density response of the orbits in these predominantly stochastic regions, and because this density response is not very sensitive to the particular choice of initial conditions, we chose to start the orbits with the initial conditions of the truly circular orbits of the axisymmetrized case. In the next section we investigate the various behaviors of stochastic orbits in the configuration space of the disk plane. Individual Stochastic Orbits In this section we examine the various types of generic stochastic behavior exhibited by orbits in our models. Since these types of behavior are the same in all three models, we shall restrict our attention to illustrative examples drawn from the model of NGC 3992 that was presented in Chapter 3. Basically, there are four distinct types of behavior that a stochastic orbit can display. The four types derive from three different intervals of Ej in which we may find the Jacobi constant of the orbit. Table 5-1 summarizes these four types, their behaviors, and the intervals of Â£7 in which they are found. The first interval only has an upper bound to Ej. This upper bound is the value of Ej at the Lagrange points Lj and L 2 . Outside the bar an orbit can exist with an arbitrarily low value of Â£7 provided that it is located at a distance sufficient to allow it to exist above the zero velocity surface. Within the bar, however, an orbit may exist only if it has a Jacobi constant in excess of the minimum Â£7 in the bar. Also, since an

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137 orbit in this interval of Ej has insufficient energy to traverse the L\ and L 2 points, it remains trapped forever either inside or outside the bar region. Therefore, this interval of values of Ej leads to two distinct types of stochastic orbits. Figures 5-11 and 5-12 give examples of these two types of stochastic behavior. Table 5-1: Summary of basic stochastic orbit types and their behaviors. Interval oi Ej Type Distinct behaviors (I) Ej < Â£y(Li) ( 1 ) confined to bar circulation in bar ( 2 ) confined to outer disk circulation in outer disk (0) Ej{L{) Ej{L^) (4) unconfined energetically circulation everywhere kpc Figure 5-11: Stochastic orbit trapped within the bar. The value of its Jacobi constant {Ej = -191307 km^ s"^) is slightly less than that of the Lagrange points L\ and L2 {Ej = -191188 km^ s"^). The darker circle represents corotation at 5.5 kpc.

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138 Figure 5-12: Stochastic orbit confined to the outer disk. The value of its Jacobi constant (Â£y = -194202 km^ sÂ“^) is also less than that of L\ and La. The bar and spiral potential minima are drawn for reference. The second interval of Â£/ is bounded by the value at L\ (or La) and the value at L 4 (or L 5 ). While orbits in this energy range cannot move freely everywhere (i.e. they are restricted from the vicinities of L 4 and L 5 ), they are free to move through the bar and into the outer disk (Figure 5-13). In our work with these orbits we have noticed three subtypes of general motion (relative to the rotating frame) which they can possess: (1) direct circulation within the bar, (2) circulation around the Lagrange points L 4 and L 5 {direct when inside these points, retrograde when outside), and (3) retrograde circulation in the outer disk. The majority of these orbits also form loops which involve (temporary) motion counter to the general motion (see Figure 5-10). We find that, given sufficient time, all of the orbits in this range of Â£7 display all three subtypes of behavior; however, over the length of time we have calculated these orbits in our models (-1 billion years), they may exhibit only one or two of these behaviors.

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139 Figure 5-13: Stochastic orbit which traverses both the bar and the outer disk. The value of its Jacobi constant {Ej = -189692 km^ sÂ“^) is greater than that of L\ and Li, but less than that of L 4 and L 5 {Ej = -187300 km^ s~^). These stochastic orbits that are able to traverse both the bar and disk constitute the so-called Â“hotÂ” populations observed in the respective 2-D and 3-D N-body simulations of Sparke and Sellwood (1987) and Pfenniger and Friedli (1991). As can be seen in Figure 5-13, they tend to make many loops near the spiral potential minima just beyond the ends of the bar (near corotation), and we find that this Â“hotÂ” population is instrumental in supporting the imposed spiral structure in this region. When in the region of the bar, however, these orbits tend to remain near their curves of zero velocity, which are rather broad and elliptical. Hence, the same orbits that support spiral structure may, at the same time, weaken strong, thin bar structure. For comparison. Figure 5-14 shows an orbit of approximately the same value of Ej but calculated in a spiral perturbation whose amplitude is ten times greater. While we see that the orbit still % traverses the bar and spiral, it spends most of its time in the outer disk at large distances

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140 and does not support the imposed bar or spiral structure. This behavior, therefore, places upper bounds on the amplitudes of structures which might be enhanced by these orbits. Figure 5-14: Stochastic orbit, whose Jacobi constant (Ej = -189476 km sÂ“*) is approximately the same as the orbit of Figure 5-13, calculated in a spiral potential of ten times greater amplitude than that of Figure 5-13. Finally, we have the type of stochastic orbit whose Jacobi constant is greater than that of L 4 (or L 5 ). Figure 5-15 shows an orbit of this type. These orbits are not restricted from any region energetically, and they tend to fill large (almost circular) regions more or less uniformly. For this reason, and since the energies of these orbits exceed those of all the circular orbits in the axisymmetrized disk, we assume that these orbits are negligibly populated and ignore them in our models as being of only very minor importance. In the following section we give estimates of the numbers of each I type of orbit, both ordered and stochastic, in our models.

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141 Rgurc 5-15: Stochastic orbit whose Jacobi constant {Ej -183000 km^ s"^) exceeds that of L 4 and L 5 and is energetically unconstrained. Proportions of Ordered and Stochastic Orbits in the Models ' Determining the relative proportions of ordered and stochastic orbits in our models is somewhat difficult. It is made so not only by the fact that we do not keep track of whether a given dispersed orbit is stochastic or ordered in the modeling process, but also by the fact that our calculated orbits form a discrete representation of a continuous distribution in velocity space (i.e. a particular choice of a representative dispersed orbit may lead to either an underemphasis or overemphasis of the relative importance of stochasticity). Other techniques of self-consistent modeling do not suffer from this uncertainty. For example, the linear programming method used by Schwarzschild (1979) and the nonnegative least-squares method employed by Pfenniger (1984b) require libraries of individual orbits from which their self-consistent models are constructed. Straightforward combination of the knowledge of the regularity or stochasticity of each

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142 orbit with its population weight gives directly the numbers of regular versus stochastic orbits. Despite these difficulties, we have devised a method by which we can roughly estimate the proportions of stochastic versus regular orbits in our models. Moreover, we can estimate how the stochastic orbit distribution breaks down into those trapped in the bar, those confined to the outer disk, and those in the Â“hotÂ” population, respectively. The data at our disposal for this estimation process are of two types: (1) the initial conditions and velocity dispersions of the model orbit families, and (2) the surfaces of section presented earlier in this chapter. The initial conditions of an orbit family are specified by radius r and radial velocity r along the x (intermediate bar) axis as a function of Jacobi constant (parameterized by the circular radius, of the orbit of the same Jacobi constant in the axisymmetric case). We first plot r versus Vc for each family. The region of trapping about a stable periodic family corresponds to a range of values of r. We estimate, for a given value of r^, the amount of matter trapped on regular orbits around this family by integrating our assumed Gaussian velocity distribution, whose amplitude and width are given by the weight (i.e. the azimuthally averaged disk surface density) and velocity dispersion a{r) = cro + rcOr respectively, over this trapping range. Also, if there are more than one family at a given value of rc, then the amplitude of the Gaussian centered on the wth family is decreased by the corresponding relative weight Wm (see Chapter 2). Outside the trapping ranges and up to the limiting curves the contributions of the Gaussians are assumed to be stochastic. Furthermore, guided by the results of the surfaces of section (given earlier in this chapter), we assume all orbits dispersed around a Â“circularÂ” orbit are stochastic. While this is not strictly true, the regular orbits that do exist at these values of rc have very large radial velocities and are

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143 not significantly populated. Finally, we break down the stochastic orbit distributions further by sorting them according to the value of their Jacobi constant. Those with Jacobi constants less than that of L\ and Vc inside corotation are assumed to be trapped in the bar. Those of similar Jacobi constants with outside corotation are assumed to be confined to the disk. Those with Jacobi constants greater than that at L\ are taken to be the Â“hotÂ” population. Table 5-2: Estimated breakdown of the massweighted orbit populations comprising the models of Chapter 3 according to type (trapped versus stochastic) and location (bar, disk, or both). The errors in the cited figures are somewhat uncertain, but are estimated to be of the order of 5%. NGC 3992 Bar Population (%) "Hot" Population (%) Disk Population (%) trapped 74 6 stochastic 5 12 3 NGC 1073 Bar Population (%) "Hot" Population Disk Population (%) (%) trapped 63 2 stochastic 14 14 7 NGC 1398 Bar Population (%) "Hot" Population Disk Population (%) (%) trapped 59 1 stochastic 10 5 25 Table 5-2 gives the results of these analyses for the three models presented in Chapter 3. Several conclusions can be drawn from these results. First, we see that the bar is dominated by trapped orbits (primarily around the x\ family). For example, of the total number of orbits confined to the bar region in the best model of NGC 3992, fully 94% of them are quasi-periodic. This agrees with the results of several other studies, for

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144 example Pfenniger (1984b), Teuben and Sanders (1985), Sparke and Sellwood (1987), and Pfenniger and Friedli (1991). A second conclusion is that, of the orbits that make up the outer disk, the Â“hotÂ” population comprises a significant fraction. In fact, in two of the models (NGC 3992 and NGC 1073) this group is larger than the total population confined to the outer disk (regular and stochastic orbits combined). In their study of a 3-D A-body bar and disk model, Pfenniger and Friedli (1991) found that fully 31.3% of the particles were of the Â“hotÂ” variety at the end of their simulation (55.9% were located in the bar and 12.8% were in the disk). Finally we note that there seems to be slightly too many estimated orbits in the bar region. For example, the mass of the model (bar + exponential disk) inside the bar radius (5.5 kpc) for NGC 3992 is 62% of the total mass inside r = 12.0 kpc. From Table 5-2 we see that the orbit population estimates give some 79%. Although the exact cause of this discrepancy is not known, we believe that it is probably due to an underestimation of the Â“hotÂ” population. It is also possible that the discrepancy is due simply to the uncertainties of the figures in Table 5-2. In any case we deem the two previous conclusions to be firm.

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CHAPTER 6 GAS RESPONSE In this chapter we examine the response of gas to the Â“bestÂ” models of NGC 3992, NGC 1073, and NGC 1398. In order to calculate the gas behavior we have used a two-dimensional smoothed particle hydrodynamics (SPH) code. Since the formalism of our SPH approach has already been described in detail in Chapter 2, we restrict our attention here to the details of the simulations and to the presentation and interpretation of the results. There are two basic questions to be answered by our simulations: (1) does a gaseous component exhibit long-lived spiral structure in the presence of the strong stellar spirals of our Â“bestÂ” models? and (2) can our model bars alone drive spiral structure in the gas? The answer to the first question has direct bearing on the overall success of our models. As mentioned in Chapter 1, the spiral arms observed in most barred galaxies seem to exist both in the gaseous component and in the underlying stellar disk (Elmegreen and Elmegreen 1985). Thus we cannot consider as ultimately successful a model in which the gas does not exhibit long-lived spiral structure along with the stars. This may occur, for example, if the stellar spirals are so strong that they either disrupt the gaseous features soon after they form or do not allow them to form at all. The second question addresses the issue of whether stellar spirals are necessary at all in order to form spiral structure in the gas. Elmegreen and Elmegreen (1985) find 145

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146 that some barred galaxies seem to possess primarily gaseous arms. This leaves open the possibility, mentioned in Chapter 1, that spiral structure is solely a bar-driven feature of the gaseous component. With these two questions in mind, we have performed the following simulations. We have run two simulations for each Â“bestÂ” model. The first employs the full amplitude of the bar/spiral perturbation (in addition to the Ferrers bar) while the second employs no bar/spiral perturbation (only the Ferrers bar). In all simulations the initial gaseous disk is represented by approximately 12,000 particles placed down uniformly on a Cartesian grid. The radius of this initial particle distribution is taken to be roughly the radius of the outer Lindblad resonance of the model. The particles are given masses such that the mass of the whole gaseous disk is 10% of the mass of the model (out to the maximum radius of the initial particle distribution). Also, the gas particles respond only to the gravitational forces of the model and to the viscous and pressure forces among themselves (i.e. the particles are not self-gravitating). The initial state of the model disk is axisymmetric. Flence, the amplitude of the bar/spiral perturbation and the eccentricity of the Ferrers bar are initially set to zero. The particles are started on purely circular trajectories with velocities whose magnitudes are determined by the radial forces of the initial axisymmetric model. Over the first rotation of the pattern, however, the bar/spiral amplitude (if used) and the eccentricity of the Ferrers bar are allowed to increase to their full values, which they retain for the duration of the simulation. The 18 pages of figures in this chapter show the various simulations after different numbers of rotations of the bar or bar/spiral pattern. We start with the simulations of NGC 3992, and proceed to those of NGC 1073 and NGC 1398 in turn. Six snapshots

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147 per simulation are given showing the evolution of the gas structure. For a given galaxy, the first six snapshots refer to the run with the bar/spiral perturbation imposed. The second six snapshots refer to the run with the Ferrers bar alone. The number of rotations of the bar or bar/spiral pattern is noted in the upper right hand corner of the figure. In all figures the frame of reference is the one corotating with the bar/spiral pattern such that the bar always lies along the y-axis (i.e. vertically). Figures 6-1 through 6-3 show the gas response in the model of NGC 3992 with the bar/spiral potential imposed. Well-defined global spiral structure is seen in the gas even in the earliest snapshot, at 1.5 rotations of the pattern (top panel of Figure 6-1). At this stage the gaseous arms exhibit thin ridges of high density, implying the existence of shocks. These ridges are particularly evident in the outer portions of the disk, from roughly 7 kpc all the way out to the edge of the disk at 12 kpc. We also notice local enhancements in the gas density near the ends of the bar, and a slight extension of the inner termination of the spirals beyond the direction of the bar, giving the impression of an angular offset between the bar and spirals. At 2.4 pattern rotations (bottom panel of Figure 6-1) little has changed except for the beginnings of an elongated inner ring surrounding the bar and a slight tendency for the gas to evacuate the regions around the Lagrange points L 4 and L 5 . In the top panel (3.3 pattern rotations) and bottom panel (5.6 pattern rotations) of Figure 6-2 we see the continued slow development of the elongated inner ring and the evacuation of the regions surrounding L 4 and L 5 . We also notice that the inner spiral structure, extending from slightly beyond the bar radius (6 kpc) to about 9 kpc, is gradually weakening. In fact, the outer spiral structure (at 5.6 rotations) is beginning to wrap back onto itself and detach from the weakening inner

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148 spiral structure. This phenomenon is more clearly visible in the panels of Figure 6-3, where we see the outer spirals forming a rather clumpy ring near the OLR (10.6 kpc). Hiotelis et al. (1993) found that a similar clumpy gas ring formed near the OLR in their SPH calculation of the response of viscous gas to a purely spiral-shaped imposed potential. Gaseous spirals still exist very near the ends of the bar, but in between the structure is significantly weakened. The gas structure in the bar is essentially unchanged from the earlier snapshots. We note that, in agreement with the observations of NGC 3992, no strong offset shocks along the bar are present at any time during the calculation. A much different result is seen in the calculation where no outer spiral potential was imposed (Figure 6Â—4 through Figure 6-6). Only weak gaseous spirals are observed at any time during this run. For example, at 1 .5 pattern rotations (top panel of Figure 6-4) we see only that a short gas bar has formed, and that gas has begun to accumulate somewhat near the ends of the bar and to be evacuated from regions just outside the bar (particularly along the directions of the bar minor axis). Only the slightest hint of spiral structure is visible. At 2.5 rotations (bottom panel of Figure 6-4) the spiral structure is somewhat stronger; nevertheless, the spirals that do appear are very tightly wound and do not emanate from the ends of the bar at all. In fact, these spirals are not discernable inside a radius of approximately 7 kpc. Figures 6-5 and 6-6 show very similar morphologies. The only noticeable change in the gas structure from 3.5 pattern rotations (top panel of Figure 6-5) to 10.6 pattern rotations (bottom panel of Figure 6-6) is the slow outward movement of the spirals toward the edge of the disk. We do notice a tendency for the gas in the bar to accumulate near the edges of the bar by 10.6 pattern rotations.

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y (kpc) y (kpc) 149 10 0 -10 10 0 -10 Figure 6imposed. 20 -10 0 X (kpc) 20 -10 0 10 20 X (kpc) : The gas response in the best model of NGC 3992 with the bar/spiral Top panel: 1.5 pattern rotations. Bottom panel: 2.4 pattern rotations.

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y (kpc) y (kpc) 150 10 0 10 20 -10 0 10 20 X (kpc) 20 -10 0 10 20 X (kpc) Figure 6-2: The gas response in the best model of NGC 3992 with the bar/spiral imposed (continued). Top panel: 3.3 pattern rotations. Bottom panel: 5.6 pattern rotations.

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y (kpc) y (kpc) 151 10 0 10 0 (kpc) 10 0 -10 20 -10 0 10 20 X (kpc) Figure 6-3: The gas response in the best model of NGC 3992 with the bar/spiral imposed (continued). Top panel: 8.1 pattern rotations. Bottom panel: 10.4 pattern rotations.

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y (kpc) y (kpc) 152 -10 0 10 X (kpc) 20 -10 0 10 X (kpc) Figure 6-4: The gas response in the best model of NGC 3992 with only the Ferrers bar present. Top panel: 1.5 pattern rotations. Bottom panel: 2.5 pattern rotations.

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y (kpc) y (kpc) 153 0 (kpc) -20 -10 0 10 20 X (kpc) Figure 6-5: The gas response in the best model of NGC 3992 with only the Ferrers bar present (continued). Top panel: 3.5 pattern rotations. Bottom panel: 5.3 pattern rotations.

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y (kpc) y (kpc) 154 10 0 10 0 (kpc) 20 -10 0 10 20 X (kpc) Figure 6-6: The gas response in the best model of NGC 3992 with only the Ferrers bar present (continued). Top panel: 7.6 pattern rotations. Bottom panel: 10.6 pattern rotations

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y (kpc) y (kpc) 155 0 0 (kpc) 0 (kpc) Figure 6-7: The gas response in the best model of NGC 1073 with the bar/spiral imposed. Top panel: 1.1 pattern rotations. Bottom panel: 2.8 pattern rotations.

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y (kpc) y (kpc) 156 0 (kpc) 5 0 -5 0 (kpc) Figure 6-8: The gas response in the best model of NGC 1073 with the bar/spiral imposed (continued). Top panel: 4.5 pattern rotations. Bottom panel: 5.4 pattern rotations.

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y (kpc) y (kpc) 157 0 X (kpc) 0 (kpc) Figure 6-9: The gas response in the best model of NGC 1073 with the bar/spiral imposed (continued). Top panel: 6.8 pattern rotations. Bottom panel: 8.3 pattern rotations.

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y (kpc) y (kpc) 158 5 0 -5 0 (kpc) 0 5 10 -5 0 5 X (kpc) Figure 6-10: The gas response in the best model of NGC 1073 with only the Ferrers bar present. Top panel: 1.1 pattern rotations. Bottom panel: 2.6 pattern rotations.

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y (kpc) y (kpc) 159 10 -5 0 5 X (kpc) -10 -5 0 5 10 X (kpc) Figure 6-11: The gas response in the best model of NGC 1073 with only the Ferrers bar present (continued). Top panel: 4.5 pattern rotations. Bottom panel: 5.5 pattern rotations.

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y (kpc) y (kpc) 160 -5 0 X (kpc) 0 -5 10 -5 0 5 X (kpc) Figure 6-12: The gas response in the best model of NGC 1073 with only the Ferrers bar present (continued). Top panel: 7.3 pattern rotations. Bottom panel: 8.3 pattern rotations.

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y (kpc) y (kpc) 161 20 -10 0 10 20 X (kpc) 20 10 X 0 (kpc) 10 Figure 6-13: The gas response in imposed. Top panel: 1.4 pattern the best model of NGC 1398 with the bar/spiral rotations. Bottom panel: 3.3 pattern rotations.

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y (kpc) y (kpc) 162 X (kpc) X (kpc) Figure 6-14: The gas response in the best model of NGC 1398 with the bar/spiral imposed (continued). Top panel: 4.1 pattern rotations. Bottom panel: 5.0 pattern rotations.

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y (kpc) y (kpc) 163 X (kpc) X (kpc) Figure 6-15: The gas response in the best model of NGC 1398 with the bar/spiral imposed (continued). Top panel: 7.2 pattern rotations. Bottom panel: 11.1 pattern rotations.

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y (kpc) y (kpc) 164 10 0 -10 10 0 -10 Figure 6-16; present. 0 X (kpc) -20 -10 0 10 20 X (kpc) The gas response in the best model of NGC 1 398 with only the Ferrers bar Top panel: 1.9 pattern rotations. Bottom panel: 3.1 pattern rotations.

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y (kpc) y (kpc) 165 20 -10 0 10 20 X (kpc) -20 -10 0 10 20 X (kpc) Figure 6-17: The gas response in the best model of NGC 1398 with only the Ferrers bar present (continued). Top panel: 4.2 pattern rotations. Bottom panel: 5.3 pattern rotations.

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y (kpc) y (kpc) 166 20 -10 0 10 20 X (kpc) 10 0 10 -20 -10 0 10 20 X (kpc) Figure 6-18: The gas response in the best model of NGC 1398 with only the Ferrers bar present (continued). Top panel: 7.5 pattern rotations. Bottom panel: 11.1 pattern rotations

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167 Figures 6-7 through 6-9 show the response of gas to the model of NGC 1073 with the imposed bar/spiral potential. As in the case of NGC 3992, the gas displays well-defined spiral structure from the earliest snapshot, at 1.1 rotations of the pattern (top panel of Figure 6-7). At this early stage the gas density is enhanced all along the spiral arms from their inward terminations near the ends of the bar to the edge of the gas disk. There is, however, no analogous enhancement of the bar by the gas. While a small amount of gas has congregated at the center of the galaxy to form a very short nuclear bar structure there, most of the gas originally located near the imposed bar potential has already begun to evacuate the region by 1.1 pattern rotations. This trend continues until a steady-state, which can be seen in the next snapshot at 2.8 pattern rotations (bottom panel of Figure 6-7), is reached. By this time there is little gas in the bar and essentially none near the Lagrange points L4 and L5. The gas that does remain near the bar region is located either in the short nuclear bar or in an elongated inner ring surrounding the bar region. The morphology of the gas changes very little and very slowly from this point (2.8 pattern rotations) until the end of the calculations at 8.3 rotations of the pattern (bottom panel of Figure 6-9). There is a slight tendency for the spiral enhancements to weaken near the ends of the bar and to become smeared out in the outer disk. These two effects combine to make the gas distribution more and more like an outer ring with time. Here also, we fail to note the development of offset shocks along the bar at any time. Figures 6-10 through 6-12 show the response of gas to the model of NGC 1073 without the imposed bar/spiral potential. As in the case of NGC 3992, we see very different gas behavior without the imposed bar/spiral potential. At 1.1 pattern rotations

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168 (top panel of Figure 6-10) no discernible spiral structure is present. The bar region has begun to be evacuated almost completely of gas. However, as in the case of the imposed bar/spiral, a small quantity of gas remains at the center and forms a nuclear bar. By 2.6 pattern rotations (bottom panel of Figure 6-10), though, we do see some spiral structure; however, it is quite weak, has a very small pitch angle, and does not emanate from the ends of the bar. No spiral structure is visible inside a radius of about 4 kpc. Figures 6-11 and 6-12 show essentially this same picture. In this case we see no tendency for the gas to evolve towards an outer ring. Figures 6-13 to 6-15 depict the response of gas to the model of NGC 1398 with the bar/spiral potential imposed. At 1.4 pattern rotations (top panel of Figure 6Â—13) the gas exhibits strong, well-defined spiral structure. As in the analogous run for the case of NGC 3992, we see local density enhancements in the gas near the ends of the bar. However, by 3.3 pattern rotations (bottom panel of Figure 6-13), and more clearly by 4.1 rotations (top panel of Figure 6-14), we see a dissolution of the spiral structure between the ends of the bar (4.8 kpc) and 7 kpc. At the same time the gas that forms the outer spirals (located from approximately 7 kpc to 12 kpc) is disrupted, and by 11.1 pattern rotations it has coalesced into a more or less continuous outer ring located roughly at the OLR (9.8 kpc). Meanwhile, the gas remaining inside corotation is distributed almost uniformly in a thick oval bar. Again, we do not observe the formation of offset shocks along the bar. Figures 6-16 to 6-18 show the response of gas to this same model of NGC 1398, but with no imposed bar/spiral potential. In this case we see an evolution very similar to the case of NGC 3992 without the imposed bar/spiral. We again have the

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169 slight enhancement of the overall gas density in the bar at early times, and the larger enhancements near the edges of the bar at later times (see bottom panel of Figure 6-18). Also, there is a depletion of gas in the regions beyond the gas bar, especially along the directions of the bar minor axis. Outside the bar we have the generation of very weak and tightly wound spiral waves that propagate slowly to the edge of the disk by the end of the calculation (11.1 pattern rotations, bottom panel of Figure 6-18). Examination of the SPH runs yields a couple of general conclusions concerning the gas response in our models. First, in cases where there is no imposed bar/spiral perturbation, only weak spirals are found. These spirals appear rather far out in the outer disk and soon after the Â“formationÂ” of the Ferrers bar (i.e. when the bar reaches full ellipticity), propagate slowly toward the edge of the gas disk, and leave the rest of the disk outside the bar almost completely featureless for the duration of the run. In the cases of the models of NGC 3992 and NGC 1398, a short, stubby gas bar forms and the remaining area within the bar radius becomes somewhat depleted of gas. Thus it seems that a bar alone is insufficient to drive strong spirals in the gas, consistent with the findings of Ball (1984, 1992), Hunter et al. (1988), England (1989) and England et al. (1990). This inability to drive gaseous spirals is due to the fact that the quadrupole field of the Ferrers bar drops off too rapidly outside the bar. In contrast, though, are the cases where the bar/spiral perturbation is imposed. Here the spiral response of the gas is very strong. In the case of NGC 1398, in fact, the perturbation is so strong that it disrupts the gas and causes it to form an outer ring after approximately 11 pattern rotations (Figure 6-15). This appears to be consistent with the observed HI distribution in NGC 1398 (Moore, private communication). Moreover, the

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170 gaseous spiral structure just outside the bar is substantially weakened and has just about dissolved long before this (Figures 6-13 and 6-14). The optical spiral structure in NGC 1398 is indeed quite long, thin, and continuous, similar to the panels of Figure 6-13. The gas response in the case of NGC 3992, though, takes the form of relatively longlived spirals (Figures 6-1 through 6-3). Therefore it appears that spiral perturbations of the underlying disk are necessary in order to generate a long-lived quasi-stationary response in the gas. The study of the dynamics of gas in barred galaxies is an area of intense ongoing research. In fact, there is current evidence from numerical simulations for a dynamical coupling of the stellar and gaseous components of bars (e.g. Friedli and Benz 1992). However, since a full consideration of this aspect of our models would take us too far afield from the present topic of self-consistent stellar models, we defer further comment on this topic to future investigations. In Chapter 7 we summarize the conclusions that we, based on the work presented in this thesis, are able to draw concerning the nature of self-consistent stellar models of barred spiral galaxies. %

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CHAPTER 7 SUMMARY The conclusions of this dissertation fall into three basic categories: (1) the existence and properties of self-consistent stellar models of barred spiral galaxies, (2) the role of stochastic orbits in these models, and (3) the response of gas to the models. We end with some general comments regarding our models and directions for future research. Self-Consistent Models of Barred Spirals Perhaps the most important conclusion derived from this work is that self-consistent barred spiral models exist. Moreover, they exist for all three galaxies considered (i.e. for a wide range of Hubble types). The parameters of the most successful models reinforce the conclusions of previous studies. For example, we find in each case that the most successful model places corotation near the end of the bar. Deviation from this condition results in worse self-consistency. The close agreement between bar length and corotation, as mentioned earlier, is also found in most A-body simulations and orbit analyses. Another common result is that our model bars are made up almost entirely of regular, elongated orbits trapped around the Xi family. Also, we find that the preferred phase difference between the bar and spirals at the bar radius is zero (i.e. the arms connect smoothly to the ends of the bar with no angular offset). While an examination of the images of actual barred spiral galaxies seems to confirm this result, Sellwood and Sparke (1988) argue that, in fact, bars and spirals represent independent patterns 171

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172 with different angular velocities. The obvious objection to this claim is that, under this assumption, the observed phase differences between the ends of the bars and the beginnings of the spirals should be randomly distributed, which they arenÂ’t. They reply to this objection by noting that (1) in fact, some galaxies do exhibit a phase difference, and (2) the bar and spirals may appear to the eye to be connected when actually a phase difference exists. The numerical simulations of Combes and Elmegreen (1992), however, support the idea of a single pattern speed. In only one case did a bar and outer spiral pattern exhibit independent angular velocities. Since we achieved success using uniform pattern speeds for the bars and spirals, we were not forced to consider independent pattern speeds and therefore did not check this possibility. While a sample of three is too small to allow us to draw firm statistical conclusions, several properties of our models appear to correlate with Hubble type. First, the magnitude of the spiral pitch angle is larger in the best models of NGC 3992 and NGC 1073 than in the best model of NGC 1398. In addition, and somewhat reassuringly, the model pitch angles agree very well with the observed pitch angles. Secondly, the fact that the Ferrers bars in the models of NGC 3992 and NGC 1398 needed to be augmented by an extension of the spiral perturbation into the bar region indicates that these bars are qualitatively different than the bar in NGC 1073, which required no such augmentation. This qualitative difference seems to corroborate the findings of Elmegreen and Elmegreen (1985), who distinguish between flat bars, seen typically in early-type galaxies, and exponential bars, predominantly found in late-type galaxies. While these authors suggest that the two bar types result from different formation mechanisms and have different extents relative to their corotation radii, we find for

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173 these two different types successful models whose bars extend all the way to corotation. Recent work (Combes and Elmegreen 1992) has given support to the idea that the exponential bars of late-type galaxies may not be limited in extent by corotation but by other factors. The Role of Stochastic Orbits Another important result of our work is that stochastic orbits seem to play a significant role, especially near and just beyond the ends of the bar, in supporting selfconsistent spiral structure. In this respect the most important stochastic population is the so-called Â“hotÂ” population, consisting of orbits with sufficient energy to be able to traverse both the bar and the outer disk. Particles of this type form a significant fraction of the total in the respective 2-D and 3-D A-body simulations of Sparke and Sellwood (1987) and Pfenniger and Friedli (1991), thus we expect their behavior to have a large influence on long-lived spiral structure. Stochastic orbits of other types are of much less important in our models. For example, stochastic orbits trapped in the bar region tend to fill the areas delimited by their zero velocity curves. Since these areas are more oval than the bar itself, stochastic orbits of this type tend to weaken the bar. These orbits, however, are not highly populated in our models and their destructiveness to the bar is small. Stochastic orbits confined to the disk tend to fill rings and not support imposed spiral structure. Again their dynamical role, destructive to imposed spirals, is minimized by the fact that they are rather minimally populated both relatively (due to increased proportions of regular orbits as the spiral amplitude decreases outward) and absolutely (due to decreased matter density at large radii). Finally, stochastic orbits with energies sufficient to allow complete exploration of the bar and disk regions seem

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174 also to be unimportant to self-consistent structure. Not only do they not support the imposed structure, but they also are undoubtedly less populated at these high energy values. A more quantitative exploration of the latter point is wanting, however, and could (and should) be included in the quoted results of future A-body simulations. The Gas Response A further result, derived from our hydrodynamical calculations, is that bars alone seem incapable of driving and sustaining gaseous spiral structure in the outer disks of barred spiral galaxies. The presence of a source of nonaxisymmetric forces in the outer disk appears to be required. Our calculations show that a spiral stellar wave is sufficient. This result agrees well with the observations of Elmegreen and Elmegreen (1985), who find that in almost all cases the spiral waves of barred spirals contain enhancements both in the young objects (0-B stars, HII regions, etc., which are tracers of the gas) and in the older disk stars. Ball (1984, 1992), Hunter et al. (1988), England (1989) and England et al. (1990) have found that oval distortions in the surface density of model disks, distortions which extend beyond the bar radius, can excite and maintain spiral structure in the gas of the outer disk. However, it is not clear at this time whether these oval distortions in the models have counterparts in real galaxies. Our results also indicate that self-consistent stellar spirals may be so strong that they disrupt the gaseous spirals (e.g. in the gas response to the model of NGC 1398). While this may be seen as a weakness of our models, it is not quite clear whether these results actually contradict the observations. For example, as noted in Chapter 6, the gas in the outer disk of NGC 1398 may exist primarily in an outer ring. In other cases (e.g. NGC 3992) the gaseous spiral structure is quite long-lived.

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175 Directions for Future Research In the end we have to address the question of whether our models represent what is actually going on in barred spiral galaxies. Our answer is a qualified Â“yes.Â” We have obtained self-consistent models (at least approximately) of our program galaxies, models which are derived from and consistent with the observables of these systems. The properties of our models are also generally consistent with the results of both N-body simulations and orbit theory. Still, uncertainties remain. How stable are our models? This question has not been addressed at all. The assumption of quasistationary bar and spiral structure is what has allowed us to represent the bar and spiral by time-independent potentials. This assumption would be invalid if the evolution of the system is rapid. One shortcoming of our approach is that it is unable to answer the questions of stability and evolution, and we must resort to other means (i.e. fully self-consistent simulations) for these answers. How important is the third dimension? Again, this question has not been fully addressed in our models. While we account for the effect that varying the z scale height of the bar/spiral perturbation has on the resulting response surface density, the dynamics which generate that response density is confined to the two dimensions of the principle disk plane. This simplification does not introduce any serious errors as long as the z thickness is small compared to density variations in the plane. If this is the case, the z motion decouples from the motion in the plane, with the latter determining the structure of the disk. We do, however, have to consider the z motions in the case of bars, where the vertical scale height is comparable to the scale lengths of density variations in the disk plane. In this case the z motion may couple with the motion in the plane

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176 and introduce important new dynamical phenomena (e.g. vertical resonances, complex instability, Arnold diffusion of stochastic orbits, etc.). The 3-D orbital structure of barred galaxies has begun to be investigated in recent years, most notably by Pfenniger and collaborators (e.g. Pfenniger 1984a, 1985; Pfenniger and Friedli 1991), but by others as well (e.g. Patsis and Zachilas 1990; Mahon 1992; Zachilas 1993). We need to generalize our approach to three dimensions, guided by the results of these investigations. Finally, how does gas affect the dynamics of barred galaxies? We have presented in this thesis the passive response of a gaseous component to the imposed potentials of our models. Surely the gas must play a more substantial role than this. There is a growing body of evidence (e.g. Thomasson et al. 1990; Moore 1992) that some means of holding down the stellar velocity dispersion is required in order to obtain long-lived stellar spiral structure. Otherwise the stars Â“heat upÂ” and spiral structure is lost. The dissipative gas component may yield this means. We know that the gaseous and stellar components are coupled via star formation and supernovae explosions. How substantial is this coupling and what are its effects on the global dynamics of the system? Also, what is the role of gas in producing the nuclear structures (bars, spirals) sometimes seen in barred galaxies? All of these questions are fundamental to our understanding of barred spiral galaxies, and it is toward answering these questions that future research should be directed.

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APPENDIX MODEL BAR QUANTITIES Here is summarized the quantities associated with the n = 2 Ferrers bar which are needed for this work. The spatial density distribution of the bar is given by = (A-I) lO, m>l, where = ^ + ^ + (a > 6 > c > 0). (A-2) 0-^ cÂ‘^ The central density pc is related to the total mass Mg and the axesÂ’ lengths by Pc = 105 Mg 327T abc (A-3) The surface density projected onto the z = 0 plane is gotten by integrating the spatial density pix,y,z) with respect to z in the interval where the spatial density is nonzero. It can be written as Â•max E(x,?/) p{x,y,z)dz, (A-4) Â•max where Zmax Â— cy 1 Â— ^ Â— |r. The resulting formula for the projected surface density IS 16 y X 5 2 ^(^Â’2/) = TTPcC 1 ^ 15 7Mg 27rab aÂ‘ 9 2 (A-5) . _ r _ ^ 111

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178 The potential in the z = 0 plane is given by = Â— -ri^GOO Â— 1^030^^ Â— 0 +3(21Tnox2j/2 WnoxS'^ Â— 1^1002/^ + Wo2QX^ + W2QQy^ Â— W^OIO^;^)}, where C = 2-KGpcabc = ^GMbThe coefficients ITp depend on the lengths of the axes a,b, and c, and on the position {x,y), and are given by ITooo = W^ioo = W^ooi = Woio = W^iio = 1^011 = W^ioi = IT 200 = VT 020 = VT 120 = M^201 = 1^210 = IT 021 = IV 300 = Wo30 = 9 y/a^Â—c ,F(4.,k) 2 [f (^, k) EU, 4)1 2 /_JiÂ±A 2E{(f>,k) V (6^ Â— c^) va^ Â— 2 Â— IT^ioo Â— 1^001 Wj^-Wm. o2-ft2 ^ ^001 Â—W( b^Â—c^ ^lOQ Â— T^ ^OOI 1 3 VA(A)(a2-fA) 1 3 A(A)(62+A) tVo 20 Â— ^110 P-ft2 Â— VFiio Â— W\Q\ Won Wno) 2sm W c^Â—a^ m. Wj^-W, a^-b^ 2M Wi Q2IL w. b^-c^ m. 1 ^ \ A(A)(a^H-A)' 1 5 VA(A)(62+A) Â— W 210 Â— W 201 j 7 Â— W 021 Â— W 120 K where k) and k) are the incomplete elliptic integrals, sin (j) = ^ = Y (a 2 Zc 2 ) Â» A is the unique positive solution of y + X + Z' (a2 + A) (62 + A) ' (c2 + A) and A^(u) = (a^ + u) (6^ + u) (c^ + u) . 1 , (A-7)

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179 In order to calculate the forces due to the bar, the partial derivatives of the potential with respect to radius and azimuth are required: = -{1^1002/^ + 2 ( 1 ^ 200 / + 2Wnoa:2y2 or r and -\-Wq2QX^) + + VFosoa:' +3(Wi2oa:^y^ + W2\QX^yÂ‘^)} = Cxy{2x\WQ2i^ l^no -\2 y^{W\iQ Â— VF200 + r^hf 2 io) +x^(3VFi 20 Â— ll^so) (A-8) (A-9) +?/^(11^300 Â— 3W210) + 13^100 Â— 13^010}Also, the second partial derivative of the potential with respect to radius is given by d'^VB{x,y) ^ ^ _ 6(^200/ 5r2 +2WiioxÂ‘^yÂ‘^ + Wq2qx'^) + 5(VF30o2/' (A-10) +ZWi2QxS'^ + 3W2ioa:^/ + WosoxÂ®)}.

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BIOGRAPHICAL SKETCH David Eugene Kaufmann was bom on the 18th of January, 1964, in WinstonSalem, North Carolina. With his parents, John and Betty Kaufmann, and his older brothers, Greg and Eric, he moved to Nashville, Tennessee, in 1968. In 1970, he returned with his family to Winston-Salem, where he graduated from R. J. Reynolds High School in 1982. In 1986 he graduated from Davidson College, Davidson, North Carolina, with a B. S. in mathematics. In the fall of the same year he enrolled in the graduate astronomy program at the University of Florida, Gainesville, Florida, and obtained his M. S. in 1989. He will receive his Ph. D. in August, 1993. 186

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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. James H. Hunter, Jr., Chair 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. Stephen T. Gottesman, Co-chair 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. Visiting Graduate Research 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. Neil S. Sullivan Professor of Physics f

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This dissertation was submitted to the Graduate Faculty of the Department of Astronomy 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 1993 4