Using Tilted-Ring Models and Numerical Hydrodynamics to Study the Structure, Kinematics and Dynamics of HI Disks in Galaxies

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1989 Using Tilted-Ring Models and Numerical Hydrodynamics to Study the Structure, Kinematics and Dynamics of HI Disks in Galaxies. Dimitris Michael Christodoulou Louisiana State University and Agricultural & Mechanical College

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Using tilted-ring models and numerical hydrodynamics to study the structure, kinematics a^d dynamics of HI disks in galaxies

Christodoulou, Dimitris Michael, Ph.D.

The Louisiana State University and Agricultural and Mechanical Col., 1989

U-M-I 300 N. Zeeb Rd. Ann Arbor, MI 48106 USING TILTED-RING MODELS AND NUMERICAL HYDRODYNAMICS TO STUDY THE STRUCTURE, KINEMATICS AND DYNAMICS OF HI DISKS IN GALAXIES

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State I lnversity and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in The Department of Physics and Astronomy

by Dimitris Michael Christodoulou B.S., University of Thessaloniki, Greece. 19S4 August 1989 A C K N O W L ED OE M I M S

The hist few years at LSE have been a delight. My work lias moved ahead smoothly m an ideal environment Many people and several resources have heen responsible. The Vi >1 k reported in this dispel t at ii >n has hern sponsored, in part, by \S F through the giants AS I So 01S-12 and ASJ Si 01 ,">03 to ,1. E.

Tohhne and. 111 part, hy the Louisiana State I_ nive] sity. lint this is a page primarily devoted to ]>eople.

First. I would like to thank my a (hi-or. Dr. Joel E. d oh line, for his continuous help and guidance and for all the things lie taught me. Indeed. only those rvho have worked with him can really appreciate the difference.

Second. 1 would like to express my appreciation to I)rs. Ganesh Chan miipiin. Arlo Landolt. Charles Perry. Detlev km -ter. and John Drilling for

Teaching me Astronomy,

Tim'd. I have to thank Monika Lee and fha tensia \ aides for assisting me m computations rind Linda Gauthier for typing many manuscripts, including this one. Jeff Anderson's work on video animat ion and movie making is also greatly appreciat ed.

Finally. I should acknowledge the fun I haic had all those nights at the com puter room and the nearby coffee shop. G<>n/alo kuster. Harold Williams. Han

Chen. Bing Y 11. Saied Andalih, and Ramon Lopez Pena are definitely guilty for sharing the fun. In addition. I am obliged to Harold for many stimulating discussions before and after he left LSI .

ii TABLE OI CONTENTS

page

Acknnwledgi‘liients ...... ii

Table of Content' ...... iii

List of In’ >!< - ...... \

List < if I- igui < s ...... vi

Abst met ...... viii

1 Int roductiou ...... 1

1.1 Kmemntical versus Dynamical Modeling ...... 2

1.2 Historical Out line of Ilelated Work ...... 4

1.2 Summary of this Wot k ...... S

'1 Ki:.< ]..•: ’,< ;•! NI

2 .1 Edgi1 on Spirals ...... 13

2.2 Barred Spirals ...... 17

2.3 HI Disks m Ellipticals ...... IS

2.4 Permission to Publish Observations ...... 20

3 Dynamical Modeling ...... 20

3.1 Introduction ...... 30

3.2 Dynamical Equations and Initial M odels ...... 35

3.3 Settling Time-Scales and Inflow ...... 37

3.-4 Model Evolutions ...... 42

a ) /„ — 0° ...... 42

b) Low and moderate inclinations 43

c) High inclinations and polar rings ...... 44

d) Strongly distorted halos ...... 47

in pn^e

i i Development of dynamicaltwists ...... 40

3.5 Linear Behavior of the PPInstability ...... 50

4 Summary of Conclusions ...... 85

Epilogue ...... SO

References ...... 00

Appendix A ...... 05

Appendix If ...... 101

Appendix ( 1 1 2

Cui'i'ieulmn Vita ...... 2S2

IV LIST OF TABLES

page

Table 2.1 Warping A twisting angles for C'en A ...... 24

Table 3.1 Estimated settling t ime-seales ...... 53

Table 3.2 Linear growth rates (^.7 1 and ]>attern

S]>eeds (j,’/f ) of the PP instability ...... 54

\ LIST OF FIGURES

page

Figure 2.1 Isodensity maps of NGC 0007 from the standard

tilted-ring model (top) and from a spiral densiry

field showing an illusionary warp (bottom) ...... 25

Figure 2.2 Isovelocity maps of NGC 572S from the

actual observation [Seliommrr el ill- 1 f)SS]

(top) and from the tilted-ring model ...... 20

2.5 Isi neloci (y maps of NGC 51 2^ ( Cen A) from

the actual observation [Bland et ah 10S7J

( top) and from the model (m iddle) and the model s

isodensity map (bottom) ...... 27

F igm f 2.4 Isovelocity maps of NGC 4275 fi oin the

actual observation [Knapp 19S2] (top) and from

t he m

F IgtUl * 3.1 (a) 3 D isodensity diagrams duiing the evolution

of the extended ring (tn„ — 4.2 j. initially ])laced at an

inclination of t,, — 1 0 ° from the equatorial plane of a

spheroidal potential with Jjiu/" =■ 0 .0 2 , (b.c) 2-D

horizontal and vertical cross-sect ions through the XY

jilane and the negatire X-axis of tin 1 grid for the

model shown in (a). Thenumbers in each diagram

indicate the time in initial centralrotations ...... 55

Figure 3.2 (a.b.c) Same as in Figure 1, but for

the thin ring (m„ = 7.2) ...... 5S p age

Figure 3.3 (a.b.c) Same as in Figure 1. but for i„ = 40° ...... Cl

Figure 3.4 (a,b.c) Same as in Figure 1, but for the thin

ring (m„ = 7.2) and for >a— 40° ...... 04

Figure 3.5 (a,b.c) Same as in Figure 1 . but for itJ ~ S0° ...... 07

Figure 3,0 (a.b.c) Same as in Figure 1 , but for the thin

ring (m„ = 7.2) and for /0 — 80° ...... 70

Figure 3.7 (a) Same as in Figure la, but for i„ = 90° ...... 73

Figure 3.S (a) Same as in Figure la. but for the thin

ring (m (J = 7.2) and for i„= 90° ...... 74

Figure 3.9 (a.Inc) Same as in Figure 1, but for ia = 40°

and for a strongly distorted potential with

Jjuo' ^ 0.088 ...... 75

Figure 3.10 {a.b.c) Same as in Figure 1 . but for /„ = 10°

and for an extremely distorted potential with

J>na- = 0.192 ...... 78

Figure 3.11 (a.b.c) Same as in Figure 1 . but for >„ — 80° and

= 0.192 ...... 81

Figure 3.12 (a) Tin 1 logarithmic amplitudes of the m = 1.2,

and 3 modes of the FP instability versus time

in rotation periods for the extended ring with

tnp ~ 4.2. (b) The logarithmic amplitudes of the

m = 1,2, and 3 modes of the PP instability

versus time in rotation periods for the thin ring

with m 0 - 7.2 ...... 84

VII ABSTRACT

We model the structure, kinematics, and dynamics of neutral hydrogen gas in galaxies. Our attention is focused on specific features that are commonly observed mainly in the 21-cm line, such as twists, warps and polar rings. The intimai e connection between t hese feat ures and t he presumed dark halos of galax ies is analyzed with the intention to discover ways to probe the dark matter of individual galaxies by simply observing the distribution and kinematics of the luminous matter. An important. iiUermediate step in this analysis is to demon strate that the twists that appear in the isovelocity maps of many spiral galaxies

(and that are reproduced by the corresponding kinematical models) have a dy namical origin, owing their existence to the differential precession of gas-particle orbits.

The main conclusions from this work are : (a) The kinematics of normal spiral galaxies ran be modeled satisfactorily by a tilted-ring model. If, in addition, the observed velocity field of a galaxy shows a strongly twisted struc ture. then unambiguous results can be derived about the structure of the im plied warp and the geometry of the superimposed dark halo. (b ) The galaxies

NGC 5033 and NGC 5055 show the signature of a prolate halo in their kine matics. This result supports the 'LDark Matter Hypothesis" while it seems to have no explanation in the framework of dynamical theories that modify the

Newtonian gravitational force in order to reproduce the observed flat rotation curves of spiral galaxies. (c) Strongly twisted warps form naturally when a gaseous disk attempts to settle toward a preferred orientation in the external potential well of a weakly distorted halo, but only if the disks start from low or moderate inclinations relative to the preferred orientation, (d) Highly and mod

em erately mdm ed disks. as well as disks at all inclinations midci the influence of a stionidv non-spherically-symmetric halo, slum' significant matter inflow. This result suggests that warps should only develop at relatively low inclinations and in weakly distorted halos. (e) Polar titles and highly inclined disks can only survive for a Hul>l>]e time if the superimposed halo is nearly spherical. This conclusion i- in agreement with the results of lecent oliservations of polar ring galaxies.

i \ C H A P T E R 1 INTRODUCTION

At the quantum level our universe can he ,'fni as an indeterminate place, predictable in a statistical way only when you employ large enough num bers. Between that universe and a relatively predictable, one where the passage of a single planet can be tuned to a picosecond. other forces come into play. For the in-between universe where we find our daily lives, that which you believe is a dominant force. Your beliefs order the unfolding of daily events. If enough of us believe, a new thing can be made to exist. Belief structure creates a filter through v’hich chaos is sifted into order. from “Heretics of Dune" by Frank Herbert

When people think about spiral galaxies, images of thin, flat, roughly circularly symmetric disks readily come to mind. The spiral arms seem to be very luminous (wave) patterns that move through the material of an otherwise fairly symmetric disk. Recent optical and radio observations have delivered two decisive blows to this simplistic picture ; (a) the dominant contribution to the gravitational field of all spiral galaxies comes from an unseen component

(the “dark halo") that is 3-10 times more massive than the luminous disk and the bulge combined; and (b) by extending further away from the optical disks,

HI observations have provided evidence that large-scale asymmetries (“warps") dominate the outer regions of the disks. Because tin* HI disks are usually much larger than the optical disks, the warps are the dominant asymmetries rather than the spiral arms.

The work in this dissertation takes the reader well beyond the above simple picture and into the complicated subject of dark halos and warped gaseous layers in galaxies. We analyze the structure and kinematics of warps with the intention to discover ways to probe the geometrical shapes of the dark halos and we study

1 2 tin* dynamical evolution of model-disks m order to observe the formation of warps and examine the physical mechanisms that are responsible for it.

The Introduction is divided into 3 parts. First, we clarify the distinction between kinematical and dynamical modeling of gaseous galaxy disks. Then, we describe the development of the entire subject of this dissertation in a chronolog ical order, emphasizing previous key papers. Finally, we summarize the results presented in this dissertation. Additional references to the work of numerous other authors can lie found in the Appendices.

Because a large fraction of this work has already been published, we suggest that the reader begin by studying not the remainder of this Introduction, but the introductory and concluding sections of the papers in Appendices A, B, and

C. Then, the reader has the option to continue with this Introduction or even jump ahead to any other chapter of the dissertation.

1.1 Kinematical versus Dynamical Modeling

The physical scales involved in problems of galactic kinematics and galac tic dynamics are enormous compared to every-day human experience. (This fact alone makes the study of galaxies look like an awesome and, at the same time, extremely interesting task,) Our own Galaxy, the Milky Way, can serve as a representative example of normal spiral galaxies. The luminous disk has a typical radius of about 10 kpc but the HI disk extends to more than 30 kpc. The mass in the disk is estimated to be 10 11 solar masses and the mass of the dark halo is at least 10 times larger. The Sun orbits the galactic center at about S kpc with an average* speetl of 250 km/s. It takes 250 million years to complete a full revolution, but this time-scale is short compared to the age of the Galaxy that is estimated to be roughly 10 10 years (a Hubble time). Within this

time, the Sun has completed a maximum of 40 revolutions around the galactic

center. Under the circumstances, the only efficient probe of such systems is

extensive theoretical modeling, coupled with earth-bound observations of the

projected images of galaxies, in 2-D on the sky.

Kinematical modeling of HI disks in galaxies is based on the fact that the ve

locity field and the matter density distribution can be observed for most galaxies

in the plane of the sky and, then, they can be deprojected theoretically in full 3-

D space. The geometrical models that perform this deprojection with a certain

number of simplifying assumptions are usually called tilted-ring models. Since

tilted-ring models provide information about only the geometry and the velocity field of galaxies, they are useful in analyzing the st met me and kinematics, re spectively. of the hydrogen gas of which their disks are primarily composed.

Observat ions of galaxies art' performed at a certain time ("now") and. there fore, they cannot alone provide information about the long-time evolution of the observed systems. Theoretical dynamical modeling is clearly needed to complete

the picture of evolving gaseous disks. During this type of modeling, one starts

from a set of initial conditions and utilizes the dynamical equations of motion

to predict the behavior of galaxy disks over astrophysicnlly relevant time-scales

(in our case, one Hubble time). Unfortunately, the equations of motion for

a gaseous medium (the equations of hydrodynamics) form a coupled, nonlinear

system that can only be solved numerically in the most interesting cast's. This is

nett a trivial task to undertake, but it is necessary in order to follow the detailed

changes that gaseous disks undergo during their evolution.

In this dissertation, we have tried to put the emphasis on the physical as 4 pects of the studied problems. As a consequence, detailed descriptions of numerical techniques that we employ have been omitted. A detailed mathe matical description of our tilted-ring model can be found, however, in the paper of Appendix C. On the other hand, we suggest the dissertations of Toliline

(19TS) and Williams (1 OSS} to those interested in the de'ails of 3-D numerical hydrodynamics.

We should also clarify the following point. In Chapter 2, the word “rings’' is meant to describe the ammli of our kinematical. tilted-ring model. These annuli can be moved independently from one another and they have no vertical thickness. If the}’ are forced to be coplanar, they form a circular disk. In Chap ter 3. the word "rings" refers to the adopted initial dynamical models. These models are truly tori and not full disks. Finally, in Chapter 3. the term “polar rings" is used to describe, in general, rings observed around galaxies that are at high inclinations relative to the symmetry plain’ of the main galaxy.

1.2 Historical Outline of Related Work

Rogstad, Lockhart, and Wright (1974) observed the galaxy MS3 in the 21 cm line of neutral hydrogen and were the first to model their observations by using a kinematical, tilted-ring model. The model, composed of twisted and tilted rings, reproduced the strong twist observed in the isovelocity map and demonstrated that the gaseous disk of M83 is also strongly warped. This type of modeling of radio observations has become common ever since and, combined with observations of nearly edge-on galaxies, has demolished the premise that the disks of most normal spiral galaxies are flat.

Bosnia (1981) published both high-resolution observations and the corresponding tilted-ring models for 5 galaxies: NGC2S41. 319S. 5033. 5055. and

7331. Tlie models showed that only NGC 319S is unwarped and untwisted. Un fortunately, the twisting and warping angles for the other 4 galaxies are inconve niently presented relative to the plane of the sky in Bosnia's paper. Complicated trigonometric transformations are required to uncover the true size and the 3-D structure of the warps and twists of these galaxies. As a result, the fact that

♦he warp and the twist of NGC 5055 are as strong as those of MS3 has not been appreciated until recently (set 1 Appendix B).

The starting point of gas dynamics in galaxies is the acquisition of large amounts of gas hy a galaxy that participates in some type of galaxy-galaxy in teraction (collision, merging, cannibalism, or accretion). The infalling gas initially responds to the zeroth order term of the galactic potential and is quickly redistributed to form an almost flat, disk-hke structure. Then, it realizes any existing distortion of the potential and attempts to readjust again. Tohline. Si monson, and Caldwell (19S2) brought back the idea that a gaseous disk that is inclined relative to the symmetry plane of an imposed, external, spheroidal, gravitational potential well will attempt to change its orientation and settle to ward that plane. Based on the precession frequencies of test-particles derived by

Garfinkel (1959), they presented estimates of the settling time-scales for inclined gaseous disks in typical galaxies.

A scries of mostly analytical calculations soon appeared (Tohline and Durisen

19S2; Durisen. Tohline, Burns, and Dobrovolskis 19S3; Simonson and Tohline

19S3; Steiman-Cameron and Durisen 19S4: David. Steiman-Cameron, and

Durisen 19S4), determining which orientations were preferred to the settling disks under a variety of conditions: spheroidal or triaxial external potential, tumbling potential about a principal or a non-principal axis. In the framework of the <;

“preferred orientation theory," the differential precession of orbits 111 the inner part of the settling disk is faster than that of the outer regions. Therefore, the inner region settles first, leading to the appearance of a smoothly warped structure. The warp itself can be transient (settling) or in a steady state (set tled), depending on tin* assumptions about the dynamical state of the external potential (halo). However, the outer warped regions should be twisted during settling if the disk is not massive enough to counter-balance the differential pre cession induced by the halo or if the external potential of an already settled disk tumbles about a non-principal axis.

Sparke ( 19S4 ) presented an alternat ive theory of format 1011 of warps. \\ arps can be the result of discrete vertical modes of oscillation in galaxy disks. Dis crete inodes are favored over the continuum ill the presence of an external po tential. (Warps formed by this process must have a common line of nodes. 111 disagreement with the results from observations and kinematical modeling. We. therefore, favor the “preferred orientation theory" in the remainder of this work.)

Schwarz (19S5) observed and modeled the kinematics of XGC 371S. This work (another one in a long series of works presenting observations and tilted-ring models see Appendix C) is particularly important. If (dearly demonstrates that the strong warp of XGC’ 371S is close to a steady state with no differential precession between orbits because the twisting of the rings is almost zero. Other galaxies with almost zero twist are not necessarily presumed to be in steady state because their warps are also very small, encouraging the belief that their disks are essentially fiat. Galaxy disks appear to be fiat if they have completed the process of settling or if the resolution of the observation is poor (see Appendix

C).

Switching from kinematics and settling disks to numerical hydrodynamics. the works of Habe and Ikeuchi (19S5. 19SS), Yarnas (19SGa,b), and Steiman

Cameron and Durisen (19SS, 19S9a,b| signal tlit' first attempts to simulate nu merically the settling process by using smoothed particle hydrodynamics ( two methods }, and the cloud-fluid approarh, respectively. Although Habe &:

Ikeuchi and Steiman-Cameron Y Durisen used fundamentally different meth ods, their results are in general agreement to each other and to the predictions of the analytical single-particle orbit calculations. The results of Varnas are not in agreement with the other two groups and. therefore, their validity is questionable.

Having outlined the basic development of this field prior to 19S5, it is rel atively simple to describe our own work. C'hristodoulou and Tohline (19SG

Appendix A ) pointed out that t he direct inn of t wist ing of t he observed isovelocity contours in a map of a galaxy's disk can be effectively used to provide the di rection of precession of the orbits and, as a consequence, the gross geometric shape of the dark halo. They proved that a twist that turns in the same di rection as the spiral arms of a galaxy (a retrograde twist to the disk rotation) develops from prograde precession of particle orbits and, therefore, signals ihe existence of a prolate-spheroidal halo with a negative value of the quadrupole coefficient. Christ.odoulou, Tohline. and Steiman-Cameron (19SS-Appendix

D). constructed tilted-ring models for NGC 5033 and NGC 5055. They con cluded that the above idea can be followed iq> to the end because the twists of these galaxies are relatively strong, eliminating inconclusive (ambiguous) re sults. The dark matter in the halos of NGC 5033 and NGC 5055 was found to be distributed in a “prolate-like'’ shape, i.e., the longest axis of the halo is aligned with the rotation axis of the disk. This particular result is important because it seems to settle the issue between the Dark Matter Hypothesis and theories of modified gravity in favor of the former: theories of modified gravity are inconsistent with overall prolate distributions of matter in spiral galaxies since the luminous matter appears to have a disk-like shape and dark halos are not in cluded at all. Christodoulou, Tohline, and Steiman-Cameron (19S9 Appendix

C) presented additional tilted-ring models for 15 normal, not-edge-on, spi ral galaxies. All models were examined for uniqueness, hut the ambiguities were not resolved for more than half of the modeled galaxies. This study is, however, systematic; it brings together a number of previous attempts to construct kinematical models and includes detailed comparisons between models of different workers in the field. Finally. Christodoulou ( 10S0 > presented pre liminary results from dynamical modeling of polar rings and rings settling from high inclinations.

In the following two chapters, we conclude the presentat ion of our kinematical and dynamical models. In Chapter 2 . the results from kinematical. tilted- ring modeling are discussed for a sample of galaxies that includes the edge-on spiral NGC 5907. the barred spiral NGC 1 572S. ami two HI disks observed in the elliptical galaxies NGC 427S and NGC 512S (C'en A). In Chapter 3, the results from hvdrodynamical models concern only massless disks that settle to ward the preferred orientation under the influence of non-tumbling, spheroidal halos. In Chapter 4, a brief summary of the main conclusions from this work is provided. Further work using hydrodynamics is now in progress. Anticipated future extensions of this work are listed in the Epilogue.

1.3 Summary of This Work

For the reader’s convenience, we briefly discuss the results contained in the remainder of this work. It is appropriate to begin with the Appendices. <)

a) Append?? A : The standard multipole expansions of a Newtonian and a logarithmic potential arc derived. The former describes the presence of a dark halo, while the latter is valid if the assumption of a 1 /r gravitational force is adopted at large scales instead of the conventional 1/r2 Newtonian force. The magnitudes and signs of the coefficients of the quadrupole terms determine the magnitude and direction of precession of test-particle orbits in each case. It is shown that, for both potentials, the orbits process in the same direction: the precession is retrograde/prograde to t he direct ion of rot at ion of an orbiting gas el ement if the geometry of the potential is that of an oblate/prolate spheroid. It is argued that, in the absence of a halo, precession can only be retrograde because the luminous matter of normal spirals appears to be distributed in a flattened, disk-like, oblate configuration. Retrograde precession of orbits results in the development of prograde twists if, as is generally thought, the precession frequency decreases with radius. It is suggested that a search should be conducted for spiral galaxies whose velocity contour maps show a twist in the direction of the spiral arms because that would signify the presence of a prolate mass distribution, if all ambiguities during kinematical. tilted-ring modeling can be resolved. On the other hand, the results of Rogstad. Lockhart, and

Wright (1974) are critically examined and it is concluded that the direction of the observed twist reveals the existence of an oblate halo potential in M83. This result is consistent with both the Dark Matter Hypothesis and theories of mod ified gravity.

b) Appendix B : New tilted-ring models are l resented for NGC 5033 and

NGC 5055. These two galaxies were chosen because Bosnia’s (1981) observations clearly show that the twists of the isovelocity contours are retrograde to the disk rotation. The tilted-ring models demonstrate that the gross geometric shape of 10 the potential is unambiguously prolate, a result that theories of modified gravity seem unable to explain. In fart, this work formulates the strongest objection to such theories to date. A brief explanation of why other theories that attempt to explain galactic warps as originating from mechanisms other than the settling process and compete with the “preferred orientation” theory seem to fail is provided. The explanation is necessary because the results of this work hold only if the observed twists of the isovelocity contours are produced by differential precession of orbits in the settling disks.

c} thi C : Results from tilted-ring modeling are presented for lb normal spiral galaxies. The fundamental mathematical equations of our tilted- rmg model are derived and many disparate observations and kinematical models that have been published by other workers over the period 1974-19S5 are brought together. A system of definitions of all parameters involved in modeling is de scribed. 111 an attempt to standardize future kinematical modeling. Parameters of older, different models are converted to this new system. It is demonstrated that non-uniqueness problems arise in deprojecting the models from 2-D to 3-D if the twist of tin* isovelocity contours is not very strong. In these cases, two models of equal quality can be produced, one with prograde and another with retrograde twisting of the rings. Under the assumption that spiral arms are trailing features, all ambiguities are resolved for 7 galaxies: M33, MS3, NGC

2805, NGC 2841. and NGC 3718 possess kinematics dominated by an overall oblate mass distribution, while NGC 5033 and NGC 5055 exhibit the influence of an overall prolate mass distribution in their kinematics. Normal spiral galaxies are classified into 3 types according to the size of the twisting of the rings (measured in the plane of the model-disk); large twisting (> 30°) usu ally means that all ambiguities will be resolved, while small twisting (< 1 0 °) 1 i is rither associated with steady state warps or is just a product of low-quality observations. Finally, a discussion is presented on interpreting the results from tilted-ring modeling in order to deduce conclusions about the ongoing dynami cal behavior of normal spiral galaxies. The interpretation of kinematical data is done in relation to standard results derived from the "preferred orientation theory."

d) Chapter 2 : Our presentation of results from kinematical modeling is concluded with a discussion of tilted-ring models for an edge-on spiral (NGC

5907). a barred spiral (NGC 572S). and two gaseous disks in ellipticals (NGC

4278 and C'en A). These galaxies represent a few ex; 30 1 *s of systems where either only isodensity contour maps can be produced (edge-on spirals) or tilted- ring models are not very successful in reproducing the observations because of

11ic' strong influence of non-circular motions (barred spirals and HI disks in ellip ticals). In addition, a different class of models is produced for NGC 5907: they do not exhibit warped rings, but the isodensity maps show a warped structure

(see Byrd 1978). We demonstrate that these illusionary warps cannot be made strong and sharp enough to match the actual radio observations.

c) Ch ayicr ,v : Numerical hydrodynamics is used to follow tin* dynamical evolution of settling disks. The disks are massless and the gravitational field is supplied by an external, static, spheroidal potential. The symmetry plane of each disk is inclined relative to the equatorial plane of the underlying poten tial. Typical models are considered at inclinations i„ — 0°, 10°. 40°, 80°. and 90°. The settling process is different between high and low inclinations, in agreement, with Habe and Ikeuchi (1985). Disks settling from moderate and high inclinations show significant inflow of matter toward the central re gion of the potential, in agreement with Steiman-Cameron and Durisen (19SS. rj

19S9a,b). but they do not collapse entirely to the center at high inclinations as the evolutions of Habe and Ikeuchi would indicate. Disks settling from low or moderate inclinations form smoothly warped structures after several rotation

periods.

If the disks, however, start initially from high inclinations, they do not feel the effects of differential precession or angular momentum loss for several

rotations and, when they do, matter inflow and settling proceed slowly. As

a result, formation of nuclear disks is favored at moderate and high inclinations,

but the time-scale for nuclear disk formation at high inclinations is very long

comparable to a Hubble time. All warps are found to be transient features

that, given enough time, disappear. Rings at the exact polar orientations

are extremely long-lived structures because the differential precession is zero

at >t, = 90°. Numerous plots of the computed models depict their time

evolution and the global changes in structure they undergo as they settle toward

the “preferred orientation."

As a side effect, two modes of the Papaloizou-Pringle (19S4) instability can be

seen clearly in the plots. These instabilities probably do not play an important

role in the evolution of real HI disks that are much colder than our models, out

their visualization is. by all means, interesting and educating. The m = 1

mode is especially interesting because it creates strongly asymmetric structures,

similar in some sense to the asymmetries seen in lopsided galaxies (e.g., M 101 in

Sandage 19C1). C H A P T E R 2 KINEMATICAL MODELING

In all of my universe I have, seen no la.it> of nature, unchanging and in exorable. This universe presents only changing relationships which arc sometimes seen as laws by short-lwed awareness. These fleshly sensoria which we call self are ephemera withering in the blaze of infinity, fieet- mgly aware of temporary conditions which confine our activities and change as our at iivit.ics change. If you must label the absolute, use its proper name; Temporary. -■from l'God Emperor of Dune" by Frank Herbert

The kinematical modeling of HI disks seen m edge-on spirals, barred spi rals, and elliptical galaxies is more complicated than that presented in Appen dices B and C' for ‘‘normal" spirals. For edge-on spirals, the velocity field cannot be observed. For barred spirals and HI gas in ellipticals, the kinematics is fre quently dominated by non-circular motions and/or large-scale asymmetries. We have attempted to model these systems by including more free parameters in the tilted-ring models to deal with the oval distortions of orbits, but this has turned out to be not a good idea. The advantage of fitting observations better is greatly overwhelmed by non-uniqueness problems and the appearance of new ambiguities that arise because of the increasing number of free parameters. We have, consequently, chosen to explore the capability of our circularly symmetric tilted-ring model by applying it without modifications to a few of these types of systems. Here, we present in detail only 4 representative examples: NGC

5907, NGC 5728, NGC 4278. and NGC 5128 < Cen A).

2.1 Edge-on Spirals

Several edge-on spirals have been observed in the 21 cm line: NGC 4244.

i:i 11

45G5. and 5007 (Sancisi 197G, 19S1 ). NGC’ SOI (Allen. Baldwin, and Sancisi

1078). and the pair of NGC 4G31/NGC 4G5G (Weliachew. Sancisi, and Guelin

1978). In a review paper, van YVoerden (1979) concluded that, on a statistical basis, the observed warps are not created by interaction with companions. In deed. only NGC 891 possesses a flat disk and only the kinematics of NGC 4G31 can be adequately explained by a tidal interaction model that includes its two companions (Combes 1978). So, what is the physical mechanism responsible for the formation of warps in isolated, non-interacting galaxies? Four such mechanisms have been proposed: a) The “preferred orientation theory”, according to which the tlisk is settling to a new steady-state orientation. The warp is either transient if the

overall potential is static or the outline of the steady state itself if tin*

potential is tumbling. In Chapter 3. we will utilize this theory in order

To interpret our results. b ) The idea that warps are modes of oscillation of the disk (Sparke 1984;

Sparke and Casertano 19SS). As is explained in Chapter 1 (and in Ap pendix C), this mechanism has difficulties in explaining twisted warps. No

tice. however, that a dark halo must he present for discrete modes of oscil lation to exist. In that respect, this theory also argues against theories of

modified gravity. e) The idea that the self-gravity of the luminous matter delicately counter

balances the influence of the dark matter so that the total differential pre cession at all radii is constant (Petrou 1980). The result is, again, a

steady state with the disk processing as a solid body. This mechanism is discussed in Appendix B. It involves unusual and highly improbable assumptions: the precession frequency due to the halo must be an in

creasing function of radius m order for the halo to counter the precession

induced by the self-gravitating matter. Petrou accomplished a constant

precession over the entire outer disk by forcing the isopotential surfaces of

the halo to become flattened only away from the disk. As a result, the

outer region would settle first in disks whose dynamics are dominated by

massive halos. Moreover, massive disks under the influence of massive

halos should necessarily begin their evolution with constant precession as

they would not have the ability to extinguish any amount of pre-existing

differential precession.

(1) Byrd (197S) proposed that the only directly observed warps (those of edge-

on galaxies) may simply be illusionary a combined result of the almost

edge-on disk orientation and the smoothing of the map by the telescope’s

beam. At the time it was proposed, this was a viable alternative because

tilted-ring modeling had not yet shown that warps are prevalent, in normal

spirals that are not viewed edge on.

It is interesting to note that the discovery of warps in not-edge -011 spirals is indirect. Therefore. Byrd’s idea requires further examination. In order to test this idea, we have constructed models of illusionary warps for specific edge-on spirals. Figure 2.1 shows the comparison between two isodensity maps of NGC 5907. The top map is produced from the standard tilted-ring model

{Appendix C). The central, unwarped disk is composed of 10 rings and is inclined to the line of sight by ta — 80°. The position angle of the blue-shifted side of the major axis is -y = 150°. The southern edge is determined to be closer to the observer than the center of the disk from the unequal division of the bulge 10

by tin 1 dust lane (Appendix C). The observed war]) is produced by the bending of the 10 outer rings. The warping angles are incremented by Air — 1°

(starting from zero) and the twisting angles are all zero. The position angle of the warj) is /„ = 2 0 0 °.

The bottom map in Figure 2.1 shows an illusionary warp. This map is produced from an unwarped, tilted-ring model by the following procedure: a)

A spiral density pattern is imposed on the unwarped disk, with the form:

A R 0 = Oo + — (n — . {2.1 ) " R« where (R.o) are polar coordinates on the plane of the disk and A.n.tf’u, and

R„ are free parameters. We have chosen ti — 2 (two-armed spiral). A = 10 and R0 = 5 ring radii. The resulting spiral structure is similar to that of our own Galaxy. The density contrast between the arm and the interarm regions ranges from 2 : 1 to 10 : 1 across the exponential disk. b) The model is brought to an almost edge-on orientation (?„ — SG°) and is convolved with a Gaussian beam whose effective width is that of the actual observation (51" by Gl" on the sky). Tin 1 bottom map of Figure 2.1 shows the strongest possible illusion ary warp that could be constructed for NGC 5007 ( 0O — 30°), This warp is clearly weaker than the observed one mainly because it lacks sharp edges. The same conclusion was reached by modeling NGC 45G5 and NGC 4G31 in a sim ilar fashion. The illusionary warp constructed by Byrd (1078) also shows no sharp edges because convolution spreads it out at the low density regions. It, therefore, seems not possible to explain the observed sharp warps of edge-on spirals as simply illusions, although some illusionary warping ran still affect the observations of edge-on spirals. 17

2.2 Barred Spirals

We have used the circularly symmetric tilted-ring model to study only one barred spiral, NGC 5728. We have implemented the rotation curves of Rubin

(1980) and of Schommer el ah (1988) and compared our models’ maps to the observed maps of Schommer el nl. (19SS) (Figure 2 .2 , top). A comparison between the maps of Figure 2.2 shows that the model is not very successful in reproducing the observations, probably because the kinematics of the disk is dominated by oval distortions driven by the bar. To test this assumption, we introduced elliptical orbits in our model and the new maps were improved, but the new models were no longer unique (different sets of free parameters produced very similar m aps).

The values of the parameters of our best model (shown at the bottom of

Figure 2.2) are: .V = 29 rings; position angle of the major axis *, = 2°; inclination of the unwarped disk i„ - 4S°; the southern edge is farther away from tlu' observer than the central region: position angle of the warp tv = 40°; no twisting of the rings, i.e.. At = 9°: the first 5 rings are unwarped: then, the warping angles increase smoothly to a value of 1 2 ° at the 1 2 th ring: outside the

12th ring, they drop smoothly, reaching 9° at 11 it' 18th ring and —19° at the

29th ring.

The geometric shape of the dark halo cannot be determined because A t =

9°. This is termed the “SP ambiguity'' and is discussed extensively in Appendix

C. We should also point out that the results of our modeling indicate that the disk of NGC 5728 may be in a steady state; as in the rase of NGC 3718

(Appendix C"), the twisting of the rings is zero but the warping is substantial. IS

2.3 HI Disks in Ellipticals

The standard example in this category is the (hist lane of Cen A (NGC

5128). Bland, Taylor, and Atherton (1987) have recently presented new.

high-quality Ha observations of the ionized gas in C'en A. Based on their

observations, we produced isovelocity and isodensity maps with our tilted-ring

model. They are shown in Figure 2.3, where the top figure is a magnified, black-

and-white copy of the observed isovelocity map. (The observed isointensity

map is not shown, because it is composed of spots of Ha emission and does

not spread out over the entire dust lane.) Tin 1 relevant parameters of our

best model are: .V = 20 rings; the position angle of the projected major

axis - 125°; the inclination of the unwarped region ir, = 73'J; the

southern edge is further away from the observer than the central region (Simonson

19S2); the direction of twisting is consistent with the presence of a prolate mass

distribution (the underlying elliptical). but the maps do not change significantly even if an oblate shape is used (as Table 2.1 indicates At = 0° over most of the outer rings, allowing for the SP ambiguity to arise): the rotation curve reaches its maximum F(, = 250 km/s at the fourth ring and stays flat thereafter: and the position angle of the warp t0 ~ 50°. The variation of the warping (ir ) and twisting (f) angles is not smooth. Their values are listed in Table 2.1. The constancy of the twisting angles after the 1 2 th ring suggests that the entire dust lane may be in a warped, steady state. Presumably, the warped gas layer has settled to a preferred orientation that is not planar because of the figure rotation of the underlying elliptical. There is a constraint, however. The figure rotation of the galaxy must be opposite to the rotation of the gas and dust for this steady-state configuration to exist (Tohline and Durisen 1982; van

Albada et_ ah 1982; Steiman-Cameron and Durisen 19S4). I't

The valuf's of Tahir 2.1 show that tin* (lust lane is roughly composed of two

disks. Tlir innrr disk extends up to the 12th ring. The abrupt warping of the

rings is implemented to reproduce the curving of the velocity contours in the inner

region but the warping angles are uncertain in this central region. The outer

disk starts at the 13th ring and is oriented orthogonal to tin' inner disk. The

warping is smooth with no twisting of the rings greater than 3°.

This result is rather surprising because the war]* of the dust lane in Cen A

is currently thought to be transient and is trying to settle to the preferred plane

(Bland el ah 19SC. 1DS7; Bayes et. ah 19SG). Moreover, it is interesting to note

that Scliwar/schlld admitted openly ( see the discussion at the end of the review by

Schweizer 19SG) that, because the outer regions of the dust lane are not considered

to be settled and because the elliptical tumbles in the prograde sense, the famous

idea that the dust lane is stabilized by the rotation of the elliptical (van Albada et ah 19S2) is not the proper explanation for C'en A! In view of the above result from our modeling, we can provide an alternative explanation: Within the limitations of our model, the outer region of the dust lane of Cen A seems to be in a warped, untwisted, steady state a result that is in complete support of the idea of van Albada ey ah (19S2). We think that tin’ observations of C’en A may have been misinterpreted. The observed twist of the contours in the isovelocity maps is simply a projection effect and can be reproduced by warping without

twisting of the rings. After all, it is not very strong and it does not seem to keep twisting at the outer regions. Since, however, the elliptical tumbles in the prograde direction relative to the rotation of the gas ( Schweizer 19SG and references therein ), a dark hah* is needed to provide the necessary retrograde figure rotation.

A system that presents fewer difficulties in modeling is the HI disk observed 20

in tin elliptical NGC 4278 (Raimond et ab 1 OS 1: Knapp 1082). Figure 2.4 slum's the velocity maps of the actual observations (top) ami of our best model

(bottom). The relevant model parameters are: .Y = 20 rings; y = 225°;

j() — 4 5 °; the southern edge is assumed to be closer to the observer; tlie direction of twisting of the rings assumes an oblate potential; the rotation curve is entirely

Hat at C, = 290 km/s; the first 8 rings are not warped; the position angle of

the warp is /„ ^ 110°; and the twisting and warping angles are incremented by

A/ — 4° and Ate =. 1°, respectively. Again, a model with elliptical orbits

fits the observations better but it cieates non uniqueness problems,

2.4 Permission to publish observations

We wish to thank Drs. ,1. Bland, N. Caldwell, and G. Knapp who have made

the task of comparing models with observations easy by granting permission to reproduce their observed contour maps in Figures 2.2 2.4 of this chapter. Copies of their letters, granting permission to publish some of their observations, are

listed below. 21

Letter 2.1

SMITHSONIAN INSTITUTION ASTROPhvSiCal OBSERVATORY

PUB l IC information BUSINESS (602> 1B6-2432 Iflon «?9-6T4i (FTSt 762-674’

FRED LAWRENCE WHIFFLE OBSERVATORY

June 19, 1989

Dimitris Chrlstodoulou Dept, for Physics and Astronomy Louisiana State University Baton Rouge, Louisiana 70601

Dear Dimitris,

This letter is to officially grant you permission to publish in your thesis the data on the galaxy NGC 5728 that 1 gave you a few years back. As well, you are welcome to use any figures, graphs, or quote any text for your thesis that appeared In the paper 1 published with others on the NCC 5728 data (the paper appeared in the Astrophysical Journal, volume 329, page 159, 1988). I have enclosed two copies of that paper, one for your own convenience, and one which 1 hope you will give to Joel Tohllne Here's to a speedy writing up of your thesis, and I look forward to seeing you here in Tucson soon.

Sincerely,

"71 afcevt Nelson Caldwell T1

L e tte r 2.2

AH610-AUSTRUIAH OBSERVATORY

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J PniKTton L’nncrsin Department of A«trophytic*l Science* Peston Hal/ PrincCKm, N** Jer»*\ 08544

June 1, 1989

Dr. Dimitris Christodoulou Dept, of Physics & Astronomy Louisiana State University Baton Rouge. LA 70803-4001

Dear Dr. Christodoulou:

Thank you tor your letter of May 24. 1989 This letter is to grant you permission to reproduce Figure 1 of the paper 'Interstellar Matter in Elliptical Galaxies' by G.R Knapp, which appeared in 'Internal Kinematics and Dynamics of Galaxies.* International Astronomical Union Symposium No. 100. edited by E. Athanassouia and published by D. Reidel Inc.

Sincerely,

Gillian R Knapp Associate Professor of Astrophysical Sciences GRK/wt Table 2.1 Warping and Twisting Angles for Cen A

Ring; f(°) f<'(°)

1 A 0.0 0.0 5 0.0 AS G O.S 9.C 7 1.5 14.3 8 2.5 19.1 9 1.5 14.3 10 O.S 9.G 11 0.0 4.S 12 S9.0 0.0 13 178 0 4 .8 14 177.0 9.G 15 170.0 14.3 1G 175.0 19.1 17 175.0 20.0 IS 175.0 21.0 19 175.0 22.0 20 175.0 23.0 2j

Figure 2.1

T T T* 20

Figure 2.2

NGC 572B Figure 2 8

Figure 2.4

too I 70C

i / y

uo f

700 t sso

s s ss h m 12 17

CU1950) NGC °i218 C H A P T E R 3 DYNAMICAL MODELING

Of the laws we can deduce from the external world , one. stands above all: the Law of Transience. Nothing is intended, to last. The trees fall year by year, the mountains tumble, the galaxies burn out like tall tallow candles . ... Against the Law of Transience may be set one. of its ancillary laws. The Law of Endurance. -from “Galaxir-s like Grains of Sand" by Brian Aldiss

We have conducted a preliminary study of gaseous disks, settling toward a preferred orientation under the simplest possible initial conditions: the disks in reality, tori or rings are inassless and the external potential dark halo is oblate-spheroidal and non-tumbling. Under these conditions, rill the warps seen in the dynamical models are expected to be transient features and all the twists that will develop during the settling process must be prograde to the disk rotation. We plan to extend this work in the near future, and include more complicated conditions (triaxiality and figure rotation of the halo and, more importantly, self-gravity of the disk). Hydrodynamical computations of settling disks ate unfortunately very expensive to carry out. We estimate that, in the absence of self-gravitv, one model evolution takes about 40 CPU hours on the IBM 3090-000E vector processor of LSU. One model evolution usually runs for 10,000 time-steps with 14.4 CPU seconds per time-step in a 3-D grid witli

64 x 04 x 64 resolution. In addition, the memory and storage requirements are

14 Mb and 125 Mb, respectively. :jo

3.1 Introduction

As we reviewed in Chapter 1 . the theoretical studies of settling disks in external, gravitational potentials with applications to HI warped disks in spi rals, disks of gas and dust in ellipticals, and rings around galaxies started with analytical calculations that determined the entire spectrum of possible ori entations that are energetically preferred to the orbiting gas or dust. Idealized external potentials (spheroidal and t riaxial. st at ic or t umbling) were analyzed and standard methods of Celestial Mechanics were used to predict the available “pre ferred orientations" and the orbital precession frequencies of set tling test-particles

(Tohline and Durisen 19S2; Tohline. Simonson, and Caldwell 1982; Durisen,

Tohhne. Burns, and Dobrovolskis 19S3: Simonson and Tohline 1983: Steiman-

Cameron and Durisen 19S4: David. Steiman-Cameron, and Durisen 1984). Nu merical integrations of orbits were used, in addition, to identify families of closed orbits in spheroidal and triaxial potentials and to examine the results of figure rotation (or “tumbling") [Schwarzschild 1979: Heiligmau and Schwarzschild

1979; van Albada, Kotanyi. and Schwarzschild 19S2; Simonson 19S2; Heisler.

Merritt, and Schwarzschild 19S2; Merritt and de Zeew 19S3; Simonson and

Tohline 1983], A complete summary of all the work that has been done in this

area can be found :n the review of Schweizer (19SG).

Although the above studies shed new light on the dynamics of gaseous disks,

they did not improve our insight regarding the details of the evolution of these

disks from initial, arbitrary orientations toward the preferred orientations. We

begin with a summary of the basic dynamical properties that a gaseous galactic 31 disk should exhibit :

(a) Settling is driven by differential precession and viscous dissipation of angular momentum. The components of angular momentum parallel to the preferred orientation are destroyed by viscous forces. The end-product of set tling is a disk whose total angular momentum is only the vertical component of the original angular momentum of the disk and there is no differential precession in the final, steady-state configuration.

(b) Given the shape or symmetries of the external potential, the preferred orientations can be identified. Figure rotation drastically modifies the preferred orientations deduced for static potentials, and generates warped and twisted disks in steady state.

(c) The precession frequencies of test-particle orbits depend on the strength of the quadrupole distortion of the potential and on oa/,,, where i„ is the inclination of the disk relative to the preferred orientation. The characteristic time-scales for settling are estimated to he proportional to tin* differential precession times between orbits which interact through viscous forces, i.e.,

r, cx . (3.1)

This expression was given by Tohline, Simonson, and Caldwell (1982). The differential precession frequency Au.'j, includes the dependence on the range of viscous transport as it depends on A?\ the “epicyclic amplitude" of interacting particle orbits.

(d) The final radius of a settled disk R f should be smaller than the initial 3 2 radius /?, by a factor of co.

R f = R, co*i0 . (3.2)

As the settling disk shrinks, matter will flow toward the inner region and trigger or enhance some type of observable nuclear activity.

(e) A variety of families of closed stable orbits is found in spheroidal and triaxial external potentials. Figure rotation helps in creating even two distinctly different groups of closer! orbits one exists well into the core of the poten tial. while the other occupies the outer regions. Gas is expected to occupy those orbits and. therefore, present a mult it ude of shapes and orientations, from simple planar disks to complicated integral sign disks.

Numerical calculat ions were employed next in order to study the details of set tling by following the dynamical evolution of gaseous disks (for a brief summary see Christodoulou and Tohline 19SS). Three different numerical techniques have appeared in the literature: smoothed particle hydrodynamics (SPH, in short; Habe and Ikeuchi 1985, 1988; Yarnas 19SG a.b), a cloud-fluid approach

(CF, in short; Steiman-Cameron and Durisen 19S8, 19S9a,b), and Eulerian hydrodynamics (EH, in short; Christodoulou 19S9 and this work).

The SPH calculations showed that only some of the disks were settling to the predicted preferred orientations within times comparable to those determined analytically [eq. (3.1)]. Habe and Ikeuchi (1985) also showed that the settling mechanism at high inclinations is different from that at low inclinations. In stead of the familiar, smoothly warped settling structures, disks starting at high inclinations showed a catastrophic evolution, with most particles flowing directly into the central region of the potential and forming nuclear disks. The

SPH calculations of Yarnas (19SGa,b) have produced results which, at least at present, are rather unacceptable. They do not agree with the predictions of the analytical theory or with our EH or the CF results. Specifically. Vanias finds ring structures forming at high inclinations while gas at the inner regions settles to a different plane. In addition, entirely new preferred orientations are identified in those computations. Without getting into details, we will simply attribute all these strange results and the points of disagreement between SPH methods to the inadequacy of the adopted numerical schemes.

The C'F calculations have led to a useful analytic description of the settling mechanism, valid for small inclinations. Steiman-Camoron and Durisen (198S) showed that the inclination of a settling disk decays as a function of time from the original inclination itJ :

,(/) ^ ,ut j-p( —tI rf j'1 , (3.3)

The decay constant is given by the equation:

i/:( (3.4) where v is the adopted coefficient of kinematic viscosity in the cloud-fluid medium. Equation (3.3) is only valid in the absence of inflow'. Numeri cal calculations show'ed, however, that inflow* could not be neglected at high inclinations. This agreement with Habe and Ikeuchi (19S5) is rather surpris ing, because in tin* CF method angular momentum is transported by shearing 3*1 viscosity, a process totally neglected in the SPH method. We know that viscous dissipation is important in set tling (the N-hody calculations of Miller et al. [1989] do not show any settling because the models are dissipationless), but it seems tlint the particular mechanism is not important as long as there is some type of viscosity to destroy the unwanted components of angular momentum. We imag ine, however, that other differences may still exist for different mechanisms in the radial distribution of the remaining momentum and the settling time-scales.

We have used yet another numerical technique (EH) in order to study the details of settling and, hopefully, better understand the aforementioned results from hydi odynamical computations. We have employed a 3-D. first-order ac curate, explicit, Eulerian, hydrodynamical computer code to follow the dynam ical evolution of massless disks under the influence of a Newtonian spheroidal potential. We have used a special class of initial “disk" models 111 hydro static equilibrium, the Papaloizou-Pringle (19S4) tori, because they are easy to construct. Because they have inner edges and are geometrically fat, these tori do not represent real HI disks in galaxies. In addition, they suffer the

Papuloizou-Pringle (PP, in short) instability, which results in an exponen tially growing, non-axisymmetrie wave bouncing between the inner and outer edges. They do, however, include all the physics of settling and the PP insta bility is quickly damped out by the viscosity of the numerical scheme. As a by product of this work, the growth rates and pattern speeds of the rn — 1, m = 2 , and tii = 3 modes of the PP instability art* determined for the computed models in the linear regime.

In §3.2, we briefly present the equations of hydrodynamics and discuss the structure of the initial models. In §3.3, we estimate (the order of magni tude of) the settling time-scales for our models using both equations (3.1) and

(3.4). We also discuss the importance of matter inflow at moderate and high inclinations. In §3.4, we present and discuss the evolutions of the models. In

§3.5, we report results of the non linear development of the PP instability. Fi nally, in §3.0, we summarize the basic conclusions regarding gaseous disks that settle under the influence of an external, spheroidal potential.

3.2. Dynamical Equations and Initial Models

We use a cylindrical grid with coordinates (/?, o~Z ) to solve the equations of hydrodynamics:

(3.56)

OA (3.5c) dt and

^ + V ' (f*v) - 0. (3.G)

These equations are supplemented by a polytropic equation of state:

(3.7)

In equations (3.5) - (3.7), p is the density, P the pressure. $fTt the external po tential, A' the polytropic constant, n the polytrnpic index, v = ( vp, e0, vz ) the velocity vector, and (5\ .4, T) are the radial, angular, and vertical components of the momentum vector per unit volume, respectively, i.e., S = />!’/(, -4 — pRv$, and T — pvz- The algorithm uses the donor-cell scheme to adveet the dynamical quantities from cell to cell in time.

For our initial models, we choose zero-mass PP tori with j; = 3/2. In these tori, the specific angular momentum /u is uniform in spare. We choose the value In — 1 and, in addition we normalize R 0 = 1 at the pressure maximum in each torus. Then, there is one free parameter that essentially modulates the thickness of each torus. We choose, equivalently, to specify the polytropic speed of sound c0 or, since r^(/?0) = 1, the Mach number m„ = 1 /cat the pre'ssurc maximum. The radial thickness of the torus is uniquely determined from the values of n and in,,. If 7?+ (/?_) is the cylindrical radius of the outer

(inner} edge, then:

n ± = (1 rp v/ 2 u /m (J}-] . (3.S)

As a first approximation, we use the standard quadrupole expansion of a spheroidal potential for $,,/ :

where 7’ — \/R~ + Z ~, a is the equatorial radius of the spheroid, and J20 is a constant that determines the strength of the quadrupole term. I 11 addi tion, the gravitational constant <7 = 1 and the mass of the spheroid M = 1 complete the normalization of units. In this system of units, one initial central rotation period ( defined at radius R„ ) corresponds to 2tt. 37

3.3 Settling Tim e-Scales and Infl aw

We have followed in detail the evolution of two models, a thin torus with rn 0 — 7.2 (/?+ — R- = 0.5) and an extended torus with m 0 — 4,2 (R+ —

/?_ - 1.0). initially inclined by i„ = 0M 0°, 40°, 80°, and 90° from the equatorial plane of an oblate potential with J ^ o 1 = 0.02. This rhoice of the quadrupole coefficient is consistent with results from recent observations of polar- ring galaxies (Whitmore. McElroy. and Schweizer 19S7) which suggest, that the dark halos in three observed systems do not deviate by much from spherical symmetry. Fully 3-D images and 2 -D cross-sections of the initial models are shown at the top panels of Figures 3.1 and 3.2. Considering test-particle orbits in the gravitational field of equation (3.9), the precession frequencies are;

3 a > — ^ “ ) COSl,, , (3.10) where is the rotational frequency. Inserting the numerical coefficient into the right-hand side of equation (3.1) [see. however, Steiman-Cameron and Durisen

19SS for a discussion of the original expression proposed by Tohline. Simon son, and Caldwell 1982]:

r, = . (3.11)

Steiman-Cameron and Durisen (1988) have demonstrated that equation (3.11) can only be considered as an upper limit to the settling time-scales. Now, Au.’p can be determined from the equation:

= ^ A r + , (3.12) Or Oi „ whore Ar ~ rA ;D is the effective range of viscous forces in the gaseous medium.

To proceed with the calculation, we have to adopt an interaction model of angular momentum transport. We adopt the simplifying assumption that “random motions” are represented by random velocities of a mean amplitude At' equal to the maximum speed of sound m the medium. (For our models, then. At' is about 14-24% of the rotational speed at pressure maximum.) Then, equation

(3 .1 2 ) can be written as:

= l ^ r + ^ ) — . (3.13) \ or Oi„ / r„r(J

where ru — 1 , va — 1 in the adopted system of units and the partial derivatives are determined from equation (3.10).

Lower limits for the decay times are derived from equations (3.3) and (3.4).

Once the decay times are known, the settling tunes are determined from the equation of Steiman-Cameron and Durisen (1 OSS):

r, = * (314)

This equation assumes that, the effective settling time is that for the disk to settle from the original inclination t0 down to 1° away from the preferred plane.

In summary, the final expressions for the settling times, expressed in central initial rotations for our models, are:

T3 , m a z ~ 2.1 rn„ ^ — J (com,, + -sju io )"1 . (3,15a) 3< J and

]/:i 1.3 (3.156) rci.i ~i„\r0/ 1 °

Numerical values, based on equations (3.15). are listed in Table 3.1,

Habe and Ikeuchi (1DS5) have commented on the difference between settling from high and low inclinations. Their argument is based on the dependencies of the differential precession frequency : if the inclination of the disk is low. A j.'ji depends primarily on the radial “epicyclic amplitude" Ar (the first term of eq. [3.12]). but if the inclination is high. A~;, depends primarily on the angular amplitude A?(, (the second term of eq. [3.12]). Here, we present yet another argument that demonstrates not only the difference between settling from various inclinations, but the onset of matter inflow at high inclinations as well.

Consider an equilibrium orbit on the equatorial plane of a disk that is in clined from the equatorial plane of an external spheroidal potential by Let

XYZ be a system of coordinates whose Z axis aligns with the symmetry axis of the potential and X'Y'Z' be another system of coordinates whose U axis aligns with the symmetry axis of the disk and X' = X. In what follows, we adopt the cylindrical coordinates (7?,0,Z) in the inertial frame of the potential and the corresponding primed coordinates in the frame of the inclined orbit. Our discussion is based on the two different physical pictures that emerge when La- grangian and, in turn, Eulerian coordinates are used to describe the same phenomenon. Test-particles moving on this orbit cross from smaller to larger radii and vice versa, seen from the frame of the potential. We assume, then. 40 that velocity perturbations At'/; will develop at least early in the evolution of the tli.sk that will be radial in the frame of the potential. Moreover, if settling of the orbit occurs, then we expect that additional velocity perturbations Any will develop in the settling direction. We imagine that Ar/j will enhance os cillations about the equilibrium orbit, successively bringing test particles from smaller to larger radii and vice versa or even prohibit this alternation from oc- curing smoothly. Similarly, we imagine that Any either enhances settling of the orbit or works against it. In either case, the velocity perturbations force the orbit out of equilibrium and a “kinematical" description that incorporates

Ar/f and A i'/ can provide a useful illustration of the onset of inflow at different indinat ions.

We begin with standard coordinate transformations in order to find how the velocity components Ar// and Ary are realized in the frame of the disk itself. Obviously, A 17; and Ary are not distributed equally between the 7?\

Ary = 0, and (c) Ae/; — Aey.

(a) If only Any ^ 0 , then transforming to the frame of the disk, we find:

At’/; = Any ^ — fmii„ , (3.1Ga)

At’Jj, = A ry ^ — ,snuv , (3.16M and

Any — A ry cos}a . (3.16c)

Since the ratio = tan4>/ cosia, the perturbation does not contribute at 41 all in the

A r^ /A r^ tani0 at 4> = 90°, 270°. It is exactly the dependence on the tam„ term that shows the perturbation is exclusively funnelled into the R' direction at high inclinations, but is equally distributed between the R' and Z' directions at moderate inclinations (i0 a; 45°).

(b) If only Ar/j ^ 0, then transforming to the frame of the disk, we find:

Av'J{ - A vf{[ — (3.17a) •(£) ■

A — Ar// sniO <<)s0 >tut,, ititn, (3.176) V W and

Ae^ = -Ar/; sit io sin i „ . (3.17c)

This perturbation is not funnelled into the o' direction either at o ~ 0°. 180° or at 0 = 90°, 270° because A i^ ~ smOcosO , Since A i^ /A t^ ~ iani0 at 0 -

90°, 270°, we conclude that, in this case, inflow is only favored at low inclinations.

A i^ = Ai (I) {1 + si nO tutu,,) . (3.1 S cj )

R Atu = Aim 757 1 ( 1 — sir> tm\i0)cos(p stnt^ (3.186) R and

A<^ = A i1 CO,S70 (1 — St no t(IT)l0 (3.18c) 4 1>

— 90°, we find: A r^ /A c ^ = tan(45° — i0) and Ai’^/Ai’Jf = 0. In this case, Ar^ is always more important than Ai’^ but its relationship to AeV. is complicated: Ai1^ peaks relative to A i^ at moderate inclinations ( i0 ~ 45°).

The above description of inflow at different inclinations is only valid for the onset of inflow and cannot follow the fully nonlinear effects that eventually dom inate in the settling disks. It is, however, directly comparable to the nonlin ear dynamical behavior of our models in fj3.4. where inflow is initially observed mainly at 4* ^ 90°, 270° in the R' direction (r f. the velocity fields on the hor izontal slices through the models, early in their evolutions). We also believe that the above discussion serves as an attempt to describe some of the details of the settling process analytically.

3.4 Model Evolutions

a) in = 0 °

Both the thin and the extended rings were evolved for about 40 central ro tations (i.e., roughly a Hubble time) in a spherically symmetric potential

<£er( = —l / 7' [he., equation (9) with Jjn — 0 ]. This test demonstrated that the rings stay in steady state if the potential is not distorted and showed that the numerical scheme remains stable for many rotations. It is worth pointing out that the numerical viscosity of the first-order rode damped considerably the 43 rr instability. (This instability grows out of numerical noise in the second- order accurate scheme, and it completely disrupts the models cf. Hawley

[1987,1959]. For this reason, the more accurate scheme is not suitable for the present computations.)

Then, each ring was placed on the equatorial plane of an oblate spheroidal potential, i.e., in its preferred orientation. The small quadrupole distortion of the potential excited the PP instability, but the numerical viscosity of the code quickly damped it out and both models preserved their steady states thereafter and up to about 40 rotations.

b) Low and mod r rale. ivi'lniatnnix

In all our computations, the initial models start out horizontal in the cylin drical grid. Then, the angular momentum vector is always initially aligned with the positive Z axis of the grid and the preferred orientation is always tilted by ia relative to the initial orientation of the model. Before we describe specific evolutions, we should point out that computations are not performed in the inner 10 grid cells. Matter and momentum density that reaches this boundary is taken out of the grid. We will, hereafter, refer to tins inner region of the grid as the “central region.”

Figures 3 .1a and 3.2a show 3-D isodensity snapshots from the evolutions of the extended and thin rings, respectively, both placed initially at i 0 = 10°. Sim ilarly, Figures 3.3a and 3.4a show 3-D isodensity snapshots of the same rings, ini tially placed at i0 — 40°. Settling in these plots corresponds to a rotation of the rings about the Y-axis. Figures 3.1 3.4 with indices b and c show the corre -11 sponding horizontal cross-sect ions through the grid on the XV plane and vortical cross-sections through the negative X-axis, respectively. Both density contours and the velocity components are plotted on each cross-section. The preferred orientation is indicated by a straight line at angle i„ from the horizontal.

The models with t0 ~ 10° clearly show smoothly warped structures during their evolut ion. In addition, no significant inflow of mat ter was observed. The warps of the models with i„ ~ 40° also develop smoothly, but large inflow is observed. After t = 12.8 G rotations. 75% of the mass and 82% of the angular momentum lias flown into the central region and is lost, respectively, from the extended ring of Figure 3.3. The effects of inflow and differential precession are more dramatic for the thin ring of Figure 3.4: the inflow is 98% and the specific angular momentum loss is 99% after only t ~ 10 79 rotations. For comparison, the angular momentum loss is 8 % (f = 28.GS) and 15% (t = 34.12) for the models of Figures 3.1 anti 3.2, respectively.

The nuclear disks that presumably form at the central region art* not shown in Figures 3.3 anti 3.4. The fact that the remaining tlisk is very close to the preferred plant* at the end of each evolution suggests that any nuclear disk that does form from inflow will have the same inclination and. therefore, form on the preferred plane.

cj High mclnialioT!* and polar ring*

Figures 3.5-3.8 with indices a, b or c are analogous to Figures 3.1-3.4, but for initial inclinations of 10 = S0° and 90°. As Table 3.1 indicates, these rings do not have the time to settle smoothly within ~ 40 or more rotations. In 4r> particular. the models with ;(J = 90° do not process differentiiilly or settle at all, as can be seen in Figures 3,7 and 3.S. The difference between those evolutions and the results of Habe and Ikeuchi (1985) is attributed to the particle nature of the SPH code used by Habe and Ikeuchi.

Some parts of the above evolutions of highly inclined rings have been pre sented also in Christodoulou {19S9). Our dynamical calculations clearly suggest that polar rings are extremely long-lived, although transient structures, if they are placed at exactly i„ — 90°. In reality, however, the so-called polar rings seem to exist at high inclinations but not at 90° (Whitmore 19S4). In this case, settling of the rings in some form toward a preferred orientation must be expected. We certainly do not expect smoothly warped structures to develop because the differential precession rates between the edges of a highly inclined ring are very large, leading to the conclusion that whenever the inner region attempts to settle, the outer region still remains unchanged. In addition, the discussion of §3.3 indicates that inflow will be more important than settling at high inclinations. Tohline (19S9) speculated that, since the inner region of a highly inclined ring will try to settle first, the ring will be destroyed slowly from the inside out and the central hole will broaden with time. That would be a natural explanation of the sharp inner edges observed in polar rings (Schweizer,

Whitmore, and Rubin 1983) and it can also serve to identify the preferred orientation from the absence of a smooth war]). Our results, shown in Figures

3.5 and 3.G, do not support this speculation. They rather indicate that, as soon as the inner region feels the effect of angular momentum loss, the central hole is flooded with inflowing matter and inflow proceeds as it floes for moderate ■1

There is a possible explanation for smooth, slow inflow instead of the broad ening of the central hole, based on the fact that the settling mechanism is different at high inclinations. If real polar rings were settling in the conven tional sense, then the hole would be broadened because matter of the inner region would be moving to a different plane in a tiiue-scale within which the rest of the ring would not react at all. In reality, however, matter of the inner edge docs not only settle toward the preferred orientation but flows toward the center continuously as well. In this scenario, a sharp disruption of gaseous layers close to the inner edge of the ring is not expected. We have observed disruption of the inner regions, however, in cases where the potential is strongly distorted from spherical symmetry {see (j3.4d below).

The above results clearly indicate that rings at very high inclinations do not feel the influence of a weakly distorted potential for at least 20 rotations. We are led to believe, therefore, that real polar rings should survive comfortably for at least half a Hubble time and further evolution should not present drastic changes for at least another one-half Hubble time (except blurring of the inner edges and some inflowr of matter toward the main galaxy). The main reason for such a slow' evolution is the weak distortion of the potential deduced by observations of polar ring galaxies (Whitmore, Me Elroy, and Schweizer 10S7). Moreover, we conclude that formation of nuclear disks is favored only at moderate inclinations •17 where inflow is important and ran make progress in only a fraction of the Hubble time.

d) SfTangly distorted halo,*

The results of the previous subsections are strictly valid if the external po tential is not very distorted. For comparison, we have calculated the dynam ical evolutions of rings under the influence of a strongly distorted halo with

J — O.OSS. In this cast*, we estimate that the settling time-srales will be shorter than those listed in Table 3.1 by a factor of 2.7 (lower limits) and 4.4

{upper limits). This, in effect, means that we only need to follow these evolu tions for about 10 rotation periods. Finally, we have studied the extreme case

• A i d — 0.102 in which the rings undergo all the important dynamical changes in less than 4 rotations.

During these evolutions, the following differences have been observed. (a)

Rings under the influence of strongly distorted potentials are unable to approach a new steady-state equilibrium without breaking into pieces or flooding the cen tral region of the potential. The total energy of each ring clearly runs away from its equilibrium value. (b) Although rings at i0 = 1 0 °. 40°, and 80° attempt to settle, they overshoot the symmetry plane of the potential but do not find any other preferred plane. As a result, they implode anti flood the central region with matter. The remaining matter forms a thin tlisk that is quickly distorted by differential precession and even breaks into pieces in low inclinations after it expands explosively. (c) Polar rings at exactly t„ — 90° lose angular momentum faster now and also flood the central region, but this inflow is not ■is catastrophic. They arc able to survive for 10 rotations and we suspect that they will survive even longer and keep feeding the central region steadily.

Figure 3.9 shows the evolution of the extended ring under tire influence of a potential with J>Q(r = 0.088 and starting from an inclination i0 = 40°. The outer region slightly “misses" the preferred plane and the entire ring is eventually destroyed by inflow. At the end of the calculation, almost all the mass has flowed into the central region and is not shown in Figure 3.9. The remaining disk assumes a very thin, flat structure at an orientation other than the preferred plane and is fatter where it crosses the equator of the grid.

Figure 3.10 is analogous to Figure 3.9 but ,/_.{lfr — 0.192 and = 10°. This ring quickly shrinks and, then, it throws out some matter in the form of an ex tremely thin, flat disk that is unstable and breaks m*o pieces. At this point, we do not know whether the later stages of this particular evolution are really phys ical or simply an artificial product of the numerical scheme (i.e.. the imposed boundary conditions at the central region of the grid).

Figure 3.11 is analogous to Figure 3.10 but = 80°. The dynamical evo lution of this ring proceeds faster than the rings with J>n

e) D< vf_lojnt it? if of dijinnrnral tun fils

It is hard to identify the presence of twisting (due to differential precession of the gas) by examining Figures 3.1 3 G or 3.9 3.11. The horizontal cross- sections are very helpful in doing so. although they are not drawn through the symmetry plane of the external potential {i.e.. the preferred orientation). As soon as the isodensity contours break on two sides (a sign that the ring settles in those regions, and, therefore matter moves away from the equator of the grid), the remaining two slices would have to be symmetric to each other if the ring were settling at a constant line of nodes (no twisting). Those contours are not symmetric, however, because of retrograde twisting of the “orbits," that leads to the development of a prograde twist as the precession frequencies decrease with radius. The different (steep and shallow) density gradients at the ends of each group of contours on the equator of the grid and the asymmetric orientation of the two “pieces" with respect to the center of the cross-section provide the only indication for twisting without calculating the line-of-sight velocities of a model relative to a distant observer. We are now in the process of projecting rid our dynamical iikk IcIs onto a "plane of the sky" in order to derive isovelocitv and isodensity contour maps.

3.5 Linear Behavior of the PP Instability

In this section, we deviate profoundly from the main stream of the disserta tion in order to report on the development of the PP instability in gaseous, polv- tropic tori. This instability is thought to be important in real stellar accretion disks in which most of the mass is locked in the central star but most of the angular momentum is in the accretion disk. A few detailed 3-D, fully nonlinear computations of PP tori have previously been published ( see Hawley 19S7, 19S9

: Tohline and Hachisu 1989 and references therein ). On the other hand, we cannot avoid this instability in the course of our galaxy simulations.

Fourier analysis of the density at constant radii on the equatorial plane of the torus yields the amplitudes and phases and. in turn, the pattern speed and growth rate of the dominant mode in the linear regime. In addition, we can attempt to measure other harmonics that seem to grow through their interaction with the fastest growing mode but the results are uncertain because the data are noisy. The growth rates must be corrected for numerical viscosity because the scheme is only first-order accurate ( Tohline, Durisen, and McCollough 1985

; Williams and Tohline 1987). We nave attempted to obtain and correct these values for the first 3 Fourier components in our models. The results are listed in Table 3.2. For the extended torus /?_//?„ — 0.707, and for the thin torus

R „ /Iiu = 0.806. For comparison purposes, we have estimated the correspond ing values from the results of a linear stability analysis of tori with polytropic index n = 3 (Kojinia 1DSC) by using the above values for /?__//?„. These esti mates, taken from Kojima's published plots, are enclosed in parentheses in Table

3 2

Both tori are dominated hy the in — 1 inode if they are placed at an inclina tion of ifj ~ 0°. But the evolution of the thin torus is dominated hy the in — 2 mode in the case ia = 90° because the non-axisymmetry of the potential excites this mode to a higher amplitude than the in = 1 mode. In the case in — 0 °, the first 3 Fourier components saturate at about the same time, f ~ 14 for the thin torus and / ~- 12 for the extended torus. At the saturation times, the amplitude of the in — 1 mode is 0,347 for the thm torus and 0.421 for the extended torus

(both amplitudes are normalized to the leading term of the Fourier expansion of the density, as is explained in detail in Williams and Tohline 19S7).

The logarithmic amplitudes of the first 3 Fourier components are plotted versus time in rotation periods in Figure 3.12a (extended torus) and Figure

3.12b (thin torus), for i(> — 0°. — 0.02. and at radii /?//?„ = 1.232 and

RjUf, — 1.075, respectively, on the equatorial plane of each torus. The solid line traces the m — 1 , the dashed line traces the in = 2 , and the dash-dotted line traces the m = 3 component. The linear regime is clearly indicated by the exponential growth of the m — 1 mode. All Fourier components are damped out after 20 and 2G rotation periods, respectively.

We should point out that the results of Table 3.2 indicate that the in — 2 and the m — 3 terms are harmonics of the in — 1 mode because their pattern speeds are approximately integer multiples of the pattern speed of the dominant inode. Finally, our second-order accurate computer code shows that the m — 2 modi' dominates the other components during the evolution of the thin ring- It seems, therefore, that this mode is wiped out hy the numerical diffusion of the less accurate scheme in the above first-order accurate calculation. The results from the second-order accurate hydro code are listed in the last footnote of Table

3.2. Again, the values listed in this footnote for the ni — 3 Fourier component are uncertain because the m = 3 data are dominated by noise. Table 3.1

Estimated Settling Time Scales'1 ■l‘

Extended Ring ( m„ = -1.2) Thin Ring (m,. — 7.2) i0(°) R - R t, /?+ R - Ru /?+

10 1.7 3.0 4.5 S.5 1S.G 42.5 3.0 7.7 0.5-14.7 13.3-33.0 40 2.5- 3.3 0.2- 9.5 25.S- 47.1 4.9- S.5 S.S-1G.3 IS.5-37.2 S0r 7.0 7.4 17.7 20.9 73.3 104.0 14.2 IS.S 25.2 35.9 52.7-S2.2 90 cx; vs x x x oc

NOTES:

b } Lower limits are derived from eq. (15b) [see also Steiman C’ameron and Durisen 19SS], E]>per limits are derived from eq. (15a) [see also Tohline. Simonson, and Caldwell 19S2],

c) The lower limits for i„ — 80° are uncertain since eq. (15b) breaks down at high inclinations. ■VI

Table 3.2

Linear Growth Rates ( ^ 7 ) and Pattern Speeds (lc/G of the PP Instahility" 1 f

Thin Ring frn,, — 7.2)

Mode u.7

1 1.008 <0.943) 0.170 (0.139) 2 2.000 (1 .SOI ) (0.230) 3 2.840 (2.S39) (0.197)

Kxtended Ring (>n„ 4.2 I

Mode j.' a ^ 7

1 0.83S (O.S37) 0.227 (0.193) 2 1.937 (1.S73) (O.loG) 3 2,914 (3,200) (0.091)

NOTES:

ft) Botli u.7 { and 1^7 are normalized to the rotational frequency at the pressure maximum u.‘„.

b) The numbers in ]»arentheses are our estimates from Kojima's (1980) pub- lislied plots for n = 3 PP tori.

e) The corresponding second-order accurate values of Ac ^.7 are : For the thin ring, 0.SS9 Ac 0.101 (m = 1), 1.84G Ac 0.234 (tu — 2). and 2.00/ Ac 0.097 (m =3). For the extended ring, 0,780 Ac 0.208 (m — 1 ). 1.079 Ac

0.114 (7/1 - 2), and 2.212 Ac 0.070 (m = 3). Figure 3.1a

f «16.50 .rj < ;

Figure 3.1b

t 16.50

t -- 4 9 8

I t --11 96 ( ■- ZG68 5 7

Figure 3.1c

\\\W^TV" '

t * 16.5O -X

t * 4 98 ( .2) 44

( - I! 96 t * 28 68 . 58

Figure 3.2a Figure 3.2b f.n

Figure 3.2c

t* 6 36 t--3 0 . 6 3 Cl

Figure 3.3a

t = 4 9 0 CSJ

Figure 3.3b

f •12.86 Figure 3.3c

( * 2.56 t * 9 .70 \

t ’ + 9 0

t * 8 ro fi2.ee!

6 T>

Figure 3.4b

t (-10.2 r! 0 0

Figure 3.4c Figure 3. Figure 3.5b

f* 16 6 0

t' 10.50 Figure 3.5c

?>j\\

l* 16 60

|^V'yV\\

JO. 50 ure 3.6a 71

Figure 3.6b

U 17 90 72

Figure 3.6c

t'2.62 t* 13.35

t* 17 90

Figure 3.8a Figure 3.9a 7G

Figure 3-9b Figure 3.9c

(■- > 89 t -3 €C

7 9

Figure 3.10b

1 i I : /

( = 0 5 7 1 . 5 7

r-1 os f * 3 02 80

Figure 3.10c

f ‘ 0 5 7

itii...

/* t 06 t ‘3 .0 ! 81

Figure 3.11a $2

Figure 3.11b

i V: L t-- t 35 S3

Figure 3.11c

f --1 92 t = 3.57 Ln 0 -IS -w o *1S O iue 3.12 Figure ft

ft ft o C H A PT E R 4 SUMMARY OF CONCLUSIONS

Because great law.* are not.

o } T1 ic direction of precession of gas- particle orbits in the disk of a galaxy that

is under the influence of the spheroidal or ellipsoidal potential well of a su

perimposed halo can be used eiiectively to predict the direction of twisting

of the isovelocity contours in the observed HI map of the galaxy. Specif

ically. an oblate-spheroidal or ellipsoidal halo forces the development of

a twist that is prograde to the disk rotation, while a prolate-spheroidal

or ellipsoidal halo forces the development of a retrograde twist. There

fore, if the spiral arms of a galaxy are assumed to be trailing features, a

prediction can be made about the gross geometric shape of the superim

posed dark halo. That is, if the observed isovelocity contours and the

optical spiral arms twist in the same general direction on the plane of the

skv. then the halo is prolate-spheroidal or prolate-ellipsoidal in shape and

vice versa. b) A kiuematical, tilted-ring model was used to analyze the kinematics of 21

galaxies from which 15 are normal spirals. From the latter subgroup, we

have confirmed unambiguously the above prediction for only 7 spirals, the ones whose velocity fields possess a strongly twisted pattern, Specifi c;dly, M 33. M S3, NGC 2505. NGC 2511. and NGC 371S show the signature of an ohlate halo in their kinematics, hut the disks of NGC 5033 and NGC’ 5(J55 seem to he embedded in a prolate halo. In addition, the dust lane in the elliptical galaxy NGC 512S ( Cen A) seems to he com posed of two orthogonal disks from which the outer disk is strongly warped hut not twisted.

Our kmematical modeling confirms the e x iste n c e of disk halos in spiral galaxies that are generally prolate in shape. This result prohahlv creates an lmsolvahle problem for dynamical theories that modify the Newtonian gravitational law in order to naturally explain the observed Hat rotation curves of spiral galaxies. These theories are no longer expected to sur vive because their basis is not wide enough to allow for the presence of substantial amounts of daik matter that can be distributed in an over all prolate shape around galaxies, and the luminous matter in galaxies is always distributed in a disk-like, generally oblate shape.

In drawing the connection between kinematics and dynamics m the gaseous disks of galaxies, we have made extensive use of t he most popular theory to- date, the "preferred orientation theory." We have performed additional dynamical modeling of gaseous disks dominated by an external gravita tional field in order to demonstrate1 how complicated structures ( warps and twists } develop during the evolution of such disks and to decipher the influence of purely dynamical phenomena ( diflerential precession and dissipation ) on the observed kineinatical behavior of real galaxy disks.

Conventional ideas about settling disks in a static, oblate-spheroidal, ex ternal potential toward a preferred orientation and formation of transient 87

warped and twisted striictuns arc c

at low inclinations in weakly oblate-spheroidal potentials. In this case,

the settling is driven by differentia] precession of gas-particle orbits and, it

seems, any mechanism that will dissipate angular momentum. Numeri

cal calculations give consistent insults and they also agree with the results

predicted by the analytical theory.

/} In the presence of weakly distorted potentials, inflow of matter toward the

center of the potential (and formation of unclear disks) becomes important

in models that start at moderate and high inclinations. The time-scales

for significant dynamical change of the models become increasingly longer

with increasing inclination. For inchnat ions typical of real polar rings, the

dynamical time-scales are well over the Hubble time and nuclear disk for

mation proceeds slower than for moderately inclined disks. That would

not be the case if the halos of i eal polar ring galaxies were strongly distorted

from spherical symmetry.

e) Rings placed at exactly 90° from the symmetry plain1 of an external po

tential are extremely long lived structures. Although they slowly lose

angular momentum, they do not settle at all because of the absence of

differential precession. Rings starting from any inclination other than 0°

or 90° are transient anti they attempt to destroy excess angular momentum

and achieve a new, energetically preferred, dynamical state.

h) In the presence of strongly distorted potentials, rings evolve away from a

steady-state equilibrium and, therefore, do not realize the existence of a

preferred plane. Some transient warps are seen but inflow eventually

dominates at all inclinations. In general, it should be hard to observe

such intense activities in real galaxies because they develop in short times (typically 1/10 of a Hubble time).

The warps oltserved 111 normal spiral galaxies do not seem to exceed 40° from the plane of the umvarped disk. This result is derived from kine- niatiral. tilted-ring modeling of normal spirals (Christodoulnu, Tohline, and Steiman-Cameron 19SS, 19S9). An explanation for this result lies in the dynamics of settling disks as smooth warps are only generated in disks of low and moderate inclinations where inflow is either negligible or, iit most, comparable to settling. Furthermore, it seems that the dark halos of these galaxies do not d e v i a t e much from spherical symmetry, com promising the abundance of warps and the frequent observations of nuclear activity that is often caused by infalling matter. EPILOGUE

It is clear now that a good qua lit at ive agi -cement has been established between analytical results and numerical calculations on the genera] topic of settling disks in galaxies. It is not at all clear, however, what kind of future studies will he most profitable and useful. Fuither research can focus on the following topics:

(o) Massive disks and the influence of self-gravit y < m theii evolution.

{/>) Inflow of matter in the central regions of a galactic potential followed by

triggering of nuclear activity 01 formation of nuclear disks.

( e ) Dynamical evolnt 1011 of gas accreted in a galact ic potential during tlie initial

stanes and before it forms a disk at some inclination.

\il) Dynamical evolution of nitet act mg, mei gnig. and colliding galaxies.

it i The impoitauce of visco-nty m the angular inoiuentum transport and its

influence on the angular momentum distribution of the final equilibrium

ol iject s.

( / ) Slow ly growing, asynmiet i ic. dynamical inst abilu ies of cohl galaxy disks.

Despite extensive (and expensive) dynamical modeling of settling disks, we have not broadened substantially our understanding of the structure and dynam ics of the dark halos of galaxies. On the contrary, kinematical modeling of

HI disks has proved to he more useful as it provides hints on how to decipher the gross geometric shapes of the dark halos. In conclusion, 15 years after the introduction of dark halos in galactic dynamics all the important pieces of this enigma are still missing. The worst part of the "dark halo mystery" may be that we do not even know what to look for in galaxies. But, then again, we should not quit trying!

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Bland. J . , Taylor, K., and Atherton, P. D. 19S7. M .N.R . A.S.. 228. 595.

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246. 70S.

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This Appendix presents n paper published in The Astmphysical .Tonrnal in 1DSG under the title: "IV/tfjt Warped Dink* Can Tell lit about Galaxy Halot mid Grartit/. The complete reference is :

Christodoulou, D. M.. ;md Tohline. .1. E, 198G. A p. T. 307, 4-19. Letter

THE ASTROPHYS1CAL JOURNAL

H t U l T A. AlT, Managing Editor A DaLG4*NO, L e f t r r t E d ito r Kxi h t k National Obaervatory O n t r r far AjgrophyiKa B o b V 7 K 60 Garden Street Turacn. Artaona KW ATS Cambndce. y • 02136 tm m : m m m is Telephone «I7.|Sm7B BITNCT eriHia.edu B I T N E T apjleffn r ft!

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July 12, 1989

Dr. D im itris Christodoulou Department of physics snd Astronomy Louisians Stats U nivarsity Baton Rouge, LA 70803-4001

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7 m al Jou»nal. J*7 449-432. I9t6 A u g u s t J 3 * l**r iM A v rn ji «|irM «*cilbtK i) 4Ji rt^his W t

WHAT WARPED DISKS CAN TELL US ABOUT GALAXY HALOS AND GRAVITY'

D i m i t r i s M C kristoooulou a n d J o e l E T o h l i n i Depannwni of P h y u a ind Astronomy. U u i i u m Stair University A t c f j u t 4 39B3 Jum tJO mcctp%*4 tW to Jarm try 31

ABSTRACT A warp on the outer fringe of a galaxy is often thought to signal time-dependent settling of gat into a preferred plane of that galaxy At settling occurs, a twist should usually be apparent in the region of the warped gas layer The direction of the I was I—prograde or retrograde with respect to the direction of the disks rotation— should identify whether the gas it settling into an oblate-like potential well or into a prolate-like potential well A study of twisted warps in a large number of spiral galaxies should show whether dark halos are as diverse in geometric shape as are elliptical galaxies. Furthermore, evidence for even a few disks residing in prolate-like potential wells would probably rule out theoretical models that have attempted to explain flat rotation curves by modifying gravity or the dynamics of gravitating systems, because these theoretical models demand that all disk galaxies exhibit oblate-like potential wells Subject headings galaxies structure — stars stellar dynamics

i introduction observed kinematical features of galaxies can be explained. It has been known for some time now that the outer fringes without invoking dark halos around galaxies, by allowing a of tome spiral galaxies exhibit warps Warps are most apparent modification of Newtonian dynamics in the presence of small in H i contour maps of nearly edge-on spirals (Sancisi 1976, gravitational accelerations In a similar vein. Tohline (I983j van W ocrden 1979, 5ancisi 1981), but their presence is also has proposed that a specific modification of Newton's law of revealed in H l velocity-contour maps of some face-on galaxies gravity will explain the observed kinematical structure of gal (Rogttad. Lockhart, and Wnghl 1974; Rogstad. Wnght. and axies without invoking the presence of dark matter Both these Lockhart 1976, Rogstad. Crutcher, and Chu 1979, Bosma proposals, which differ rather drastically from conventional 1981spiral galaxy is distributed largely in a disk spiral disks are embedded in nonsphencal dark halos, mecha these theories carry with them the implicit prediction that the nisms exist to sustain warps as permanent features of the disks kinematics of warped gas layers in isolated spiral galaxies must A steady state warp can be maintained tf a halo is sufficiently always reveal settling into a disklikc potential well What is not flattened (Petrou 19801. if a halo it tumbling (Btnney 1978. immediately obvious, however, is whether the kinematical sig 1981, van Albada, Kotanyi. and Schwarzschild 1982. Tohline nature of gas settling into an oblate (disklikc) potential accord and Durisen 1982. Lake and Norman 1983. Simonson and ing to the rules of these non-Newtonian theories should be the Tohline 1983; de Zeeuw and Mernt 1983. Steiman-Cameron same as the familiar Newtonian signature In § 111. we detail and Durisen 1984. Sparke 19846). or if the symmetry axis of the prediction of these two proposed modifications of Newto the halo is permanently tilted at a finite angle to the rotation nian dynamics as they pertain to gas settling into disk galaxies axis of the disk (Dekel and Shlosman 1983. Toomre 1983. An outline of an observational means by which the theories Sparke I984a| However, not all observed disk warps demand can be tested is included in $ IV a steady state interpretation As most of the cited authors have II INTERRUPT A1TON Of KINEMATICAL TWISTS noted, it it reasonable to expect that many disk warps are transient features They result from the simple time-dependent Rogstad. Lockhart, and Wnght (1974. hereafter RLW| were process of gas settling into a preferred plane of the galaxy As the first to convincingly demonstrate in print that strictly cir we discuss in ft II and IV of this paper, transient warps are of cular motion of gas in a warped gas layer can provide a reason particular interest because a careful observational analysis of able kinematic model of a galaxy disk. They initially used their their kinematic structure can reveal important information warped model to explain the observed motions of H i gas in about the geometric structure of the dark halos in which they M83, then later applied it to M33 (Rogstad, Wright, and Lock reside A point similar to this has previously been briefly men hart 1976). In their model, a gas disk is divided up into a series tioned by Sparke (1984o| of concentric, circular rings which rotate ngidly in their own At the same time, an observational analysis of the kinematic planes The innermost rings lie in a single plane, defining what structure of warps can also provide a crucial test of non- we refer to here as the "central disk," but the outer rings are Newtonian dynamical models of galaxies Milgrom (1983a, 6) lilted at various angles to this plane in such a way as to and Bekenstein and Milgrom (1984) have proposed that describe a smoothly warped structure Similar models have been fitted to observations of other galaxies (e g . Newton and Emerson 1977; Rogstad, Crutcher, and Chu 1979. Bosma ' Sun# t'nirrriify Obserteiori CnAfnSufionj. No 193 19816) The shape of the warp tn an RLW model is uniquely 449 9 8

450 CHRISTODOULOU AND TOHLINE voi K r specified once the two angles Kr) and fhr) are given as a func The rale of precession will be tion of nng radius r, where i is the inclination of the ring with lu, respect to the central disk and (1 is the position angle of the IUJ, I » | ix i, — to* J * — I f J - t „ I 4 3 ) ascending node of the nng as measured m the plane of the central disk In order to compare a given model's ifr) and Or) The precession frequency uip must generally be treated as a with observations, it is of course necessary to specify the orien function of the orbital radius r If u>, happens to have the same tation of the central disk relative to our line of sight, transform value over all radii in a warped gas layer, then a steady state each nng's t and O to that reference system, and then project warp can be achieved All the models of steady state warps each nng and its circular motion onto the plane of the sky mentioned in g I present specific combinations of halo and disk fSome published results from tilted nng models show only structures that force to, to be independent of r Usually these fmal angles that arc referenced to the sky } This pro though. u>, and all other angular frequencies in a galaxy cedure is implicit in the work of RLW and is discusied explic approach zero at large radii Hence, radial differential preces itly by Tohline and Steiman-Cameron 11986) and, in part, in sion will usually lead to the natural development of a smooth g 84 of Mihalas and Bmnev f 19811 twist in a warped layer of gas and the twist will develop in a Although the RLW nng model is, strictly speaking, only sense (looking from the inside of the disk, outward) that is kinematic in nature, it has a direcl analog in a realistic dynami opposite to the direction of precession of individual rings cal model As gas settles into the equatorial plane of a spher Therefore, as the motion of gas in different galaxies is examined oidal gravitational potential (Kahn and Wolljer 1959. Tohline. observational I y, discovery of a prograde twist should imply Simonson, and Caldwell I9B2| or. more generally, into any that the Z-oscillation frequency is larger in magnitude than is preferred plane of a gravitational potential well (Dunsen ri al the ft-oscillatton frequency (ujz uj* > II. w-hile discovery of a 198.1. Steim an-Cam eron and Durisen 1984|. it should assume a retrograde twist should imply that the Z-oscillation frequency smoothly warped structure Because it simplifies our discussion is the smaller of the two ( I On the other hand, a warped gas layer that the mass distribution, and the angular momentum vector of twists in the same direction as the central disk s spiral structure the particle s orbit tilted at a finite angle r to the Z-axis. we can identifies a system in which taz iua < I generally describe the lest particle's orbu as execuimg oscil lators motion both in the cylindrical radial coordinate ft and 111 EXPECTATIONS FROM DYNAMICS Z When building a self-consistent d y n a m i c a l model of a spiral ft ■= - m ,1 R . (1 at galaxy, what do we expect the ratio ruz uia to be in its outer t - . (Ibl regions'* In Newtonian dynamics, equations (la) and (1b) arc derived Trom the fundamental dynamical equation where in general cu* * uj, When considering nearly circular - , (41 m otion |r m [R: ■+ Z1] 1 1 constant) about a mass distribu m - tion that departs only slightly from spherical symmetry, we will where a is the acceleration and the Newtonian gravitational also find it useful to express and in terms of the circular potential K at any point a due to a nass distribution p it) is orbital frequency uiD that would be obtained in the case of given by the well-known expression spherical symmetry, and two functions. *„ and that only describe departures from spherical symmetry (t) m -G f * * dix (5) J lx - x | ru,,1 » cup1! 1 <»l . (2aI For a massless particle orbiting well outside a spheroidal mass ru*1 - tuu:U + 1*1 , (2b) distribution (or for one orbiting near a spheroidal mass that where | ef I 4 I and | e^ | 4 1 The exact values and functional deviates only slightly from spherical symmetry). MacCullagh s behavior of the two frequencies cu* and tuz depend on the formulation (Ramsey 1981, p 87) allows us to write the details of the mass system and of the dynamics being con Newtonian potential tb^tft. Z) as sidered Independent of the specific system, however, if cuj/tu„ > I (i.e., - t* > 0), the particle will complete each «ur. zi * ^ j -1 + [(4 - orj - ha- o z 1] ! . Z-osctllation before it finishes one full oscillation in ft. so the orbit will precets in a direction that is to the direc retrograde ( 6 ) tion of orbital motion lie.. the position angle Cl of the line of nodes of the orbit will regrets) Similarly, if att < I where Af is the total mass of the spheroid. C is the gravitational («j — «a < 0), prograde precession of the orbit land of Cl) will constant, and. as in § II. the Z-axis of the cylindrical coordi occur For a given galaxy model, when the size of the ratio nate system is aligned with the symmetry axis of the mass tuj/ru, is determined at a particular distance r from the center distribution In this expression. A is the moment of inertia of of the mass distribution, the direction of precession for an indi the mass distribution about an axis lying in its equatorial vidual particle orbit or for a nng of particles executing circular plane, while C is its moment of inertia about the symmetry motion in an orbit of radius r is automatically defined axis Both A and C are intrinsically positive quantities From 99

N o 2, 1986 WARPED DISKS 4*1

this expression. one can readily derive (he function* and (ion Td>t needed in equation Ml to describe dynamic* using and. by combining equation* 11), (21, and Ml. *how (hat Tohline* modified gravity From the derivation we obtain tnj - GMd" ’r" 1 and IC - A) (7( 112) For a Katie oblate mass dm n but ion, in which C > A, we aee tha< - c„ > 0 In this case, therefore, the Z-oscillation fre Hence, although the ratio | uif/tug I it smaller than in the New quency cjz must be greater in magnitude than u ( for panicle tonian case by the factor j. ihc direction of precession i* again orbit* (i.e.. tuz /tuK > If Foi a prolate mats distribution, on the identical to that found in Newtonian dynamic* for a given type oiher hand. C < 4 . so — «, < 0 and, dynamically, panicle of mass distribution orbit* mutt have < 1. Thi* result for Newtonian This qualitative agreement extends to ellipsoidal mas* dynamic* i* well known models at well It can be shown that in the dynamical models Milgrom ()983o. bf ha* proposed that, in many cases, a proposed by both Milgrom and Tohline. an ellipsoidal mas* massless test panicle orbiting in the gravitational field of a distribution having moments of inertia 4, < At < 4 3 m ea galaxy will feel an acceleration a given, not by equation Mj, but sured about the three principal axes will produce orbital oscil by the following dynamical equation lation frequencies foi, I < < ( iuj | This agrees with the dynamical behavior derived from Newtonian dynamic* («1 (£>■ IV DISCUSSION! In Milgrom'* formalism, Ogi * 2 * 10 ' cm s 1 if the Hubble In Newtonian gravitational systems, we have emphasized constant H0 - SO km s " 1 Mpc ' * I is the value of the acceler the fact that massless particles orbiting about a static oblate ation below which equation (8) dominates ordinary Newtonian spheroidal mass distribution will experience a frequency ratio dynamics Inserting from equation (6) into (8|. one can tuj/iu, > t Therefore, gas which settle* into a galaxy whose construct two equations like I la) and 11 b| governing the structure is dominated by the presence of a massive disk musi motion of massless particles orbiting outside a (weakly) spher develop a prograde twist Since much of the light from MB' i* oidal mass distribution and show, for example, that in obviously emitted from a flattened disk, RLW were probably M ilgrom s dynamic*. oi<,* « (GMa„l' 1r J, which is different not surprised when their kinematical model of H 1 motion* in from the Newtonian value uj03 • GMr J For our purpose*, M83 demanded a tignificam prograde twist in its warped gas however, it it sufficient to point out that since, in Milgrom'* layer* In fact, they performed a simple numerical simulation to dynamics, a always point* in the same direction a* Vv we are illustrate that a twined warp, whose properties were in agree guaranteed'' that for an arbitrarily specified mas* distribution. ment with their kinematical model, naturally develop* from (he dynamical motions of massless rings orbiting “an oblate non- (9) homogeneous sphenod " In light of our current thinking on spiral galaxies, however, Therefore, with regard to the direction of precession for particle it is reasonable to assume that the dynamics of the gas in the orbits, the expectation* from Milgrom'* dynamic* are identical outer region* of M83 is being controlled by a dark halo and to the familiar behavior of Newtonian dynamic* not by the luminous material (see Tubbs and Sanders 1979. a* Tohline (1983) ha* hypothesized (hat at distances r a- d well as the numerous citations mentioned in § 1) Therefore, the id * l-S kpc) from any mass distribution, gravity deviates results of RLW should be looked at with renewed interest from a Newtonian behavior and adopts, instead, a force pro They reveal that M83's dark halo is not prolate in structure portional to 1 r Such a force can be derived from a logarithmic We encourage detailed studies of twists in the warped layer* of scalar potential other galaxy disks as a tool for probing the geometric structure of other dark halos One such study involving published data on H t warps is already underway (Tohline and Steiman- a ) In | jr - x | dix ( 10) ♦if* I Cameron 1986). A few examples of twists that illustrate settling into a prolate-like potential well will help substantiate the con that then replaces in the dynamical equation (4) Adapting jecture that dark halos, like elliptical galaxies, are not smelly MacCullagh's formulation, as was used for the Newtonian dis oblate spheroidal structures but should be treated, generally, cussion above, one can show that, at a point well outside a as being tnaxial ellipsoids Obscrvationally. the signature of a spheroidal mass distribution lor, again, neat a mass whose prolate-like structure should be a warped H i ga* layer that structure deviates only slightly from spherical symmetry). twists in the tome direction as the optical spiral pattern of the 1 central disk «M R. Z\ * r -+ * [(24 - OR1 In order to avoid misinterpretation of these last statements, 4 Mr we emphasize that if a galaxy is found whose warp carries (he signature of a "prolate-fike" structure, one cannot unam - (24 - 3C\Z (HI biguously conclude that its halo is truly prolate As was men tioned in i HI, an ellipsoidal halo can also produce precession where G. M. 4, and C have the same definition* as in expres patterns thai are “ prolate-like " Instead, the important point sion (6). From this expression, one can readily derive (he func- that will be uncovered by this finding is (hat the potential well in which the galaxy sit* is definitely not oblate Likewise, we

* W r are indcfcied io L inda Sparke for brinfinj titu raiher o b w o u i p o i m in emphasize that our analysis has ignored the self-gravity of the our jncntron W c h ad over looted n in the original diafi of U rn paper warped gas layers, since we have treated them as massless rings 100

452 CHRISTODOULOU AND TOHLINE

of test panicle* The signature of a nonoblate halo may be tion that all warped gas layers in the outer regions of spiral somewhat camouflaged by a warped laser s self-gravity if the gala sies miisi exhibit pr&gradt twins The kinematical structure halo is nearly spherical or not sufficiently massive of the disk in M83 is consistent with the predictions of these The modified dynamical models or Milgrom (1983a. f>) and models, but if well-defined retrograde iwtxis are ever found in Tohline (!983| were both proposed as an alternative to the other warped tal**y disks, these models would fail to explain dark halo hypothesis. Implicit in both ihcse models is the state the warped structures as being due to the natural settling of gas ment that the gravitating material in spiral galaxies is distrib into a self-gravitating disk uted largely in a disk, that is, in an oblate geometric structure As we have shown in § 111, both of these non-Newtonian We ha vc benefited from k v c t i I enlightening diicuuions of models demand that oblate mass distributions produce orbital ihii tubjeci with Thomas Y Sieiman-Cameron This work has frequency ratios u>z'ujK > 1 Hence, the models proposed by been supported in pari by the Council on Research at Loui Milgrom and Tohline both carry with them the implied predic siana State University and in part by NSF grant AST-8217744

REFERENCES

Arp H 1466 A p J S v p p / 1 4 I R o p led D H W n| h l , M C H , end lock hen I A 1976. A p J . » 4 7 0 3 Retention J , end Milgrom M 19*4 Ap J , 2 M . " Seer E 1979 in M L' Sympoifum &4 T h e L a r # e S c o tf CAaracieriJiii: t o f the R i n n t > . J ) 9 7 £ \ ft A 5 1*3 * * 7 9 G ain* > ed W g Burton (Dordrschi Reidel I. p 513 1911, M A R 4 5 . 1 4 * 4 * 5 Sannik, R 197*. 4lfr Ap , S 3 . 1 5 9 lotma A 1981a 4 J U P 9 1 1 9 1 1 , m The Structure L w / u i u x i of Soemai Galane* ed S M Fell I 9 t ( h 4 J U I 8 2 5 and D L ynden-ReU iCsmbodge Cun b nd g e Uniterm t Fret*i p |e9 -— ---- 1 9 8 3 . m 141 Siffipfriiyfft !(H> internal AmrmaHrj and D ynam m of Sendift A 1961, The Hubbit Altai of ind*nr \ o f (sateor.i ed E Aihineuouia (Dordrecht Reidel > p I®' Sparke L S l9S4a. Ap J . 2B0 I P de Z f t u « . T . end M r r m . D 1911 A t J 3 * 7 . 3 7 1 I9M6. M A ft A S, 2 1 1 9 1 1 Dun ten R H Tohline J E , Burnt J A , ind E>obtovoltki*. A R 198), ep Sinmen-Ceiifeeron T V . and D u n ten. R H 19*4 Ap J . 2 7 * 1 0 1 J U 4 1 9 : Tohline J E 1983, in M l Sy^iponwnr tOU, inirrna! kmemoHfi a n d P m a m ti > K ihn.F D . and W olijrr. L 1939 4? J 1 ) 0 7 0 3 o f G a ta x m . *d E A than Aioouta (Dordrecht Reidel I. p 305 Lake. G R , and N or m an C 191) 4 r J 1 7 0 . 4 1 1 T o m i n e . J E . end Duneen. R H 1913. Ap J , 2 * 7 , 9 4 Mihelif D and Binrifk I 198J ina/j.rn A sim n o m i (Sen F rencitco TohUne. J E Simoown, G F , and C » U *^npisium J00. Internal Ktnematk i and D ynam a i o( ( 9 $ ' n At J 3 7 0 m Cd/jxiei.ed E Atheneuoul*(Dordrechi Retdclkp 177 Nekton. K . and Emerurn D T 197- % A A S , I I I Tubhv A D end Seodert R H I9?9 Ap J , 2 J 0 7 J 6 Pelrou M 19*0 M S 8 1 S . Itt ^ ven AI bode T $. Koien>i t G , end Schwirachild. M 19B2. M A ft .4 5 Riinu) A S I 91 1. A grw ir-nifln 4 1 " ^ rum K a m b n d f c C e m b n d f e I niverm f 1 f t . 3 0 3 Ptmi ven WpeTdrn H 1 979. in M l 5impoj;vm 44, The t^rge-icaie CHarat trm m i R Of Mid D H Crutcher R M tndfhu k I97v A t J 2 3 9 S * N of the Galaw ed W B Burion iDordrechi RfxJeD p 501 Ropied D H Lockhen I A . end M C H 19*4 Ap J 1 9 ) > 1 * < R L 9 k i

Diutrais M Chuistodolttu and Jon t Tohlim Departmrni of Physics and Astronomy. Louisiana State University, Baton Rouge. LA 70805-400! APPENDIX II

This Appendix presents a paper published in The Astronomical Journal in 19SS under the title: " Tilted ■ Iim

Christul<>u. D. M.. and Tohline. J. E.. and Stennan-Canieron, T. ^ . 19i>S. Ast i on. .T. 96, 1307.

1 a 1 Letter

THE ASTRONOMICAL JOURNAL Depart men! of Astronomy, FM-20 University ofVMihta|tDii Saattle, Vfeshlngton 96198 (206) 545-2150 ASTKQK&uwaphast.bitnct ASTRQJ@ pheat.phya.washlngton.edu

July 25, 1989

Nr* Dimitris Christodoulou Department of Physics and Astronomy Louisiana State University Baton Rouge, Louisiana 70603 Daar Mr. Christodoulou; This letter confirms our telephone conversations of this morning and last week regarding the reproduction of your Astronomical Journal paper, titled "Ring Models of the Prolate Spiral Galaxies Nfic 503^ and $0$5W. You need the permission of the author (yourself) to reproduce this paper as part of your thesis. 2 am happy also to give my permission as Editor. Yours sincerely,

Paul Hodge Editor in-'.

THE ASTRONOMICAL JOL'RN a L VOLIME « . KLMIf » ) OCTOiES 111*

T1LTED-RING MODELS OF THE PROLATE SPIRAL GALAXIES NGC 5033 AND 5055"

D i m i t r i s M. Christodoulou and Joel E Tohline D rptnm m l of Ptiyuc* and Astronomy, Loiuuans Stslt University. Baton Rouge, Louisians 7OT0 J-400I

Thomas Y SteimanCameron Theoretical Siuttm Branch, NASA/A n a R n w r t t Cater. Mail Stop 245 3. Moffett Field. California MOD R m iia f 2 ftbnary /MR miaaff 12 M*jr 2 MR

ABSTRACT

Ob*erva(iom of the kinematics o f H I in tin disks of spire! galaxies have shown that iso velocity contours often exhibit a twined pattern The shape of a galaxy's gravitational potential well (whether due to luminous matter or dark nutter) can be determined from the direction of the twist If this twist is a manifestation of the precession of a nonsteady-state disk, we show that the twists of NGC 5033 and 5035 imply an overall prolate shape, with the mayor axis of the potential well aligned along the rotation axis of the disk. Therefore, the luminous disks of these galaxies mutt be embedded in dark halos that are prolate spheroids or prolatelike triaxial figures

I INTRODUCTION fined potential well. We will make use of this dynamical- model interpretation here, so it is worthwhile to outline Observations of spiral galaxies in the 2 1 cm line of neutral briefly the key points of the model It is well known that hydrogen have not been consistent with the simple picture of orbits of test particles in a nonspherical potential well will a Rat, self-gravitating, constant-mass-1o-light-ratio gaseous process and that the direction of their precession depends on disk in circular, differentia! rotation. A typical derived rota the shape of the potential well (cf Gaiifinkel 1939; Kahn and tion curve does not show a Keplerian behavior at large dis Woltjer 1939; Tohline, Simonson, and Caldwell 1982) CT tances from the galaxy nucleus; instead, it stays almost con pointed out that in a spiral galaxy's differentially rotating stant or increases slowly with radius even out to two or three gaseous disk, precession and dissipation of the particle orbits Holmberg radii (Rogstad and Shottak 1972; Roberts and will cause the disk to form a smoothly warped and twisted Rots 1973; Knimm and Salpcter 1977). Moreover, isovelo- structure, and that, reflecting this structure, the pattern of cuy and surface-in tensity diagrams often indicate (hat a the isovelocity diagram of a disk galaxy will appear to turn large-scale disturbance dominates the kinematics of the gas, (twist) in a direction that depends on the precession direc especially in the outer regions of a galaxy (Rogstad, Lock tion Hence, kinematical twists (whether prograde or retro hart, and Wright 1974; Roberts and Whitehurst 1975; Rog grade to the direction of rotation of the spiral disk) can be stad, Wnght, and Lockhart 1976, Newton and Emerson used to decipher the shape of a disk galaxy's underlying po 1977, Reakes and Newton 1978, Rogstad, Crutcher, and tential well (whether oblatelike or prolatelike, respectively) Chu 1979; Newton 1980b,b; Boama 1981, Schware 1985) A similar point has also been made by Sparke (1984). A In s number of spiral galaxies, an obvious signature of a model that explains kinematical twists as long-lived struc "disturbance" in the H I disk is the twist that appears in the tures whose signatures infer an underlying potential-well contour lines of an isovelocity diagram (tome of the moat shape that is entirely different from the one being used here pronounced twists observed to date are thote seen in M83 has been presented by Petrou (1980). As we discuss in Sec. and NGC 300). These kinematical twists strongly suggest IV, howeWr, Petrou's model is probably not applicable to that the outer part of each galaxy disk is warped Rogstad, moat warped galaxies, Lockhart, and W right (1974, hereafter referred to at RLW ) It is important to emphasize that when a prolate shape is showed, in a pioneering work, that the kinematics of the disk indicated by the CT analysis, the reference is not to a barlike of M83 can be easily explained using a tilled-nng model to distortion whose major axis lies in the plane of the spiral represent the disk's warped and twisted structure. In such a disk, but rather to a global distortion whose major axis is model, each of a number of concentric, circular rings is al o riented normal to the spiral disk, i.e., along the disk’s rota lowed to change its orientation in space by performing two tion axis. Furthermore, we should point out that the CT rotations one around its protected mayor axis onto the plane analysis cannot distinguish between a perfect spheroidal po of the sky (warp), and another around its symmetry axis tential well and a triaxial potential well with roughly the (twist). Rogstad. Wright, and Lockhart {1976) and Rog same symmetry (for example, an oblate spheroid with axes stad, Crutcher, and Chu (1979) successfully used the same a — 6>c from an oblatelike triaxial figure with axes type of model to explain the kinematics of M33 and NGC a 2 8> c). so in CT the terms oblatelike and prolatelike were 300, respectively. Boama (1981) used a similar mode) to used to describe objects with a , canda.fi, d r, respectively. match his higb-resolutkxi obeervatioos of the gala ties NGC Hereafter, for grammatical convenience, we will adopt the 5033, 3035, 2841. 3198, and 7331 terms oblate and prolate instead of oblatelike and prolate Christodoulou and Tohline (1986, hereafter referred to as like, but the render should keep in mind that in the context of CT) showed that an observed kinematical twist in a warped this paper the simpler terms are meant to describe a wide disk can be related to a realistic dynamical model in which class of objects that includes triaxial figures as well as spher rings of massless, dissipative teat particles settle into a prede- oids We should also emphasize that a kinematical twist in a ■' LSU O torvaiory Pub) No 212 warped galaxy disk will provide a signature of the shape of

130T Astron J M 141, OcloWr 1911 0004-02 54/11 AM I JOT-01 *00 90 ® IMS Aw Aalroo Sot 1J0T 104

1308 CHRISTODOULOU ET AL T1LTED-RING MODELS 1308

that galaxy's potential well as described by CT only if the well, a "best-fit" tilted-nng model of the galaxy's disk must warp in the galaxy’s disk is transient While we cannot prove be found Then, by simply ascertaining whether the best-fit in any individual case that the identified warp indeed pic model includes a warp that twists prograde or retrograde tures a disk at an instant during a phase of lime-dependent with respect to the direction of rotation of the galaxy, one settling toward a preferred plane, it is not unreasonable to can identify whether the underlying potential well is oblate suspect that many warps are transient. This idea is reasona or prolate, respectively. ble because galaxies are relatively young systems, dynami The basic tilted-ring model used by CTS has the following cally; interactions between galaxies are not infrequent, caus properties: ing many disk systems to undergo a phase of time-dependent (a) Before introducing a warp, a "reference" disk of radi settling more than once since the era of galaxy formation, us R is divided into (V concent nc, circular rings of equal and, as has been pointed out by Tubbs and Sanders (1979) radial width Ar - R /N, each representing gas of surface and Steiman-Cameron and Dunsen (1988), in the presence density S(r) in uniform circular motion with velocity v[r) of dark halos, settling times can be quite long. As Toomre Each ring's physical properties are assigned such that, in the (1983) has reviewed, it may be the case that the warps in reference disk, the surface-density profile is some galaxy disks arc not transient features but arc. instead, 5(r)=5„e (1) steady-state structures identifying either the tilted orienta tion or the time-varying nature of underlying dark halos. and the rotation curve is Indeed, we have been party to some of this speculation (Toh- hne and Durisen 1982; Durisen et al 1983; Simonson and i\ Ir/r ) for r < r , Tohline 1983, Steiman-Cameron and Durisen 1984, 1988). i', forr r > r, , (2) For the reasons just enumerated, though, it is doubtful that all—or even most—warps that are observed on the outer where S„ and t\ are the central surface densiiy and maxi fringes of disk galaxies are steady state. Moreover, certain mum circular velocity, respectively, and a marks Ihe scale galaxies that are observed to have significant kinematical length of the exponential disk. We have chosen here a — 2 4 twists— such as the ones we present here—individually are so t hat t he outer boundary of our models at radius R is traced difficult to explain as steady-state structures; steady-state by the surface-density value S(R) = 0 IS,, warps generally require a dynamical line of nodes whose (b)The outer region of the disk is warped so that, in the orientation m space is constant as a function of radius, reference frame of the galaxy itself, the warp begins at radius whereas the twists in these galaxies simply cannot be mod r„ such that the ascending line of nodes of the innermost eled by sets of rings whose lines of nodes are constant with tilted nng is at an azimuthal angle t0 as measured in the radius. In this paper, we will follow CT and adopt the basic reference disk from the axis defined by y (see item c, below) assumption that the galaxy warps we are modeling are tran Henceforth, the angle f„ will be referred to as "the position sient features; this is being done with the understanding that angle of the warp." Notice that our definition of the position the validity of our conclusions hinges on the correctness of angle of the warp is different by exactly 90* from earlier tilt this assumption. ed-nng models (Rogstad, Wright, and Lockhart I97fi; Rog CT pointed out that the RLW tilted-ring model reveals stad, Crutcher, and Chu 1979) in which the position angle that the potential well of M83 is definitely not prolate Inter was specified for the point of each nng that was displaced to estingly, this result for M83 is consistent not only with the the maximum linear height with respect to the reference dark matter hypothesis, but also with theories that do not disk. The innermost tilted ring, number n0 — N (e,/R ). is assume any dark matter, but that modify Newtonian dynam tilted out of the reference disk by an angle Au> and each ics on large scales in order to explain fiat rotation curves successive ring at larger disk radii is tilted an additional (Tohline 1983, Milgrom 1983a,b,c, 1984, I986a.b; Beken* equal increment of A w degrees. As a result, the maximum stein and Milgrom 1984; Sanders 1984; Kuhn and Kruglyak extent to which the disk at radius R warps out of the refer 1987; for a review see Bekenstein 1987). ence disk is u 1 = (N — n0 + 1) Aw degrees. A twist is intro The galaxies NGC 3033 and 3035 are members of a group duced into the warp by also specifying a different ascending of 14 unbarred, "normal." spiral galaxies for which high- line of nodes for each rotating nng of gas outside of ring resolution 21 cm observations and good isovelocity contour Specifically, we int roducc a constant shift in the line of nodes maps exist in the published literature. In order to examine by At degrees per nng. Hence, the ascending line of nodes of the implications of the idea put forth by CT, we have chosen the outermost ring in our model is ai « t0 + A ltN — n0). to model the kinematics of this entire group using a tilted- T he sign chosen for the twisting increment At is th e key nng model similar to the models that have been used by element that distinguishes between a prograde and a retro RLW and by other authors. Our exact procedures, the rel grade twist for the warp. IfapaM(/ur A rit required to explain evant equations, and our complete set of results will be given the observed kinematics of a disk, then the twist is, by deuni- in Christodoulou, Tohline, and Steiman-Cameron (1988, tion, prograde, and we will deduce that the shape of the un hereafter referred to as C IS). In this paper, we summarize derlying potential well is oblate; if a negative At is required, the results for NGC 3033 and 3055, the only ones from our then the deduced shape is prolate group of 14 galaxies whose kinematics identify a gravitation (c) The orientation of the unwarped reference disk rela al potential well that is unambiguously prolate in shape. The tive to the line of sight and, hence, also of the fiat portion of implications of our results, in relation to the dark matter the warped disk inside radius r^ is specified by two angles— hypothesis and the theories of modified dynamics, are dis the inclination angle to of the disk's symmetry axis to the line cussed in Sec. IV, o f sight (^ — CT indicating a face-on disk); and the potition angle y of the semimajor axis of the blueshifted (i.e., ap II PROCEDURE proaching) side of the disk, as measured from north due east Before any conclusion can be drawn about the underlying on the sky. It is particularly important, in order to avoid geometric shape of a spiral galaxy's gravitational potential ambiguity in the "best-fit" model, that we know the exact 105

1309 CHRISTODOULOU ET AL TILTED-RING MODELS 1309 position of the projected image on the sky of each galaxy in Table I summarizes, for the two galaxies NGC 3033 and three-dimensional space In order to eniure a clear tpecrfita 3035, all the parameter* that were held fixed during our lion o f this poti(ton, we have restricted the inclination angle search for a "best-fit” tilted-ring model. These parameters ^ to values 90*>i0>0* and have specified. in addition, basically refer to the structure of the unwarped reference whether the toutbern edge of the galaxy image on the sky u disk, as defined in Sec 11 above, and have been taken from •carer to ui or farther from u» than the center of the refer Botina (19811 (tee CTS for details). Table II summarizes ence disk the "best-fit” parameters that we have determined for the For the purposes of the present discussion, identification warped structure of each of the two p in * * of a "best-At” tilted-nng model has not included finding a NGC 90)3. The best model demand* a set of tilted rings model that reproduces in detail all the anal I-scale features whose relative orientations twist retrograde, i.e., the incre that appear in typical observed iso velocity contour maps or ment At is negative We deduce, therefore, that the underly iaointensity diagrams Rather, our aim for each galaxy has ing potential well has a prolate shape Specifically, the model been to find a model that best explains general trends that are consists o f 20 rings, with the ten innermost ring* composing present in published observational maps—a model whose a central, unwarped disk: in the eleventh ring the warp be xenMh-order velocity contour bends in the correct direction gins at r0 — 313* and incorporates twisting and tilting angle and to the correct degree, for example, and whose projected increments of A( — — 6* and Aw — 1*. The isovelocity and column density map shows steep radial gradients in the ap surf ace-density diagrams for this model are shown in Fig* propriate locations on the sky. Furthermore, our focus has 1(a) and 1(b) [Plate 811- Figures H e) and 1(f) [Plate 82] been on matching features that arise in theouter regions of are reproduction* of Boama'* (1981) actual observation* each galaxy disk, where the dynamics of gas particles in the For comparison, the "best-fit” oblate model is shown in disk arc presumably most sensitive to the gravitational influ Fig* 1(c) and 1(d). The parameter* for the oblate model ence of a dark halo. In this context, we have not tried to differ from those listed in Table II only in that t0 — 170* and model H i holes, local disturbances, or noncucular motions A/ — + 6*. Notice that, even though the general direction of driven by oval distortion* in the central region* o f the galax the observed kinematical twist is correct in this oblate mod ies in this sample. el, the high-order velocity contour* shown in Fig 1(c) do In our search fora “best-At" tilted-ring model, an attempt not reproduce the observation* Furthermore, the steep den also has been made to include as few free parameters a* pos sity gradient* that appear at the NE and SW side* of the sible. After considerable experimentation, we have found galaxy in the observational map (Fig. 1(01 are not repro that, for the most part, the general trends exhibited in all duced by the oblate model (Fig. 1(d)), where the steepest available H i maps can be accounted for by adopting a very gradient* extend directly east and west. Figures 1(c) and simple, standard form o f the rotation curve and by assuming 1(d) also show that, when projected onto the sky, the outer smooth variation* in both the warping and twisting angles— contour* of the oblate model produce a smaller axis ratio i.e., values of Aw and dr that do not vary with radius A l than is required by the observations. Therefore, in an overall though more complicated models using a larger number of sense, the tilted rings of the "best-fit” ablate model are not free parameters can certainly be used to obtain an even high correctly inclined relative to the line of sight er degree of agreement with the observations in every case, NGC 9095. The best model demands a prolate shape and we are confident that our simple model allows an unambigu warping angles r„ — 26(f. At — — 10*, and Aui — 1 ’ 1 The ous interpretation as to the general properties of the warped warp begin* outside of a central disk composed of only two and twisted structure of a significant number of galaxy disks rings. The isovelocity and mrfsee-density diagram* for this III RESULTS model are shown in Figs. 2(a) and 2(b) [Piste 83) Figures 2(e) and 2(D [Plate 84) are reproductions of Bosma’s In practice, unless the galaxy being studied exhibits isove (1981) actual observation* For comparison, the rejected locity contours that are significantly distoned from the “best-fit” oblate model is shown in Fig*. 2(c) and 2(d ). The clearly recognizable pattern of pure circular rotation— as it parameters for this model differ from the ones listed in Table the case for M83— e reasonably good fit to the observation* II only in that f„ — 2TQT and Ar — + 1(T. This model does can be obtained with either a prograde or a retrograde tilted- not compare well al all with the observed maps The reason ring model (This possibility has not been explored in pre is that, in order to achieve roughly the correct degree of vious studies that have employed the kinematical tilted-ring kinematical curvature in the velocity map, we were forced to model.) As a result, an unambiguous decision regarding the introduce very strong twisting of the ring*. As a result, as is shape of the underlying potential well cannot be made. In illustrated in Figs 2(c) and 2(d), the oblate mode) project our study (tee CTS for detail*), are have generally construct ed cm the sky is much thinner than the observed disk and it ed both a “bast-fit" oblate model—meaning ■ tilted-ring fails to generate the two H J "bridges" running from east to model possessing a warp with a prograde twist— and a north and west to south in the observed surface-denxity map “best-fit” prolate model—meaning a model having a warp On the other hand, the prolate model appears to be as thick with a retrograde twist—then, between these two, the model as the observed disk and does show the H I “bridges " At thi« that provide* the better fit to the observation* is selected as point we would like to stress once again the importance of providing the best overall description of the underlying po- tenual-well shape. At this point, we should emphasize that it Is often difficult to unambiguously select the best overall Tabli I Fuad puuMtn model by examining only modeled velocity maps. The sur- faoe-density maps can play a critical rote in identifying the Southern best model, as our discussion of individual maps below indi Qalaiy r(*> t.n v, (km i " 1) r./H ■tr S R<‘> cates. Of the 14 galaxies are have modeled, a clearcut choice NGC sou It) 42 220 0 1 ocar 20 14 between oblate and prolate models can be made easily in NGC SOS) IT* SS 21) 0 0) star 20 24 3 seven cases— five are oblate and taro are prolate 100

1310 CHRISTODOULOU ET AL. : TILTED-RING MODELS 1 3 1 0

T a k e II Determined parameter! in Sec I, this conclusion is based on the theoretical model outlined by CT and is predicated on the assumption that the G alaxy r^/R 1„C> Arl’J AvC) Shape twitted warp in each galaxy is a transient, rather than

NOC 5033 055 Ti 3 0 ~ To p r o l i l t steady-state, structure. In both cases, the major axis of the NGC50S5 01? 3 360 - 10 II prolate underlying potential well is peqsendicular to the principal plane of the spiral disk Other investigators have constructed similar models for these and other galaxies but have not drawn a clear connection between their models and the dy examining the surface-density maps: in the process of trying namics of the gas in the outer regions of the galaxies. There many different models, we constructed a prolate model with fore, the prolate nature of these two galaxies has previously all parameters as given above except At - -5*. This mod not been noticed. el’s velocity map was practically no different from the map Our results not only give support to the idea that a dark shown in Fig. 2(a) but, as was the case with theoblate model halo surrounds the luminous disks of NGC 5033 and 5055 just discussed, the surface-densily map did not produce the but, as is signified by the halo's clear domination of the dy Observed H 1' ‘bridges " On the basis of presenting the wrong namics, they also indicate that, in each of these systems, the surface-density map, therefore, the prolate model with dark (prolate) halo has a much larger quadrupole moment i f = — 5* was rejected in favor of the one with if = - 1CT than that of the visible (oblate) disk As mentioned in Sec II, we believe that additional im Petrou ( 1980) has presented a model that, at first glance, provements can be made in the kinematics] models of these appears to be able to explain the presence of a retrograde two galaxies by fine tuning the various parameters For ex twist in the tilted-ring model of a warped disk without de ample, one could develop a more sophisticated specification manding that the disk reside in a prolate halo She has ar of the variation of the twist and the warp from nng to nng. gued, in effect, that a retrograde twist can develop naturally rather than using a simple increment, as we have done, or in a gaseous disk that settles into an oblate halo if, contrary one could specify a more complicated rotation curve or sur to the assumptions adopted here and in CT, the halo is struc face-density profile for the unwarped reference disk Indeed, tured such thai the precession rates of panicle orbits in after examining many different types of models it has be crease. rather than decrease, wilh distance from the center of come clear to us that for NGC 5055 in particular, a rotation the disk This situation can arise if the halo exhibits a shape curve that falls in the outer regions (such as the one shown that is nearly spherical al all radii r < R but flattens rapidly at by Bosm a I9BI) would help in reducing the size of the radii outside the disk Petrou's model is a rather unorthodox "looped" velocity contours in Fig 2(a), making the map one but it is, nevertheless, potentially admissible given the more consistent with the observations However, for both of weak constraints that direct observations currently place on the galaxies presented here, we have searched the available the distribution of dark matter around individual galaxies. If parameter space sufficiently well to state with confidence: Petrou’s model proves to be a viable alternative for either (1) Fine tuning of the free parameters, such as employment galaxy, we will be unable to conclude unambiguously that of a sophisticated rotation curve or varying the increments of the halos surrounding NGC 5033 and 5055 are prolate At an d Aw, cannot affect the global features (general trends) Based on settling time argumenls (Tohline, Simonson, shown in the maps of our current models, it only can gener and Caldwell 1982, Steiman-Cameron and Dunsen 1988 > ate tome differences locally exactly wherever the fine tuning one can show, however, that Petrou’s model is unlikely to be is implemented (2) For each galaxy there is a well-deter applicable to most galaxies. In the halo model discussed by mined position around the disk at which the warp starts (3) CT and employed here, precession rates decrease fairly rap In both galaxies, the warped portion of the disk twists retro idly with radius and, as a result, the time it takes gas to settle grade with respect to the direction of the disk's rotation to a preferred plane in the halo is an increasing function of Based on CT's discussion, we therefore conclude that the radius As a gaseous disk seeks its preferred plane orienta shape of the overall potential well is prolate for the two gal tion, then, it should settle from the inside out. This picture is axies NGC 5033 and 5055 consistent with the image that is portrayed by galaxies hav It is important to note that our prolate models described ing a warped disk that can be viewed edge-on—the inner here not only provide the best fits for NGC 5033 and 5055, region of the disk appears to define a preferred plane and. but that also the general structure of these models is not therefore, appears to be the region that settled first. In con substantially different from Bosnia's (1981) models. Differ trast to this, because it exhibits orbital precession rates that ences in detail between our best-fit model and Bosma’s are increase with radius, Petrou's chosen halo structure estab described in CTS, where we also show how Bosnia's param lishes settling times that are a decreasing function of radius eters can be transformed to our system of parameters and (see Cbnstodoulou and Tohline 1988 for a derivation). Jn vice versa. Boama did not elucidate the dynamical implica such a mode), the inner region of a galaxy disk is not the tions of his own tilted-ring models, however, so the evidence region that settles first to the symmetry plane of a spheroidal that these galaxies are embedded in prolate halos has hereto halo. We conclude, therefore, that the general class of mod fore gone unnoticed. els introduced by Petrou does not present a viable explana tion for how retrograde twists develop in warped galaxy tv DISCUSSION disks. From a group of 14 unbarred, “normal" spiral galaxies Theories that have rejected the idea of the existence of whose individual kinematical structure has been analyzed by dark matter (such as the ones cited in Sec. I) have had to using a tilted-ring model (CTS; Christodoulou 1988). two modify Newtonian gravity or Newtonian dynamical laws in spiral galaxies— NGC 5033 and 5055—are of particular in order to be able to generate flat rotation curves in spiral terest because their kinematics reflect an underlying gravita galaxies These theories, however, are unable to incorporate tional potential well that is prolate in structure. As discussed the essence ofa prolate potential well in modeling spiral gal- 107

1311 CHRISTODOULOU ET AL TILTED-RING MODELS 1311

axy dynamics since (he gravitating matter that defines the ics is that they do not allow for a sufficient amount of free potential well is just the luminous matter of the galaxy, and dom in the distribution of gravitating material around the the luminous matter distribution is clearly oblate (disklike) galaxy Our results add to the growing body of evidence that in shape. These theories, therefore, appear to be ruled out on indicates that theories of modified gravity or modified dy the basis that, from their construction, they cannot allow for namics cannot be used to adequately explain the observed a prolate distribution of a spiral gahmy’s material. (This con dynamical properties of galaxies clusion remains unchanged even if one adopts Petrou's (1980) model because, in her mode) as well as ours, the We would like to thank Dr. A. Bosma for hit permission existence of retrograde twists in spiral galaxy disks cannot be to reproduce tome of his observations and for providing us explained in the absence of a halo.) By modeling the shell with his original figures We would also like to thank the structure around the elliptical gaUxy NGC 3923 in a dy referee, whose comments helped us construct a better Fig 2 namically consistent manner, Hemquitt and Quinn (1987) This work was supported in part by grants not. AST 87- have reached a similar conclusion. Their analysis shows that 01303 and AST 83-31997 from the National Science Foun a major weakness in theories of modified Newtonian dynam- dation.

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P L A T E 81

(d)

Fio 1 N G C SOJJ (i.b) liovclwiiy ind w r f « N dauity contoun foe (he "bnt flt" prolate m o W I w I k i u p i r u M t m ire tAbutilcd in T ib l n i ind II The

cooiour intfrvikin 20 k m i ' in d O l^rOfMctively

B m m i ' i m a p (e.f) IwvfkKity ind H i lurfacc dntiiy contour map* from Boama'i ( 1 9 1 1 ) o b u n i n o n i

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l?UI APPENDIX C'

This Appendix presents the preprint of a paper that has been submitted for publication to The Astrophvsjcu] Journal Supplement Series under the ti tle: " Kn> cTnaiicfil Modrlnttf of Uct/is m iht HI DiHs of Gala sit1*." The current reference1 is :

Chi istodoulou, D. M.. and Tohline. ,1. E., and Steiinan-Cameron. T. Y. 1980, Apr T tSuppl.J, submitted.

11? 113

KINEMATICAL MODELING OF WARPS IN THE HI DISKS OF GALAXIES1

Dimitris M. Christodoulou and Joel E. Tohline

Department of Physics and Astronomy

Louisiana State University

and

Thomas Y. Steiman-Cameron

Theoretical Studies Branch

NASA/Ames Research Center

^SU Observatory Contribution No. xxx. 114

ABSTRACT

Almost all spiral galaxies that have been observed 1n the 21-cm

line of neutral hydrogen appear to have a warped disk. Warps are obvious

in HI surface intensity contours of edge-on galaxies as well as in the optical photographs of some galaxies but, for many galaxies, their existence is only inferred from kinematical modeling of the observed HI velocity field. If a disk is warped, for example. Its two-dimensional velocity map will often exhibit twisted isovelocity contours. In many

instances, 1n fact, observed contour maps are twisted so severely that they can only be reasonably understood 1f the warped gas layer Itself has a twisted structure. In order to gain an appreciation for the general structure of warped gas layers in galaxies, we have constructed kinematical, tilte d -rin g models of 21 galaxies for which detailed HI observations already exist in the literature. In this paper, we present resu lts for the 15 normal spiral galaxies in this sample that are not viewed edge-on. A comparison between our models and tilte d -rin g models of the same galaxies that have been constructed previously by other authors shows that there is generally good agreement. We have made an attempt to unify the notation of different authors who have previously published radio observations and/or kinematical models of Individual galaxies in this sample. We have also suggested how, 1n future work of this nature, model parameters should be presented and referenced 1n order to maintain a reasonable degree of consistency in the published

11terature.

When viewed from the perspective of dynamical models, a twisted warped gas layer can be understood as arising naturally from orbiting gas which is in the process of settlin g to a preferred orientation in the 115

nonspherical, gravitational potential well of the galaxy. As the gas layer settles* differential precession should force it to twist in a direction that is dictated by the gross geometric shape of the underlying potential well. In the outer regions of a galaxy disk, where the rotation curve is basically flat, the potential well is presumably dominated by a dark halo. Hence, detailed kinematical modeling of a specific galaxy disk can provide not only information regarding the orientation and structure of Its warp but also information about the shape (whether oblate or prolate) of the dark halo in which the disk is embedded.

By examining a large number of galaxies in a consistent manner, we have deduced some general characteristics of warped disks that have heretofore gone unnoticed. We have also identified uniqueness problems that can arise in this type of modeling procedure and can considerably cloud one's ability to completely decipher an individual disk's structure. For 14 out of 15 spiral galaxies modeled here, we have been able to determine the local kinematical structure of the warp. The overall position of the warp and the gross geometric shape of the halo, however, has been determined unambiguously only 1n cases where the twisting of the warp is relatively strong. Examples of galaxies whose disks sit in an oblate halo are M33, M83, NGC 2805, NGC 2841, and NGC

3718; prolate halos appear to surround NGC 5033 and NGC 5055; and ambiguous cases, at present permitting equally good oblate and prolate halo models, are M31. NGC 300, NGC 3079, NGC 3198, NGC 6946. NGC 7331, and IC 342. The existence of prolate halos supports the "Dark Matter

Hypothesis" and appears to rule out all theories of modified gravity. 116

I. INTRODUCTION

Neutral hydrogen gas is distributed on a luminous, disk-like

configuration of negligible thickness in spiral galaxies. The disk

component, along with the bulge of a spiral galaxy, is believed to be

embedded within a third, massive but non-luminous component--the dark

halo. One-dimensional rotation curves derived from optical and radio

observations of spiral galaxy disks helped trigger the idea that dark halos exist, as the observed rotation curves are nearly flat or slowly

increasing with radius, as far from the centers of the galaxies as they can be measured (Rogstad and Shostak 197?; Roberts and Rots 1973; Rubin

1980; Rubin et al. 1980, 1982). As probes of the dark halo, 21-cm observations are often more useful than optical observations because

neutral hydrogen disks generally extend to larger radii than the optical disks in spiral galaxies. They extend into regions where the dynamical motions are almost certainly controlled by the potential well of the dark halo. In addition, HI observations usually provide more than just one dimensional information about gas motions in spiral disks. The work

reported in this paper strives, in part, to see what additional

information can be learned about dark halos by analyzing the properties of two-dimensional velocity and surface density maps of HI disks 1n

spiral galaxies.

In their outermost regions, almost all spiral galaxy disks appear

to be warped to some degree. Warps can be directly seen in the 21-cm

surface-density diagrams of some edge-on galaxies (Sancisi 1976;

Weliachew, Sancisi, and Guelin 1978; van Woerden 1979; Sancisi 1981) as well as in the optical photographs of some systems {NGC 3190 in Arp 1966; 117

NGC 476? and NGC 5866 in Sandage 1961). In many cases, though, the

existence of warps can only be inferred from kinematical modeling

(Rogstad, Lockhart, and Wright 1974; Rogstad, Wright, and Lockhart 1976;

Emerson 1976; Reakes and Newton 1978; Rogstad, Crutcher, and Chu 1979;

Newton 1980a; Bosma 1981a,b, 1983; Schwarz 1985; Christodoulou, Tohline,

and Steiman-Cameron 1988; Christodoulou 1989). Regardless of how th eir

presence is discerned, warped disks are of particular dynamical interest

because the motions of gas in such disks cannot be described as a simple

two-dimensional flow. Warped gas layers in spiral galaxies are,

therefore, potentially valuable three~dlmensiuna1 probes of dark halo

Structures.

Our primary objective in the present study is to ascertain what

the dynamical properties are of individual warped galaxy disks, then to

use this information to decipher the basic shape of the dark halo that

surrounds each galaxy. Before the dynamical properties of these disk

systems can be discussed, however, it is imperative that th eir k inemd t i i properties be we 11-understood. That is, the observed two- dimensional velocity and surface-density images of each system must be de-projected back into a three-dimensional model that is unique, within a given reasonable set of physical assumptions. Toward this end, we have

compiled, in as uniform manner as possible, a set of data that

lllu cid ates the kinematical structure of warped disks in a large sample of nearby galaxies, and have attempted to Interpret the entire dataset based on a cohesive dynamical picture. 118

a) Kinematical Models

A lot of work has already been published on the kinematical

interpretation of HI observations of spiral disks. Rogstad. Lockhart,

and Wright (1974, hereafter RLW) were the first to convincingly demonstrate that their radio observations of M83 could be reproduced by a

tilte d -rin g model of a warped HI disk. This model, or a sim ilar one, has been used at many later times by the same or other Investigators (see references in §IV, below) to reproduce the observations of d ifferen t galaxies and, thus, to extract information about the kinematical structure of spiral galaxy systems. As we began this study, we naturally

turned to the published literature to extract the kinematical information that we needed. We soon realized, however, that the previously published kinematical models of warped disks provided an inadequate database. We discovered that we had to improve on three main aspects of the previously published models before proceeding to dynamical interpretations of these models: a) The structure of the warps was not known accurately enough, especially for galaxies viewed at large inclination angles or for galaxies that have been observed at relatively low spatial resolution, b) No model had previously been examined for uniqueness of its parameters. We discovered that non-uniqueness problems could arise and that we had to resolve the resulting ambiguities, c) The results from previous modeling efforts had been presented in different ways by different authors, prohibiting direct comparisons; we had to find a consistent basis for presenting the results and, then, convert other authors' results to this basis for comparisons. 119

Ultimately, we were forced to reanalyze all of the previously published observational data. Following the lead of RLW, we used a tilted-ring model to accomplish our analysis. We modeled the structure of 21 galaxies, of which 19 are spirals and 2 are e llip tic a ls with HI disks. In §IV of this paper, we present results for the 15 normal (non barred) spiral galaxies from this sample that are not viewed edge-on:

M31, M33, M83, NGC 2805, NGC 2841, NGC 300, NGC 3079, NGC 3198, NGC 3718,

NGC 5033, NGC 5055, NGC 628, NGC 6946, NGC 7331, and 1C 342. Results for three edge-on spirals (NGC 4565, NGC 4631, and NGC 5907), one barred spiral (NGC 5728), and two e llip tic a ls (NGC 4278 and NGC 5128) are briefly described in §V, but are presented in their full extent only in

Christodoulou (1989). The observational data used in this study was obtained by a variety of workers and published over the past 15 years.

In presenting our models, we attempt to represent these disparate observations in as uniform a manner as possible.

b) Dynawical Interpretation

Our own modeling efforts have been not only motivated, but also strongly guided by a particular dynamical picture of evolving, warped gas layers in galaxies. The d etails of this picture have been given 1n

Christodoulou and Tohline (1986, hereafter CT) but it is worthwhile to briefly review the picture here 1n order to explain why certain nomenclature has been adopted throughout our modeling e ffo rt.

It is expected that many warps in the HI disks of spiral galaxies are transient features. They result from the time-dependent settling of gas (Kahn and Woltjer 1959; Tubbs and Sanders 1979; Tohline, Simonson, and Caldwell 1982) in the nonspherical potential of a halo, toward a 120

preferred orientation in the underlying halo's potential well (Durlsen et al. 1983; Steiman-Cameron and Durlsen 1984, 1988, 1989a,b; Habe and

Ikeuchi 1985, 1988; Varnas 1986a,b; Christodoulou 1989). The settlin g process should naturally cause twisting of a warped gas layer as a result of d iffe re n tia l precession—which, when coupled with gas dissipation, ultimately drives the settling process. As CT carefully describe, the direction of the dynamically generated twisting, relative to the direction of rotation of gas in the disk, will depend on the sign of the quadrupole moment of the underlying potential well. Because oblate- spheroidal structures exhibit quadrupole moments that are opposite In sign to prolate-spheroidal structures, the two different shapes should

lead to entirely oppositely directed twists in a settling gas layer.

Specifically, in systems for which the magnitude of both the orbital frequency and the precession frequency of particle orbits monotonically decrease with orbital radius--the behavior most naturally realized for spiral galaxy disks exhibiting flat rotation curves--a settling gas layer will develop a twist that is prograde with respect to galaxy rotation if the potential well is oblate in shape, but a retrograde twist will develop in a prolate potential well.

If the kinematical analysis of a given galaxy indicates that Its disk is not only warped but 1s also smoothly twisted, then, following the discussion in CT, we will conclude that its warp 1s transient in nature. Furthermore, we will use the direction of its kinematical twist to Identify the geometric shape of the underlying dark halo potential.

CT actually predicted that the geometric shape of a dark halo can be deduced from observations without resorting to full, three-dimensional, 121

kinematical modeling. By simply comparing the direction that the optical spiral arms turn with the direction that isovelocity contours turn in a

2-D, HI map, one should be able to deduce whether the underlying halo potential is oblate or prolate. (If they turn the same direction on the sky, the halo is prolate; if they turn in opposite directions, the halo is oblate.) In situations where the 3-D kinematical models of a twisted warp allow us to uniquely identify the shape of a dark halo, we will check the re lia b ility of this CT prediction.

Our dynamical interpretation of kinematical warps will not be restricted to transient warps because we realize that not all warps have to be transient features. A warp can be maintained as a long-lived or

"steady-state" feature, for example, if the massive halo in which an HI disk is embedded has a certain orientation, shape, or dynamical state

(see the reviews by Saar 1979 and Toomre 1983; for additional references see CT). In contrast to the transient warps just described, however, comnonly modeled steady-state warps display an untwisted structure; In a kinematical, tilte d -rin g model, an untwisted warp will be represented by a sequence of tilted rings that all have a common line of nodes. In this paper, we pursue the idea that, if a unique kinematical interpretation can be secured for an individual HI disk, transient and steady-state warps can be distinguished from one another. Specifically, warps that clearly display twists will be considered transient ones while untwisted warps will be identified as steady-state structures.

Recent theoretical work (David, Durisen, and Steiman-Cameron 1984) has shown that if the dark halo of a galaxy is tumbling about an axis other than a principal axis, then a twisted, steady-state warp can 122

develop. This result certainly obscures the distinction between transient and steady-state warps as outlined in the previous paragraph because discovery of a twist in a disk will not necessarily imply that the warp is transient. A halo that is tumbling about a non-principal axis is. however, an extreme case. For clarity and simplicity, we will not take into account this case in this paper.

Sparke (1984) and Sparke and Casertano (1988) have taken a somewhat different approach toward Interpreting warps in gas disks: they have proposed that warps can be understood as vertical inodes of oscillation in galaxy disks. For this theory to work in its simplest form, the warps must have a common line of nodes, i .e ., they must exhibit no twisting at a ll. Because we find that, in most cases, kinematical models of untwisted warps fail to reproduce the observations, we prefer to interpret our models in the framework outlined by CT, that is, in terms of gas layers that are settling toward, or that have already reached, a "preferred orientation" in the halo potential well.

Finally, as has been discussed by CT, a large part of the motivation for this study stems from the opportunity that it affords to test certain predictions of recently proposed theories of modified

Newtonian dynamics (Tohline 1983; Milgrom 1983a,b,c, 1984, 1986a,b;

Bekenstein and Milgrom 1984; Kuhn and Kruglyak 1987). Proposed as alternatives to the Dark Matter Hypothesis, each of these Ideas entails modifying the gravitational dynamical laws in such a manner so that the visible matter produces a flat rotation curve, without the need of a superimposed dark halo. When adopting any such theory, one implicitly 123

assumes that a spiral galaxy is composed of only the two luminous components—the bulge and the disk--and that the potential of the disk dominates the kinematics and dynamics of orbiting HI gas. Because disks are oblate structures, these theories implicitly require that the potential wells of all spirals be oblate. Therefore, they cannot explain the structure of galaxies whose warped disks are found to twist in a

“retrograde" direction, i.e., disks (such as the ones found 1n NGC 5033 and NGC 5055, see §V, below) which appear to be settlin g in a prolate potential well. The above arguments are crucial in relation to the subjects of galactic structure and galaxy dynamics. We briefly discuss them in this paper, but we have extensively presented them in a separate study (Christodoulou, lohline, and Steiman Cameron 1988).

c) Simplifications. Qualifications and Terminology

We would like to simplify our terminology at this point. The CT analysis was restricted to only axially symmetric structures (oblate or prolate spheroids), but, as they mentioned, it is easy to demonstrate that the basic relationships found between direction of twisting and the shape of the underlying potential well extends beyond simple spheroids to oblate-like and prolate-Hke ellipsoids. By these terms we mean: if an ellipsoidal halo having semimajor axes a, b, and c envelopes a spiral disk in such a way that its c-axis aligns with the rotation axis of the disk, then an oblate-like halo has a ^ b > c and a prolate-like halo has a a •: b < c. In order to emphasize this generalization to ellipsoidal figures, CT used the terms "oblate-1 Ike'1 and "prolate-like'1 throughout their paper. Here we will simply use the terms oblate and prolate, but 124

one should bear In mind the wide variety of halo shapes that each of these two terms really refers to.

We also emphasize that each ring element in our simplified model

(as well as in all tilte d -rin g models previously published by other authors) is azimuthally syrnnetric and, as a result, the model cannot incorporate a kinematical description of any local disturbances or noncircular motions in HI disks. (These are phenomena that occur frequently in real galaxies and, in some cases, even dominate the kinematics of the system). Our model, therefore, does not reproduce eith er small scale asymmetries—which can be found in nearly every galaxy—or large scale asymnetries. We have attempted to derive models for three galaxies with large scale syrnnetries (NGC 2805, NGC 3079, and

NGC 6946), but in doing so we have assumed that one side of each galaxy is basically undisturbed and have produced symmetric models based upon this assumption.

The models presented in this paper are, by no means, the best that could have been constructed; all could be Improved by fine-tuning the various free parameters of the basic tilted-ring model. Extensive fine- tuning was not done, however, because (a) within the present constraints of our modeling, we have been able to derive definite conclusions about the structure and positions of warps in the modeled galaxies; and (b) more refined modeling of each galaxy would not be justified since details would only apply to the current resolution of observations and would probably be rendered useless by higher resolution observations of the same galaxy. 125

On the other hand, we have constructed two types of 3-D models for each galaxy in an attempt to determine and possibly resolve amgibuous cases: a best-fit "oblate" model in which the tilted rings are twisted

in the direction representative of differential precession in an oblate gravitational potential, i.e., twisted prograde with respect to the direction of rotation of the disk, and a best-fit "prolate" model 1n which the rings are twisted retrograde with respect to disk rotation.

{In the former case, the twist that appears In the contours of the 2-D, projected velocity map is also usually prograde while in the latter case the twist is usuajjy retrograde |CT].) If the best oblate and the best prolate model show significant differences from one another and only one of them really matches the observational results, then we call that model the best model for the galaxy. Otherwise an ambiguity between oblate and prolate geometries persists. We emphasize that the entire parameter space is examined for the best models to be found. Additional non uniqueness problems will arise only if the parameter space is expanded to

include more free parameters (for example, non-circular orbits).

fin a lly , in attempting to d ifferen tiate between transient and

steady-state warps, a judgement must be made regarding when a given observed or modeled twist 1s strong enough to rule out the "steady-state"

Interpretation. As we will demonstrate presently, the actual strength and importance of "twisting" in a 3-D, warped gas layer can be eith e r camouflaged or artificially exaggerated when its image gets projected onto the plane of the sky, depending on the gas layer's orientation with

respect to the observer. With this 1n mind, we will generally adopt the

following terminology throughout our discussions: an observed twist in 126

the 2-D velocity map of a galaxy will be considered "strong" if it turns

by more than 20° in the plane of the sky; in the frame of the galaxy

Itself, a warp will be considered "strong" if it bends by more than 15°

out of the plane defined by the central regions of the galaxy disk and a

twist will be considered "strong" if, overall, it extends over more than

a 30 angle.

d) Outline

In §11, we present the basic equations for our tilted-ring,

kinematical model. These equations define the component of the velocity of an orbiting particle along the line of sight and the particle's

projected position on the plane of the sky. We also describe the parameters of this model and the procedures by which either their values are chosen to be fixed for each Individual galaxy or they are considered

variables and are therefore to be determined.

Since only circular motions are examined, certain symmetries are built into the equations of the tilted-ring model. These synmetrles naturally lead to ambiguities in the interpretation of the correct

kinematical model for some galaxies. That is, models which appear to be exactly the same when projected on the plane of the sky, but which have different intrinsic parameter values in 3-D space. In §11 we discuss

these ambiguities and ways to resolve them. We also define and discuss

the three classes of warps that we model: oblate-shape, prolate-shape,

and steady-state warps.

In §111, we discuss the kinematical models of other investigators, analyze their differences from our model, and describe how these 127

differences may account for different results. We also propose a coimton system for defining and referencing the different parameters included in tilted ring models. We present a detailed comparison between the models of Bosma (1981a) and those of this paper for the galaxies NGC 2841, NGC

5033, NGC 5055, and NGC 7331. We demonstrate that our models are in basic agreement with Bosma's and show how Bosma1s parameters can be transformed to our system of definitions and vice versa.

In § 1V, we present our detailed models for each galaxy, compare them directly with 21-cm observations, and with modeled results presented by other authors. All these results are given as isovelocity and surface density contour diagrams. Previously published observations and models, kindly provided by many authors, are also reproduced 1n §1V to facilitate direct comparisons with our models.

In §V, we summarize the values of the model parameters used or determined during our modeling effort. We tabulate the values found by other authors, given, however, in the contnon "language", that we describe in §111. We also discuss our results in relation to the general structure of the warps and tw ists, the deduced structure of the halos, and the HI radio observations of spiral galaxies. At the end of this section, we consider it helpful to also briefly describe the results that we have obtained on edge-on and barred spirals and ellipticals with HI disks. Finally, the main points of the en tire paper are summarized in §V. 128

1 1 . A KINEMATICAL T IL T E D -R IN G MODEL OL A WARPED GALAXY DISK

In an effort to present our tilted-ring model as clearly as

possible, we first describe its structure in the reference frame of the

model galaxy (§lla). We then describe its projection onto the plane of

the sky of a distant observer (§IIb). Because, by necessity, numerous mathematical symbols and definitions are introduced in this section, as

an aid to the reader we include in Appendix A a brief sum-nary of all the

symbols and definitions used here.

a) Description of the Tilted-Ring Model

We take the following standard model of an unwarped disk as the

starting point for all our tilted-ring models. The disk is flat,

inf initesima1ly thin, perfectly circular, has a total radius RQ, and has a simple rotation curve given by the equation:

V(R) - (la) V R s R R C C u

Although simple, this rotation curve has been adopted because it

represents, at least qualitatively, the type of curve observed in many galaxies, lo be sure, the inner part of the rotation curve of real galaxies takes on much more structure than has been adopted here, but the outer part is almost always observed to be nearly flat (see §1), and

since warping of disks occurs primarily 1n the outer portions of galaxies, a more general representation of the inner part would present 129

here only unnecessary complications. On the other hand, we have performed numerous experiments with detailed rotation curves given in the

literature and we have essentially confirmed that no Important features are lost from the maps if the simple rotation curve of Eq. (la) is used

(see, e .g ., our reproduction of Bosma's 1981a model for NGC 5055 1n §IV).

For modeling purposes, we divide our standard disk into N concentric, circular rings numbered n-1 to N from the center of the disk, outward. Each ring has a radial width aR ^ RQ/N and an outer-edge radius

Rn = nAR. The Innermost segment, which is actually a circle of radius Rj

- R0/N rather than a ring, will henceforth be called the "reference" disk because it defines a natural plane in the galaxy's reference frame against which the orientation of all other ring segments can unambiguously be referenced. In the standard model, which is unwarped, all rings are coplanar with the reference disk. For additional clarity, we also define Jj as the angular momentum vector of the reference disk.

The hemisphere Into which Jj points will be referred to as the "upper" hemisphere of the galaxy, irrespective of its eventual orientation with respect to the observer's line of sight. Similarly, the hemisphere into which -Jj points will be called the "lower" hemisphere of the galaxy.

Following RLW and Bosma (1981a), we model warping 1n the disk by allowing each discrete circular ring to lie in some plane other than the reference disk. In general, then, for the n*h ring segment, we specify:

(a) the circular velocity as defined by Eq. (la); (b) the inclination

(warping) angle wn of the angular momentum vector Jn of the ring measured with respect to Jj; and (c) the (twisting) angle t measured in the plane of the reference disk trom a reference axis located in the reference disk 130

(defined below), counter-clockwise to the ascending line of nodes of the tilted circular orbit defined by ring n. The angles wn and tn are defined graphically in Figure 1 (see § 11b >.

For a number of reasons it is useful to identify ring "m" with the radius of the disk at which the gas first starts warping out of the plane of the reference disk. I .e .,

w - 0 , for n < m n > 0 , for n > m .

By definition, ring m cannot be the reference disk itself, so 2 < m <

N. The innermost (m-1) unwarped rings define an extended reference disk, the orientation of which can be specified by the vector Jj. We will henceforth refer to this extended reference disk as the "central" disk.

In the reference frame of the galaxy, the line of nodes defined by the intersection of ring m with the reference (or central) disk provides a sensible axis against which the twisting angle tn of subsequent tilted rings can be measured. We therefore define tm * 0 and will, henceforth, refer to the ascending line of nodes of ring m as the "warp node." (It is important to realize th at, in general, a remote observer will find that the orientation on the sky of the warp node will not coincide with the projected major axis of the galaxy. This generalization has not been realized or appreciated in a number of previous Investigations.)

Projected surface density maps are produced from the tilted-ring model under the assumption that the standard, unwarped model has its mass distributed 1n an exponential density profile of the form

S(R) = SQ exp(-A R/Ro ) , (lb) 131

where SQ is the surface density at the center of the reference disk and ft is the scale-length of the exponential disk. As the individual rings are allowed to change their orientation, mass particles on those rings are forced to change their projected position on the sky. The resulting projected density profile, therefore, describes the warped disk. Since our numerical models are usually constructed with higher spatial resolution than the real observations presently achieve, each of our surface density maps is convolved with a Gaussian beam pattern so that the convolved image can be directly compared with the observed density maps. An exponential profile for the density distribution 1s consistent with the distribution of light in the disks of spiral galaxies

(RLW; Krumm and Salpeter 1977), but, since real galaxy disks are full of local disturbances, results obtained by the use of equation (lb) should be compared to the actual observations only in relatio n to the overall patterns they exhibit, and not relative to any local, small scale characteri sties.

b) Basic Coordinate Frawes

For the sake of complete generality In projecting our tilte d -rin g model onto the sky, we will consider four different Cartesian coordinate systems. The first two are defined in the model galaxy Itself:

a) DOUBLE PRIMEO SYSTEM; A coordinate system embedded 1n the

plane that contains the tilted circular ring of radius Rn-

This coordinate system will generally vary from ring to ring

and is represented by double primes. It is oriented such that 132

the X"-Y" plane lies In the ring plane with the X"-axi$ through

the point In the tilted ring where the velocity 1s to be

evaluated. The Z"-ax1s is orthogonal to the ring plane and

coincides with the Jn angular momentum vector,

b) SINGLE PRIMED SYSTEM: A coordinate system embedded in the

reference disk of the galaxy and represented by single

primes. This system 1s defined such that, for all rings, the

positive Z'-axis coincides with Jj and the X'-Y' plane

coincides with the plane of the reference disk. For each

orbiting ring of gas, the positive X'-axis is oriented such

that it Identifies the ascending node of the ring under

considerat ion.

The relationship between these two coordinate systems Is shown in Figure

1. A particle orbiting in ring n has its orbit inclined (warped) by an angle wn relative to the reference disk and rotated (twisted) by an angle t n with respect to the warp node. (Because the "warp node" is defined, as stated in § 11 a, by the ascending line of nodes of ring m, the angle t n shown in Figure 1 will be, by definition, t - 0 if the tilted particle orbit being depicted is in ring m Itself.) The particle's position, with respect to the ascending node of its own orbit, can be specified 1n the

X"-Y" plane by the polar coordinates (Rn,$n), where $ 1s the azimuth measured from the X'-axis In the direction of the particle's motion.

The definitions of the other two coordinate systems depend on introducing a remote observer and the familiar concept of a "plane of the sky" which lies perpendicular to the observer's line of sight. The two systems are: 133

c) UNPRIMED SYSTEM: An unprimed coordinate system oriented such

that the observer's line of sight is down the Z-axis and the X-

Y plane lies in the plane of the sky. The positive X-ax1s 1s

aligned with the blue-shifted side of the projected major axis

of the disk and marks the intersection between the X'-Y' plane

of the reference disk and the plane of the sky.

d) OBSERVER'S SYSTEM: A coordinate system resulting from the

rotation of the (X,Y,Z) frame by an angle -y about its Z-

axis. This system is denoted by the coordinates (t.n.t) and

is defined such that the c-axis aligns with the Z-axis and the

e, axis always points due North on the sky. This system 1s used

to identify the position angle y of the galaxy's projected

major axis on the sky.

The relationship between these latter two systems and the reference disk of the remote galaxy Is Illustrated in Figure 2. The X'-Y' galactic plane, which coincides with the galaxy's central disk, is inclined by an

angle iQ with respect to the plane of the sky. (Alternatively, iQ is the

angle measured between the Z'-axis and the Z-axis.) In general, neither

the line defined by the X'-axis nor the warp node shown in Figure 1 will

lie in the plane of the sky because these two lines are derived from

intrinsic properties of the tilted-ring galaxy model and do not depend on

the existence of a remote observer. Therefore, the line locating the

intersection between the galaxy's reference disk and the plane of the sky

( i . e . , the X-ax1s) will mark a th ird , unique reference Hne lying in the galaxy's reference disk. Because we are Interested here in projecting orbiting particle positions and velocities onto the plane of the sky, 1t becomes convenient to temporarily bypass any reference to the warp node 134

and relate the location of the X'-axis directly to the X-axis. In Figure

2, marks the angle between the X- and X'-axes measured ccw (as defined

by the Jj vector) in the plane of the reference disk. If we

define t ^ as the angle measured in the reference disk between the X- o m axis and the warp node, then we can always retrieve the twisting angle t n

from w via the relation n

wn = l o + l n • for n - m * In reality, t Identifies the position around the galaxy's reference disk

at which the warp f ir s t begins to occur, so we will often refer to t as

the "position angle of the warp," or PAw, and use it as a primary

variable to be fit in all our tilted ring models. Henceforth we will

p refer to both and tn, interchangeably, as the "twisting angle". p Rogstad and collaborators (cited in the Introduction) use a different definition of the twisting angle. In their tilted-ring model, an angle o is measured on the central disk but to the point above which ring n reaches its maximum linear height from the plane of the central disk. Obviously, then, n = 90“ + u , where u is the twisting angle of n n n ring n in our model.

c) Projected Image Equations

Given the circular velocity V = Y" of a particle located at the point (X",0,0), and orbiting on a circle of radius Rn, we want to determine the radial velocity V along the observer's line of sight and

the corresponding projected position of (X",0,0) onto the plane of the

sky. lo do th is, we need merely perform the relevant rotations of the 135

gas particle's positional coordinates, and differentiate the transformed coordinates with respect to time:

a) To obtain (X'.Y'.Z'J and (V'.V.V') we rotate by an angle x y 2 about the Z"-axis, and by an angle -w about the X'-axis*

b) lo obtain (X,Y,Z) and (VX,V ,VZ), we rotate by an angle -w

about the Z'-axis, and by an angle -iQ about the X-axis.

c) lo obtain (t.n.f,) and (V^, V^, V^), we rotate by an angle

-r about the Z-axis.

The complete matrix of rotations can then be written as a product of five matr ices:

cost -Sint 0 1 0 0 coSw -sinw 0

siny cosy 0 0 cosi -sini • sinw cosw 0 o o 0 0 1 • 0 sini cosi 0 0 1

1 0 0 cos -sin 0 0 cosw -sinw sin cos$ 0 0 sinw cosw 0 0 1

Using the convention that the radial velocity will be negative if it is directed toward the observer, we find that

^ =■ R{cosy (cos$ cosw-sin sinw cosw)

- siny!(cos$ sinw + sin$ cosw cosw)cosiQ - sin* sinw sin1 q 1 > , (2) 136

n = R{sinY(cos* COSw - sin* sinw cosw)

+ cosY|(cos* sinw + sin* cosw cosw)cos1Q - s 1 n sinw sin iQ|} , (3)

and

V = -V{(-sin«t> sinw + cos* cosw cosw)sin1o + cos* sinw cosiQ} . (4)

It is worthwhile noting that these expressions reduce to a much simpler

form if one is concerned only with examining the kinematical features of a

fla t (unwarped) disk whose projected major axis points due North on the

sky (e.g ., Mihalas and Binney 1981, pp. 499-500). For a fla t disk, w = 0

and (w + *) + *; if the major axis points North, Y = 0. Under these

conditions, equations (2) - (4) become:

t. = R cos* , (5)

n = R Sin* CosiQ , (6)

V = -Vcos* s1niQ . (7)

Defining, then, a particle's projected radial position and polar angle on 2 2 1/2 the plane of the sky as p ; (t + n ) ' and tan6 ; n/t. respectively, we

further derive, from equations (5) and (6),

R2 = p2(cos26 + sec210 s1n26) , (8)

tans = cosiQ tan* , (9) 137

■3 in agreement with Mihalas and Binney* The radial velocity given by

JThere is clearly a misprint in equation (8-62) of Mihalas and Binney

(1981). Either the angle symbols e and

expression or the coefficient of the tane term should be cosi rather than

sec i.

Mihalas and Binney (their equation (8-60)) reduces to our equation (7) if

the particles in the Mihalas and Binney disk are forced to move 1n

circular orbits (without any radial or z-motions) and if the circular

velocity is assumed to be only a function of radius R.

c) Classes of Harps

Being guided by the theoretical discussion presented in CT and

reviewed in §1 above, only smooth warps and tw ists, which present

reasonable representations of gas disks trying to settle to their preferred orientations 1n spheroidal gravitational potential wells, will be primarily examined here. We assume that the warping angle wp

Increases in a monotonic fashion with a relatively small (perhaps zero)

Increment in angle as we progress from the center of the disk, outward.

We abandon th is tactic only for extremely disturbed galaxies like NGC

3079. Sim ilarly, we assume that the twisting angle t n either Increases or

decreases in a monotonic fashion with a relatively small (perhaps zero)

increment 1n angle as we move outward from the center of the disk. Three

specific classes of warps are modeled in d etail: the steady-state warp;

an oblate warp; and a prolate warp. Their respective definitions follow. 138

1.) Steady-state warp: A steady-state warp 1s one that, very simply, has t = 0 for all n > m, independent of the warping angle wR. Since all rings have the same line of nodes but, in general, different inclinations, this type of warp 1s qualitatively characteristic of a steady-state structure that no longer shows differential precession of particle orbits. It should be found, for example, 1n a tumbling, spheroidal potential well once gas has completely settled to its preferred orientation (such structures are plotted and discussed 1n Habe and Ikeuchi

1988).

2.) Obi ate-shape warp: An oblate-shape warp is defined as one 1n which

*n+l > ^ ^or n > m* ^dependent the warping angle wp. In this case, the ascending line of nodes shifts in a retrograde sense relative to the direction of rotation of the galaxy, as we progress outward from the center of the disk, but the magnitude of shifting decreases with radius.

Therefore, the overall pattern appears to be twisted 1n a prograde sense. This type of warp should be found In a differentially precesslng gaseous disk (with the precession frequency decreasing with radius) that is still in the process of settling to Its preferred orientation 1n a nontumbling, oblate potential well (Steiman-Cameron and Durlsen 1988,

1989a,b; Christodoulou 1989).

3.) Prolate-shape warp: A prolate-shape warp is defined as one 1n which t i < t < 0 for all n > m. Independent of the warping angle wp. In this case, the ascending I1r»e of nodes shifts in angle In the prograde sense to the direction of rotation of the galaxy, as we progress outward from the center of the disk, but the magnitude of shifting decreases with radius. 139

Therefore, the pattern appears to be twisted in a retrograde sense. This type of warp should be found in a differentially precessing gaseous disk

(with the precession frequency decreasing with radius) that 1s still 1n the process of settling to its preferred orientation 1n a nontumbling, prolate potential well (Habe and Ikeuchi 1985; Steiman-Cameron and Durlsen

1988, 1989a,b).

d) Description of the Parameters of the Model

It has become obvious from the above discussion that both constructing the model and orienting it correctly with respect to the observer is a complicated process. Below, we describe all the important model parameters from a practical point of view and explain how they are measured or determined.

1. Rotation curve: The simple rotation curve of equation (la) 1s completely specified by the maximum circular velocity Vc and the radii Rc and R0. With the single exception of NGC 2805 (an extremely disturbed galaxy) these parameters are obtained for each modeled galaxy from previously published observations (see references 1n §§I and IV).

2. Gas and galaxy's frames of reference: In keeping with the relatively low spatial resolution of most published HI maps, we have chosen to divide all models into only N = 20 rings. Each ring 1s further divided into 80, equal-sized azimuthal zones. Before the projected position and velocity on the sky of all 1600 zones is computed, we have to specify:

the extent of the central disk by choosing the radial location

of ring "m;" 140

the position angle of the warp (tD) at ring m;

the direction of shifting of the line of nodes of each ring

outside ring "m," i.e., the precession direction and, hence, the

adopted shape of the underlying potential well;

the angle by which each Individual ring outside of the central

disk is tilted out of the plane of the central disk; and

the amount by which the line of nodes of each tilte d ring 1s

shifted relative to tQ.

We will refer to these parameters as, respectively, the "central disk" size, the position angle of the warp (in short, "PAw"), the "shape" of the potential well, the warping angle "w," and the twisting angle "t".

In general, the "central disk" can be composed of anywhere from 1 to 20 rings (the latter case being a totally flat, unwarped disk). The

"shape" can only be oblate, prolate, or undetermined, according to the class of warp being modeled, but its specification is only meaningful if the angle w is nonzero for at least two rings.

As defined earlier, the twisting angle "t" is measured from the

PAw. In general, we are free to choose the twisting angle of each ring, tn, independently. For all but a few galaxies in this study, however, we have not allowed the twisting angles to be specified independently of one another. We have Instead demanded that, for an individual system, the twisting angles of adjacent tilted rings differ from one another by a constant increment, at. That 1s, for any ring n outside of ring m,

tR - s(n-m)at , (10a) where |s| * 1 and the sign of s depends on the chosen direction of the tw ist. Because It forces a monotonic change in the twisting angle, this prescription is in keeping with the idea, highlighted by CT, that a smooth 141

twist should develop from the dynamical settling of a warped gas layer.

In addition, employing relation (10a) greatly reduces the number of free parameters that can or need to be adjusted for each modeled system. (We cannot justify the choice of a linear relationship between the tw isting angles of different rings other than to say, at the present time, a more complicated relation is uncalled for on either observational or theoretical grounds. This simple relationship appears to be adequate for now because, as our results show, it allows us to match the current HI maps of most disk systems quite well.) When expression (10a) is used, the parameter at provides a quick measure of the degree to which twisting is

important 1n a given warped disk system and the sign of s Immediately

identifies the chosen "shape" parameter -- s = +1 means an "oblate" shape has been chosen and s = -1 means a "prolate" shape has been used. By setting at = 0 in expression (10a), all the rings will have a contnon line of nodes and our tllted-ring model will assume the structure of a "steady- state" warp, as defined above. In this case, the sign of the shape parameter s is undetermined.

As defined e a rlie r, w is measured with respect to the central disk as the angle between the vectors Jj and Jn. As with the twisting angle,

in general we are free to choose the warping angle of each ring

independently. Again, though, in an attempt to confine our models to ones having smoothly warped structures, we have adopted the following prescription for wn for most of our kinematical models;

wn = (n-m+l)aw , (10b) where aw is independent of n for a given mode!. In this manner, as with the specification of tn above, we have greatly reduced the number of free parameters in our models. The single parameter aw gives a measure of the 142

degree by which a given galaxy's entire warped HI layer is bent out of the

plane of the central disk. Since the model 1s symmetric, only one warped

side will be located In the upper hemisphere. Hereafter, we will call

this side the "positive" warp, and will present our results on warping

relative to it.

The five parameters m, tQ, s, a w and At represent the free

parameters for each modeled galaxy. Their values have been determined

essentially through trial and error by comparing the observed velocity

contour maps and 1sodens1ty maps with models generated from many different

possible combinations of these parameters {see § 11g below). The basic

properties of these parameters and how their changes usually affect the

contour maps of a model are described in §llf.

3. Projected Image on the sky: The true orientation of the projected

Image of a model onto the plane of the sky is specified by the inclination

angle (iQ) of the central disk and the position angle y of the projected major axis of the central disk. Both parameters can be determined either

from an optical photograph of the galaxy or, more accurately, from the

central isophotes of the HI surface density maps provided by radio

observations. The position angle y 1s measured from north through east to

the blut^ hifted side of the projected major axis. Because north is at

the top and east is on the left 1n conventional photographs of a galaxy,

> is measured ccw on the sky. The Inclination angle 1s determined by the

equation

i' = cos-1(b/a) , (11)

where a and b are, respectively, the lengths of the projected major and

minor axes of the central disk. Unfortunately, equation (11) alone does 143

not specify completely the true inclination of the central disk because the arccos introduces an ambiguity of n radians, i.e., the true inclination is either iQ = i ' or iQ = 180u-i'. This ambiguity can be resolved if we know the orientation of some other part of the galaxy, relative to the observer. We choose to specify the orientation of the southern edge (hereafter, s.e.) of the galaxy. If, when viewed as a projected image on the sky, the portion of the galaxy that points due south is a segment of the galaxy disk which, in 3-0 space, lies closer to the observer than the galaxy center, then we define the s.e. to be "near" the observer. Otherwise, we define the s.e. to be "far" from the observer. Once the position angle ^ is measured and the orientation of the s.e. is specified in this manner, the value of iQ can be determined unambiguously from i ‘. This is shown in Table 1. For example, if the blue-shifted side of the central disk's projected major axis points east on the sky (i.e., 0U i t < 180°) and the s.e. is "far" from the observer, then iQ = ISO11-! '.

In practice, resolving the ambiguity that arises 1n the determination of 1Q from the axis ratio b/a 1s not always a trivial task because determining the orientation of the s.e. of the galaxy can Involve some subjective judgment. For example, when working with an optical photograph of a disk galaxy, the location of a dust lane (arising from dust assumed to be fairly uniformly mixed with hydrogen gas in the disk) can be used to determine whether the s.e. is "near" or "far" from the observer if the dust lane obscures a substantial portion of the star light that 1s being emitted by stars 1n the disk or if it appears to divide the nucleus Into two unequal parts. If, for example, the smaller part of a partially obscured nucleus points south (as in the case of NGC 3079--see 144

Fig. 5i below), then the s.e. of the disk must be closer to the observer than the galaxy center. This technique for identifying the orientation of a galaxy disk is most useful in systems where the disk is highly Inclined to the line of sight, but it can also be used effectively in many systems having only moderate inclination.

Alternatively, the orientation of the s.e. can be uniquely specified for a spiral galaxy if the direction in which the optical spiral arms turn (clockwise or counterclockwise on the sky) can be Identified and if, at the same time, the assumption is made that the spiral arms are

"trailing" features. Figure 3 is very helpful in illustrating this technique. Consider a galaxy oriented as in Figure 3 relative to the observer at 0. If the spiral arms are trailing features, then the angular momentum vector of the galaxy's disk points in the general direction of the observer ( I .e ., the "upper" hemisphere of the galaxy is the one containing 0). Now, suppose that the western side of the galaxy is red- shifted relative to its center. This setup can only arise 1f the s.e. of the galaxy is "nearer" to the observer than the galaxy center. One can easily verify this conclusion by using the standard right-hand rule — pointing the thumb in the direction of the angular momentum vector Jj, the other fingers of the right hand will show the direction of the rotational velocity. The same line of reasoning leads to the conclusion that, 1n the above case, the s.e. would be "far" away from the observer If the western side were blue-shifted. Similar arguments are easily developed for galaxies whose angular momentum vector generally points away from the observer. In the lower part of Table 1, we classify the different cases of determining the orientation of the s.e. of a spiral galaxy, assuming that optical spiral arms are always trailing features. 145

The parameters iQ and y are assumed not to be free parameters

here. Their values have been taken from other published works and are

held fixed in our models.

e) Ambiguous cases

Ihe three basic equations (2)-(4) which define the appearance of a

tilted-ring, kinematical model as projected on the sky remain invariant if we change simultaneously three parameters: the orientation of the s.e.

(1 * 180“ - 1Q), the shape of the underlying potential well (s - -s) ,

and the position angle of the warp. This substitution is mathematically

described by the transformation iQ » 180‘-io, w ♦ 180'-^, and $ - 180"-*.

Under this transformation, resulting contour maps remain unchanged,

thereby giving rise to an ambiguity as to the true structure of the disk

and the true shape of the underlying potential well. Henceforth, we will

refer to this as the s.e.-shape-PAw (in short, SSP) ambiguity. The shape

and the twisting angle are free parameters, therefore the ambiguity cannot

be resolved from them. If we are unable to uniquely specify the orientation of the s.e. of the galaxy (eg., if the dust lane technique 1s

inconclusive and, at the same time, we cannot deduce the direction at which the spiral arms turn, as for some nearly edge-on galaxies), then the

global parameters of a "best-fit" model will be uncertain, as to the true

orientation of the disk, to the true shape of the dominant potential, and

to the exact position of the warp 1n the disk. Locally, however, the

structure of the warp can still be found. I.e., the angles t and w can be determined.

Besides the SSP ambiguity, there is no other precise ambiguity

Imposed by the symmetry of the model that can be demonstrated 146

mathematically In equations (2)-(4). We have discovered, however, another

"approximate" ambiguous case, that originates in weakly twisted and/or weakly warped models, and its mathematical explanation is based upon an approximate form of equation (10a). Suppose that two models, one with parameters Sj, tQj and another with $ 2 * t Q 2 have the same local structure of the warp, i.e., aw^ = aw^,, at^ = at^ and T°r a1^ n

l o2 ~ lol = 2(n‘m) At (12>

We can see that t Q 2 of the prolate model is larger than tQ^ of the oblate model. For a fixed value of the Increment at and a fixed central disk size, the difference between the two PAw's is still varying, because the index n (number of the ring) 1s variable. In our models the largest value of n is N = 20, and at is always much smaller than th at, thus, for all practical purposes, the variation of the difference t 0 2 - t Qj can be negligible (within the uncertainties for the values of the PAw's).

Therefore, equations (2) - (4) may remain approximately invariant under a simultaneous change of the shape and the position angle of the warp PAw, in such a manner that the new value of PAw is relatively close to the old one. In the subsequent discussions, we will refer to this

(approximate) ambiguity as the shape-PAw (SP) ambiguity. The SP ambiguity will appear only In models that are both weakly warped and weakly twisted. It may be resolved by better observations (especially of the outer regions of the galaxy) and by high-resolution theoretical modeling 147

in order to accurately determine both the structure and position of the

warp. As will be made clear in §IV, the SP ambiguity does not arise for

galaxies with very strongly twisted warps, such as M83 and NGC 5055.

f) Properties of the free parameters of the Model

Our discussion of the free parameters of the model would not be

complete unless we described each of their properties and how their

variation changes specific features of the Isovelocity and surface density

maps. The following discussion is based on numerous experiments that we

have performed in order to examine the variation of all free parameters.

1) Warping angle and central disk: a) Velocity maps: The Influence of

the warping angle and the central disk is normally seen on the zero-

velocity contour, along the projected minor axis of the velocity maps.

This contour remains straight for the entire extension of the central disk. Then, it bends in a direction depending on the direction of

twisting. The degree of bending increases as the values of the warping

angle increase. A significant part of the contour will appear nearly

straight even if the central disk is equal to only the reference disk.

The direction of bending is important because models can be generated with

the overall pattern twisting in the correct direction, but with the zero-

velocity contour bending in the wrong direction. The extent of the

central disk also determines the shape of those parts of the contours that

are parallel to the straight part of the zero-velocity contour and that

are associated with "solid body" rotation: the contours develop sharp

spikes near the galaxy center for small central disks, i.e., they appear

to have no "solid body" rotation part at all. 148

b) Surface density maps: In the region of the central disk, the contours form concentric ellipses without abrupt density gradients. The presence of "spiral arms" In an HI surface density map usually suggests very strong warping. HI arms are seen, for example, 1n the surface density map generated from RLW's model of M83, or 1n the map generated by

Schwarz's (1985) model of NGC 3718. It is Important to emphasize that the

HI spiral arms observed in M83 are not physically related to M83's optical spiral arms. In fact, the two sets of arms turn 1n opposite directions.

2) Twisting angle and shape: a) Velocity maps: The influence of the twisting angle and the shape is seen from the direction that the pattern twists. The twisting of the observed contours increases with increasing twisting angle, but the PAw has to be known (see subsection 3 below). The direction of twisting 1n the model does not have to be the same as the direction of the observed twist 1n the velocity map (see all the SP ambiguous models 1n §IV for which one of the two best models has the observed twist turning in the opposite direction from that of the twisting of the rings). The different kinds of twists and their relationship to the classes of warps have been discussed in §IIc.

b) Surface density maps: Very strong twisting displaces the steep gradients 1n a surface density map from position angles on the sky

(y ± 90“) at which they would automatically occur If an unwarped, standard model was inclined with respect to the line of sight.

3) Position angle of the warp: a) Velocity maps: For most galaxies that are n°t viewed directly edge-on, there Is no direct hint about the location around the disk at which the warp begins to occur. The main 149

characteristic of PAw is that it gives us freedom to move the entire warped pattern to different locations around the disk. As a resu lt, strongly warped patterns can be camouflaged from observed maps, as no hint of warp or twist can be seen there; conversely, relativ ely weak warps can be exaggerated in the maps. The latter case accounts for the fact that some of our models have smaller warping and twisting angles than those found by other authors (see §IV).

b) Surface density maps: Strong twists also force the surface density contours to turn and develop "bridges," "wings," or "HI spiral arms," but only 1n the presence of strong warps. This behavior, however, also can be camouflaged if the PAw is chosen properly. In real galaxies, significant twisting of density contours does not appear to occur very often (but see NGC 3718 [Schwarz 19851 and the surface density maps of NGC

5055, M83, and M33).

4) Rotation curve: The rotation curve of Eq. (la) influences the velocity maps in two distinct ways: For the same velocity interval, higher values of Vc result In more contours at the outer regions of the disk. Steeper central rotation curves (smaller Rc’s) force the central part of the contours to stack up closer to each other and closer to the zero-velocity contour. A rotation curve that drops at the outer regions normally forces some of the high-velocity contours (I.e., contours of

1soveloc1t1es with values absolutely larger than the velocity of the zero- velocity contour) to loop around and close. If it rise s at the outer regions, the ends of the high-velocity contours are forced to open. 150

g) Determination of the free parameters of the model

The procedure followed to determine the five free parameters for each model is iterative and based on trial and error and visual comparison between results. In the first step, we arbitrarily choose the warping and twisting angles and try both shapes and different PAw's. A grid of about

40 models needs to be constructed before we sufficiently know the PAw's for each shape, if equally good models exist for both shapes. (Models with no warping and/or no twisting are always constructed but they fail to reproduce the observations in most cases.] Then, we improve each model by readjusting At and a w . fin ally , we keep PAw, At and a w constant and vary the extent of the central disk and/or the rotation curve. If the result is still not satisfactory, we readjust PAw, with all the other parameters constant, then vary At, aw again with the rest of the parameters constant, or, in some cases, we vary the inclination of the central disk from its published value. The entire procedure takes about 100 models for each galaxy's velocity maps, but iteration may continue if certain features

(for example HI spiral arms) need to appear in the surface density maps.

In practice, the entire parameter space is explored, i.e., s = ±1, m = 2 to 20, t Q = 0° to 360“, At * 0“ to 15°, and aw = 0“ to 10". Values of At > 15" or a w > 10“ usually lead to extremely distorted contour maps and should not be considered seriously.

As we mentioned in §1, some galaxies (for example NGC 2805 and

3079) are extremely disturbed. Large scale asymmetries dominate and the contour maps are not symmetric. We attempted to model these galaxies using a somewhat different procedure, and our resu lts can only be compared to the observations of a certain side or part of the galaxy. For NGC

2805, we had to ite rate between all the above parameters and the unknown 151

rotation curve parameters in order to additionally determine Rc and Vc.

For NGC 3079, the warping angles are not smoothly varying. In order to determine their values locally, we had to construct many additional mode Is .

III. A COMPARISON WITH KINEMATICAL MODELS BY OTHER AUTHORS

In 1974, RLW were the first to construct a tilted-ring model for

M83, and Rogstad, Wright, and Lockhart (1976; hereafter RWL) used the same approach to model M33. Roberts and Whitehurst (1975) used a sim ilar model to explain their observations of M31. Newton and Emerson (1977) used a model to derive the warped structure of M31 based on observations of the extreme NE and SW sides of the galaxy. Reakes and Newton (1978; hereafter

RN) modeled their own observations of M33. Rogstad, Crutcher, and Chu

(1979; hereafter RCC) used RLW's original model to analyze the structure of NGC 300. Newton (1980a,b) used a tilte d -rin g model, described by

Emerson (1976), to model his observations of IC 342. Bosma (1981a) used his own version of a tilted-ring model to determine the structure of the warp for NGC 2841, 5033, 5055, and 7331. Schwarz (1985) used a model similar to Bosma's in reproducing results from his observations of NGC

3718.

a) General remarks

The relevant literature on klnematical, tilted-ring modeling 1s scattered and, as a whole, difficult to digest. Most authors have neither discussed the details of their modeling techniques nor carefully defined the different parameters Involved. In addition, although all published models are sim ilar to one another, the results have been given in many 152

different ways making 1t d iffic u lt to make a direct comparison between models. Some of the above authors have not specified how they defined the different parameters or how they referenced them; others have not published velocity maps derived from the models; still others have not oriented the observer correctly relative to the projected model on the

sky, or have published the model's velocity maps after smoothing them out by using a beam, in some cases, or unsmoothed, in other cases.

Observations of the same galaxy, done at different resolutions, also affect the resu lts from modeling, as in the case of IC 342.

We thought that it would be useful if we presented the specifics of those differences when we present and compare our results with those already published, in §IV. Below, we only discuss the differences between our model and those previously published, putting the emphasis on the definition and referencing of all the parameters of a tilted-ring model.

To simplify this discussion, the differences can be classified 1n six categories: differences in the position of the warped structure, in the overall orientation of the model relative to the line of sight, in the magnitude and direction of twisting, in the rotation curves, in the actual values of warping and twisting angles, and in the definitions of the model parameters.

Our model is the only one to study different positions of the warp around the disk, utilizing the PAw. Some authors start warping from the projected major axis (It worked the very first time for M83), others do not even know the true value of tQ since they did not search the entire parameter space (Newton 1980a; Bosma 1981a). The parameter PAw turns out to be very important because i t can to tally change the velocity and density maps of a model even if it is varied by as l i t t l e as 10U-2CT. 153

No previous work has resolved the ambiguity 1n the inclination of the unwarped disk. Schwarz (1985), because he was studying a complicated system composed of two disks with different orientations, described the ambiguity but did not resolve 1t. Our model, based on the assumptions described in §IId3, easily resolves this problem. The position angle of the major axis of the model is referenced differently by the above authors. No other models have predicted the SSP and SP ambiguities either, because nobody was looking for them.

In their approach to twisting, some authors decide to present the angles of their models relative to the plane of the sky in order to demonstrate that they match the observed tw ists (Bosma 1981a). Others present the angles in the reference frame of the galaxy in order to provide information about the true structure of the warps (RLW). They a ll, however, tend to twist the rings of th eir models in the direction they see the pattern of the velocity map twisting. In our model, both the magnitude and the direction of twisting are free parameters and have to be found independently by the procedure outlined In §IIg. Our results also show that the amount of twisting is not related to the observed deviation of the galaxy's major axis from a straight line, and the direction of twisting Is not always the same as the direction of the tw ist in the velocity map. In both cases, the overall orientation of the model relative to the observer can change the results, therefore, It also has to be taken Into account before any comparison between structure of the model and structure of the contour maps Is successfully made.

Instead of using a tr1al-and-error, eye-bal1 fitting technique like ours, some authors determine the rotation curves for their models (1f they can) In addition to the warping and twisting angles, using an error 154

minimization technique of each ring's mean velocity value. This technique may often fail because either the mean value cannot independently be determined at the warped region {as for NGC 300, in RCC), or the error

(dispersion) around the mean value is not truly a minimum, if the inclination is high (as for NGC 7331, 2841 in Bosma 1981a). On the other hand, numerical experiments have convinced us that our models do not require a highly refined rotation curve (see §IIb). Thus, we decided to use the simple rotation curve of Eq. (la) with values taken from published data. In the case of NGC 2805, for which a rotation curve is not available, we determined Vc and Rc/R0 by trial and error (see §IV).

Roberts and Whitehurst (1975) realized that, because of the symmetries of the model, there were many equivalent families of values for the twisting and warping angle. They decided that the best model for M31 would be the one with the smallest t's and w's with respect to the central disk, and, at the same time, with a smooth variation of those parameters. In 1976, RWL realized that their model of M33 was not unique, and RN presented a different model for the galaxy. Both models did have, nevertheless, a smooth varation of t ’s and w's. This is also true for many of the other published models (RLW, RCC, e tc .), but not for most of

Bosma's (1981a) models. We have the following comments to make with respect to the above two points: a) Apart from local disturbances, It is preferable for the global variation of the twisting angle to be smooth, after all this was one of the main conclusions of CT (i.e., the particle orbits all precess 1n the same direction). This is not necessarily true, of course, for the warping angle (cf. NGC 3079 In §IV). We will discuss the dynamical implications of this point 1n §V. b) A good model of a galaxy is certainly not always unique, neither does it have to include the 155

smallest t's and w's. This assumption has to be tested before any

conclusion is drawn. Specifically, we find that, for most galaxies, the

true t's and w's may be taken to be the smallest determined for the model,

but there are cases (M83, NGC 5055) where we have to choose higher values

simply because they better reproduce certain features of the contour maps

(see § 1V).

b) A scheme for defining the parameters of a tiIted-rinq model

Differences exist in the literature concerning the definitions of

the parameters Involved in the tilte d -rin g model. This 1s simply a resu lt of the fact that each investigator is accustomed to using his own system of definitions. In an effort to standardize published and future tilted-

ring models, thereby allowing for easy comparisons, we propose the

following system of definitions for the model's parameters:

1) The position angle on the sky of the projected major axis be measured from north through east to the blue-shifted side of the axis.

2) The inclination of the central disk be given as the smallest

angle iQ for which cosiQ = b/a (see equation 1 11)). In addition, the orientation of the s.e. of the galaxy (I.e., near or far), relative to the observer should be tabulated. Otherwise, the position on the sky of the

nearest side of the galaxy should be stated explicitly.

3) The position angle of the warp for ring "m" be measured from

the blue-shifted side of the major axis 1n the direction of disk rotation

and to the position of the "positive" warp (see §IId2).

A) The orientation of the "upper" hemisphere of the galaxy (see

§IIa) be given 1n order to clarify the orientation of the observer

relative to the projected image on the sky. 156

5) The direction of twisting be given in terms of the Implied

shape (oblate/prolate) of the potential, according to the definitions of

lie.

6) The twisting angle (t) be measured from the position angle of

the warp, positive in the direction of disk rotation.

7) The warping angle be measured from the central disk and be given as a positive value corresponding to the "positive" warp.

We use the above definitions 1n §IV to present our results and those of some of the above cited authors. All the results are summarized

1n the Tables of §V, also in the above format.

c) Discussion of Bosma*s mode 1s

As we mentioned e a rlie r, most of Bosma's published resu lts are described in terms of model parameters projected on the sky rather than in the frame of the galaxy. In the remainder of this section, we present the basic trigonometric equations that allow us to transform Bosma*s (1981a) results to our "language," and we apply these equations to four of his models. (Petrou 11980) has performed a similar transformation for two of

Bosma's models -- NGC 2841 and 5055 -- but did not publish the d e ta ils .)

In figure 4, the notation 1s as follows: w and u are the warping angle and the twisting angle, respectively, as we measure them; 1 and a are the inclination and the position angle of an orbit, respectively, as

Bosma defines them; and iQ 1s the Inclination of the galaxy's central or reference disk to the plane of the sky. By using the laws of sines and cosines of spherical trigonometry, we find (see also Schwarz 1985)

cosw = cosi o cosi + slnl o sini cosa , (13a) x ' 157

and

sinw = sini sina/slnw . (13b)

In order to define his tilte d ring models quantitatively, Bosma has usually drawn diagrams showing how 1 and a vary with radius for an

individual galaxy. We prefer to Illustrate, principally, how w and ui vary with radius because they reflect an Intrinsic property of the modeled galaxy disk. Only If a * 0 at all radii can we use Bosma's diagrams to read off the angles w and u> directly--1n this case, w = i - i and

= 0. In all other cases, as equations (13) Indicate, w and u cannot be retrieved triv ia lly from i and a. Moreover, since we do not know whether the determined values of u correspond to the ascending or descending line of nodes for each ring, equations (13) admit two sets of solutions that correspond to a "positive" and a "negative" warp. To make matters worse, equation (13b) permits two distinct solutions for w, independent of the sign of w. So, in mathematically transforming from the plane of the sky back to the frame of the modeled galaxy, four Independent solutions for each ring are formally allowed: If we assume, from equation (13a), that w

> 0, then both l, and lBCT-w from equation (13b) are possible solutions; 1f we assume that w < 0, then we obtain both -ui and 180°+w. Finally, the position angle of the warp can only be determined after we know the first non-zero value of w = t„;o this value must be subtracted from all other values of u to produce the twisting angles t relative to PAw. Therefore, starting from Bosma1s published parameters for each galaxy, we can construct four models, each having a different value of tQ; two of them are oblate and two are prolate. 158

The ambiguity of the true position of the ascending line of nodes

1s resolved easily 1f we plot the velocity and density maps for all four models: two of the models produce patterns on the sky that are en tirely inconsistent with the observed maps. For example, for NGC 5033 we obtain the following four models from Bosma's resu lts: a) t Q = 148:7 with t < 0, b) tQ = 328:7 with t < 0, c) tQ = 3K3 with t > 0, and d) tQ = 211:3 with t > 0 (w > 0 for all models). The derived maps reveal that models a and c are inconsistent with observations. We therefore conclude that the values of w that we get from Bosma's model for NGC 5033 refer to the descending line of nodes. (We also find this to be true for the other 3 galaxies that Bosma has modeled.) We are left, then, with two generally consistent models for NGC 5033: model b, which 1s prolate and model d which is oblate in shape. This is how the SP ambiguity arises in Bosma's models.

Using this method of analysis, we conclude that Bosma's models for NGC

5033 and 7331 are SP ambiguous, but NGC 2841 and 5055 are uniquely oblate and prolate in shape, respectively. (A similar analysis of Newton's 1980a model for 1C 342 shows that his model is SP ambiguous, but Schwarz's 1985 model for NGC 3718 turns out to be uniquely oblate.) All the above results are consistent with ours, except those concerning NGC 5033. The main reasons we resolved the SP ambiguity are that we Insisted on stronger and smoothly varying twisting angles (for example Bosma's t ^ ^ = 36:6 as opposed to our tM)( = 54°) and we also adjusted our parameters

(tQ, at, aw) to get a better fit to the observed maps after we knew the two values for t Q. In other words, even though Bosma's model for NGC 5033 is SP ambiguous, 1t is not the best that could be constructed. Moreover, after readjustment of the parameters, our values for tQ are uncertain by

+10“ while the values presented above for Bosma’s models b and d are 159

uncertain by ±25°.

Bosma's parameters for four galaxy models, NGC 2841, 5033, 5055, and 7331, as extracted from his published plots, and the transformed values of the parameters are listed 1n Table 2. The corresponding parameters from our modeling are also listed to facilitate a direct comparison. Even though our modeling was totally independent from Bosma's we have found results that are in good agreement with Bosma's resu lts. In the case of NGC 7331, however, there is a clear disagreement of resu lts. We have reproduced Bosma's model and confirmed that it is really an acceptable alternative to our model. The reason that both models are good is the high inclination of NGC 7331 and the extremely small warping angles of both models. In other words, just because all the rings of the model are at an inclination of 75°-80u, even 1f they are twisted by - 90“, still the twist can be camouflaged.

At this point, we would like to point out that equations (13a) and

(13b) are very useful since observers can easily measure 1 and a by measuring the b/a ratio and the position of the projected major axis of different isophotes in their observed contour maps. Those who use equations (13), however, should experiment with models 1n order to find out the true position of the "positive" warp, and should keep 1n mind the p o ssib ility of another, SP ambiguous model.

The Inverse transformat ion of equations (13a) and (13b) may also be useful in converting our results to the quantities i and « measured directly from the observed Isophote maps. The relevant equations for this transformation are:

cosi = cosi„ cosw - slni sinw cosw , (14a) o o 160

sino = sinw s1nw/s1ni . (14b)

These equations can be used to construct inclination and position angle diagrams similar to those given by Bosma (1981a). We emphasize, however, that the set of parameters (1,a) does not describe the true structure and position of a warp, but simply its projection on the plane of the sky.

IV. RESULTS FOR 15 NORMAL SPIRAL GALAXIES

Our results are presented for each galaxy separately. If the method for the determination of the parameters 1s different than that outlined at the end of § 11f. we briefly describe it. We discuss and compare our models with previous models, 1f any, for the same galaxy. For the reader's convenience, we also present the results from each galaxy's observations (with the permission of several authors). We emphasize, however, that there are not many high resolution HI observations of spiral galaxies, especially of their outer fringes where the warps occur. For that reason, we only try to achieve a certain degree of agreement with the existing observations; better quality observations are needed before we can try to determine the free model parameters with much higher accuracy. We are able, nevertheless, to draw some definite conclusions

(presented 1n §V and §VI), about the structure and kinematics of spiral galaxies, even with the limitations imposed by the observations' current quality. In the following description of models, the galaxies are ordered with Increasing difficulty of modeling. Optical photographs for most of the modeled galaxies are shown 1n Figure 5. Beyond Fig. 5, Figures with indices a,b present the observations. Figures with Indices c,d present our 161

best models, and other results from our or other authors' modeling are shown with indices e, f, g, etc. Most of the modeled galaxies have been included by Bosma (1981b) in a sample of galaxies for which high resolution radio observations (ratio of galaxy's Holmberg radius to beam size larger than about 5) have been published. In order to limit the extent of this paper, the discussion of the last 5 galaxies--NGC 3198,

6946, 3718, 3079, and 628--1s not accompanied by figures. [NGC 3198 and

6946 are not significantly warped; NGC 3718 has been extensively modeled by Schwarz (1985); for NGC 3079 we obtained only a moderately good fit; and for NGC 628 the tilted-ring model failed.!

1, NGC 5033

NGC 5033 has been observed and modeled by Bosma (1981a). On the sky, its pattern is obviously twisted clockwise (Figure 6a), an Indication that a kinematlcal warp (large scale, syrnnetric deviation, Bosma 1981b) exists in the galaxy. The projected major axis deviates by - 12° from the inner to the outer regions of the galaxy. The maximum warp and twist in

Bosma's model are wmax = 11“9 and tmax = ±36?6 (see Table 2), and tQ =

328?7 (or 211?3). There are two problems with Bosma's model, shown 1n

Figure 6e: a) The pattern is not sufficiently twisted, on the contrary, the high-velocity contours appear to close. This is a combined result of

Bosma's using a rotation curve that falls at large radii and his

Implementing only a weak twisting of the rings, b) Twisting by a maximum of 36?6 cw from tQ = 328?7 results in a steep projected density gradient that is located at the western and eastern sides whereas Figure 6b shows that steep gradients are observed at the NE and SW sides. 162

In order to generate a model whose projected density contours show steep gradients where they are observed, we found 1t necessary to construct a model whose rings are strongly twisted cw--a smaller PAw than

Bosma's also helped. Isovelocity and density maps derived from our model, shown in Figures 6c,d, satisfy these requirements and resolve the above problems. The relevant parameters are t Q = 3 1 5 “ , At =■ - 6 ° , aw = 1 ° , Vc =

220 km/s, Rc = 2* the central disk is composed of 10 rings, wmax = 10°, tmax = -54“ and the shape is prolate. Figures 6c,d clearly show that both the direction of twisting and the locations of the steepest gradients are c o r r e c t .

As Is discussed at some length in CTS, we managed to construct another model with all the parameters unchanged but with tQ - 170“ and an oblate shape. Even though the direction of the observed twist is correct in this alternate model, the maps do not show the main features discussed above. We conclude that only the prolate model adequately describes the kinematics of this galaxy. Its parameters are summarized in §V.

2. NGC 5055

Observed and modeled by Bosma (1981a), NGC 5055 possesses an obvious kinematical warp in Its outer regions. Its projected major axis turns in a cw sense by 4“, then 1t turns back in a ccw sense by about 20°

(Bosma 1981b), a sign that its central disk is either disturbed or dominated by non-circular motions. The velocity map from Bosma's own model 1s reproduced here 1n Figure 7e. Bosma attempted to fit the observed turns in both the inner and outer velocity field and, as a result, the warping and twisting angles he used do not vary smoothly

(monotonically) with radius. In order to more completely judge the 163

adequacy of Bosma's model we generated a surface density map from his published tilted ring parameters. (The relevant parameters are listed here in Table 2.) After reconstructing Bosma's model, we found that the surface density map does not reproduce the two "bridges" of HI observed to be running roughly east to west both on the northern and southern sides.

Our reconstruction of Bosma's model 1s shown in Figures 7f,g to facilitate a direct visual comparison.

The projected Images of our own best-fit model for NGC 5055 are shown in Figures 7c and 7d. We were interested 1n modeling only the outer regions well, so our model possesses smoothly varying warping and twisting angles. Warping angles somewhat larger than Bosma's are needed to reproduce the main features of the observations. The two "bridges" of HI are clearly evident in Figure 7d. The parameters of our best model are tQ

= 260“, at = -10“, aw = K l Vc - 213 km/s, Rc = 1, the central disk 1s composed of only 2 rings, and the shape is prolate. The two looped contours in Figure 7c are somewhat larger than the corresponding contours of the observation (Figure 7a) due to the lack of a falling rotation curve at the outer parts of the model. As can be seen in Figure 7f, this problem is easily corrected by using Bosma's rotation curve, but we did not proceed to do so in order to maintain a degree of consistency throughout this paper. The best oblate model (see CTS) that we constructed, with tQ = 270°, does not compare favorably with the observations. Actually, this model 1s not SP ambiguous with our best prolate model. The ambiguity 1s entirely resolved due to the very strong twisting of the rings. We conclude, therefore, that the halo surrounding

NGC 5055 1s prolate 1n structure. 164

Another point should be mentioned, however, relative to the best prolate model. We succeeded in generating a model with all the parameters unchanged, but a much smaller twisting angle (At = -5°). This model satisfied the usual assumption of minimum twisting (see §111, for the relevant discussion), but it was ruled out because its surface density map did not compare well with the observations.

3 . NGC 3 0 0

This galaxy was originally observed in HI by Shobbrook and Robinson

(1967). In 1979, RCC provided both higher resolution observations of the galaxy (Figures 8a,b) and a tilted-ring model. They realized that no model with a warp starting from the galaxy’s projected major axis (as was customary) produced a good fit to the observations. Instead, the warp had to start about 60° away from the major axis. Like us, they also measured the position angle from north through east to the blue-shifted side of the major axis and referenced the twisting angles with respect to this axis, negative in the cw direction. (They claim that the warping angles were referenced to the near (southern) side of the galaxy, therefore, 1n this case, their Inclination angles should be considered as negative (not positive as given In RCC) because they are referenced with respect to the blue-shifted side of the major axis and correspond to the negative warp.| The 1soveloc1ty contour map that was published from their model 1s reproduced here 1n Figure 8g.

In modeling NGC 300, we found that the best model Is SP ambiguous and has Vc = 94 km/s, Rc = 4, At = ± 2 “ 5 , Aw = 2“, and a central disk of 8 rings. The best prolate model has tQ = 120° and the best oblate model has tQ = 80°. The prolate model Is shown In Figures 8c,d and the oblate model 165

in Figures e,f. As has been discussed, our density maps do not reproduce the asymmetry seen in the southern side nor the structure of the central region. Our prolate model, however, 1s in very good agreement with RCC's oblate model (compare Figure 8c and 8g), so we cannot determine the shape of the halo even though the initial guess, based on the results of CT, would be oblate because the observed twist turns in the opposite direction from the spiral arms. The two models that we have constructed are in fact

SP amgibuous but the contour maps are not exactly the same because of the fairly strong twisting angles that are required (tmflx = ±27:5).

4. HGC 7331

Observed and modeled by Bosma (1981a), NGC 7331 possesses a weak kinematical warp (seen in the outer regions). The published velocity map from Bosma1s model Is reproduced here in Figure 9g and was discussed in §1 lie.

We were unable to produce a unique model for this galaxy because of the SP ambiguity, originating from the high Inclination and very small twisting. Our best oblate model is shown in Figures 9c,d with tQ =

240", At = +4U, aw = 0:5, Vc = 250 k m / s , Rc = 3, and a central disk of 12 rings. The best prolate model Is shown 1n Figures 7e,f with tQ = 300° (1n agreement with the theoretical discussion of §IIe), At = -5", an d a l l other parameters the same as the oblate model. The observational results show an abrupt increase of the twist at the outer region, particularly at the s.e. This feature does not appear 1n any model. Attempts to generate

1t by increased twisting and warping 1n the outer two rings failed. We conclude, therefore, that this disturbance is a local asymmetry. 166

5. NGC 2841

Observed and modeled by Bosma (1981a), NGC 2841 possesses a kinematical warp (seen at the outer regions). The published velocity map from Bosma's model is reproduced here in Figure lOe. The twisting angle is, once again, not smooth. This model was presented 1n §111. Me would only add here that the twist in Figure lOe is not convincing becauseit does not keep twisting at the outer edges along the major axis.

He found only one acceptable model. It has tQ = 40°, at =

+5°, aw ^ 1", Vc = 300 km/s, Rc = 3, a central disk of 5 rings, and an oblate shape (Figures 10c,d). Notice that the maximum twisting angle here is t = 70“, but, still, the resulting velocity map shows only a weakly twisted structure (I.e., the real magnitude of the twisting is camouflaged). This relatively large degree of twisting, however, correctly forces the steepest gradients to appear close to the ends of the minor axis 1n SE and NW (compare Figure 10b to lOd).

6. H 63

M83 was observed and modeled by RLH, in 1974. Their observations and the results of their modeling are shown 1n Figures 11a,b and llg.h, respectively. RLH start twisting from the major axis, and the last ring of their model B has t = +141°, w = 37°. The large warping required to explain the kinematics of this galaxy 1s also responsible for generating

HI spiral arms that twist 1n a direction opposite to the optical spiral arms (see Fig. llh). To illustrate this, we constructed a model with tmax

= + 1 9 5 “ , b u t wmax = 16“ at the outer ring. This model (Figures lle,f) produces velocity contours that are very similar to the observed contours but it produces only a hint of Hi arms. Its parameters are tQ = 30°, at = 167

+ 1 3 " , aw = 1 ‘ , Vc = 180 km/s, Rc = 3, the central disk has 4 rings, and an oblate shape.

Our best model (Figures llc.d) includes a stronger warping (see

§111), to reproduce the spiral arms seen in Figures llb.h. Its parameters are tQ = 30", At = + 1 1 : 5 , Aw = 1:6, Vc = 180 km/s, Rc = 3, the central disk has 4 rings and it 1s oblate In shape. No adequate prolate model exists for M 83, as was predicted by CT. We confirm, therefore, the quality of RLW's model and the fact that the warp of M83 starts very close to the major axis.

7 . M 33

This galaxy was observed and modeled by RWL In 1976 and by RN in

1978. RWL discovered that they had to start twisting from the major axis. They did not publish the model's velocity map and, for comparison purposes, only their model1s surface density map is available. Hence, for this galaxy our classification of diagrams is different from the one followed elsewhere 1n this section. Figure 12a is a reproduction of the

(RWL) observed map for the primary velocity component, but Figure 12b Is a reproduction of their mode 11s density map. Figures 12c,d show the maps of our best model. To produce Figure 12c, we have followed RWL and have subtracted the secondary velocity component from the map (eliminating thus contour twisting).

Our best model has tQ = 180“, At = 4 : 5 , aw = 3:6, Vc = 107 km/s, Rc

=■ 2, and a central disk of 9 rings. These model parameters are in complete agreement with the RWL model, and support the following arguments of RWL: a) A strong warp Is present 1n M33. b) The model's orientation 1s such that the western side 1s closer to the observer (the 168

s.e. Is "far" from the observer, to put it within our terminology; see

§1 Id3 and the Tables of §V). c) The model generates "wings" close to the ends of the major axis, d) The surface density drops off very abruptly at the ends of the minor axis, as a result of the specific orientation of the rings. We would only add that the above results show that the halo of

M33 is oblate in structure.

RN presented composite contour maps for M33, by combining all existing observations* Their own observations were of the outer regions of the galaxy. By observing farther away from the center than RWL, they managed to reveal a strongly twisted pattern in the velocity map. This result, together with their successful modeling, 1s very Important, because it clearly demonstrates a general point that we made earlier: observations of the outer parts of a galaxy should easily reveal twists and, therefore, warped structures, which may be kept hidden otherwise.

We have also produced a model of M33 based on the observations and modeling of RN but we will not present its maps here because our results are 1n complete agreement with the results of RN. For this model we find that t Q = 270'*, At =■ 3.*3,Aw = 5?3, a central disk of 10 rings, and oblate shape. RN claim, however, that their results disagree with RWL's results, a fact that can be seen from the different parameters as they presented them (projected angles on the plane of the sky) or as we present them 1n our own terminology (see also Tables 4 and 5 below for a summary). We believe that this disagreement should not be taken seriously since the region of the disk modeled by RWL 1s well within the extended region modeled by RN, and can be entirely resolved by a model that Incorporates

RWL's values for the Inner disk (tQ = 18C°) and, then, 1t uses somewhat different values of the twisting and warping angles than RN's, in order to 169

be extended In the outer regions that were observed by them. This change in the values of t and w would not be in conflict with the values derived by RN since the latter can be varied by ±5" or more with no noticeable change of any general trend (twisting, steep gradients) being produced in the contour maps.

8. M 31

M31 has been observed and modeled several times: by Roberts and

Whitehurst (1975), Newton and Emerson (1977), Brinks (1984), and Walterbos

(1986). The results of their modeling are not given as velocity and density maps. However, all authors predict that the outer regions of M 31 are warped and Roberts and Whitehurst actually compute a tmax = 30° and a wmax = ^"* the^r "individually smoothed" model. This warp is camouflaged due to the high inclination of the disk. Figure 13a reproduces the velocity field published by Emerson (1976) while the density map of Roberts and Whitehurst (1975) is shown in Figure 13b. A first look at Figures 13a,b shows no hint of a warp. Emerson (1976) modeled his observations using a totally flat disk 1n order to show how the maps of the galaxy could be matched without even taking into account any warping.

The direction of the optical spiral arms of M 31 cannot be clearly seen from Figure 5h. This provides us with an opportunity to demonstrate the SSP ambiguity. Assume that the arms follow a ccw direction outwards, therefore the s.e. is closer to the observer. If this turns out to be the wrong assumption, all we have to do 1s change the shape and tQ to (180°- tQ). The demonstration becomes even more interesting as soon as we realize that the SP ambiguity also exists in this system. We have 170

constructed two equally good models with At = ±6", aw = 4“, Vc = 230 km/s,

Rc - 5: one oblate with t Q = 160“, and another prolate with t Q = 210“.

Our results on M 31 are tabulated in §V after the SSP ambiguity 1s

resolved, however, since the widely accepted direction of the arms of the

galaxy is cw outwards (i.e., the s.e. Is away from the observer).

Resolving the SSP ambiguity results in tQ - 20“ and a prolate model or t0

= 330“ and an oblate model. The contour maps from the prolate model are

shown in Figures 13c,d. In this model, wmax = 20“, but no Important

features are lost from the maps If we only use wmax = 5“ that was

suggested by Roberts and Whitehurst (1975).

9. 1C 342

1C 342 was observed by RSR 1n 1973. More recently, Newton

(1980a,b) has published higher resolution observations and a tllted-ring model of the galaxy, clearly indicating that a warp exists 1n Its outer regions. |The RSR observations did not reveal a twist in the velocity contours while a twist is obvious in Newton's (1980a) observations.) We determined the following parameters from modeling: At = ±3°, aw = H5, Rc

=■ 3, and a central disk of 8 rings. The best oblate model has tQ = 270° and the best prolate mode! has tQ = 300°. Figures 14a,b are reproductions of Newton's (1980a,b) observed maps: Figures 14c,d present our best prolate model; Figures 14e,f show our best oblate model; and Figure 14g is a reproduction of the velocity map from Newton's model. The observed asymmetry of the western side cannot be reproduced by the models. We have c la ssified our results as SP ambiguous, but one might argue that the prolate model 1s best (as Is also predicted from the direction of the observed twist) because no oblate model shows the high-velocity contours 171

twisting in the correct direction (cw) on the SW and NE sides.

!0- NGC Z8Q5

Observed by Bosma et a l. (1980), NGC 2805 is extremely disturbed and its "circular velocity" rotation curve is not known. Our modeling is based on the southern side of the galaxy. We first determined its

Inclination (iQ = 41“) from the b/a ratio of the axes. Then, Vc, Rc/R0, and the extent of the central disk were kept fixed, and the PAw was determined for the two shapes. Once the PAw's were approximately known, we went back to determine the rotation curve parameters. Finally, we

Iterated between the rotation curve parameters and A t, aw . We repeated the procedure, with th e new values for A t, aw , Vc. The determination of the rotation curve was based on the properties described in §IIf4. We find that tQ = 60u, At = +3“, aw = 3°, Vc = 60 km/s, Rc = 4, the central disk is made of 12 rings, and the shape can only be oblate. Of course, our maps (Figures 15c,d) do not reproduce the observed asymnetry (in

Figure 15a the zero-velocity contour points NW at both ends of the minor ax i s ) .

11. NGC 3198

NGC 3198 was observed by Bosma (1981a) and modeled as a fla t disk. Our models use an extended central disk, and reproduce a zero- velocity contour that bends at large radii (north at the western side of the minor axis), and a slight twist in the cw direction. Because of the

SP ambiguity, two equally satisfactory models were found. Both have Vc =

150 km/s, Rc = 3, and a central disk of 12 rings. The oblate model has tQ

= 60°, and the prolate model tQ = 75°. The twisting and warping angles 172

are given in Table 6. After a t max = 12" the warp seems to have a constant t = 6" in the last 3 rings. The value tmax = 12" seems to help the outer velocity contour bend abruptly, but it should not be taken as a global feature.

NGC 6946

Observed by RSR in 1973, NGC 6946 exhibits an asyrmietry between its northern and southern sides. We based our modeling of NGC 6946 on its northern side. A model was constructed in which the zero order contour bends in the ccw direction on both ends of the minor axis. Its parameters are tQ - 50", at = O', aw = 2", Rc = 3, and a central disk of 15 rings.

Because at - 0", there are two important properties of this model: a) We cannot tell its shape (SP ambiguity), b) The warp may be steady-state, high resolution observations, however, may overturn both of these properties (as Newton's 1980a,b higher resolution observations did in the case of IC 342), if they reveal a twist in the outer regions of NGC 6946.

1 3 . NGC 3 7 1 8

Observed and modeled by Schwarz (1986), this galaxy is very

interesting because its disk is composed of two components. A disk in the central region at an inclination of almost 90" (edge-on), and an outer, warped disk. Schwarz's model indicates a maximum warp of - 44°, a maximum twist of - 15" in the inner rings, and a steady-state twist (I.e., at ■= 0") in the outer rings. Our best model has Vc - 255 km/s, Rc = 3, tQ

= 125", fit = i : i , aw = 3:4 and an oblate shape. Equally good maps are produced by Schwarz's detailed data for the warping and twisting angles that are listed in Table 6 after they were transformed to the galaxy's 173

frame of reference. Because the warp is strong (w = 42:4) but not very J ma x much twisted (tmax = 14:5), we believe that the disk may be indeed close to attaining a steady-state warped structure. Schwarz (1985) also concluded that the disk is in steady-state. The strong warping of the rings seems to conspire with the small twisting in resolving the SP ambiguity for this galaxy.

1 4 . NGC 3 0 7 9

Recently observed by Irwin et al. (1987) and Irwin (1988), this galaxy is extremely disturbed. Its rotation curve is asymmetric between the northern and southern side, and the zero-velocity contour of the intensity-weighted mean isovelocity map shows the most complicated bending among all galaxies of our sample. The disturbances in the outer regions can, however, be reproduced to a certain degree by a tilted-ring model.

We based our modeling on the appearance of the northern side of NGC

3079. A fairly good fit to the observations was achieved using a model with tQ = 120', Vc = 195 km/s, Rc = 6, and no twist (i.e., at = 0).

With at = 0, one might be tempted to categorize the warp as steady- state. It is unlikely that this is the case, however, because of the overall disturbed appearance of the galaxy and because the warping angles required by our b e s t-fit model produced an unsmooth, complicated structure. Outside of the central disk composed of 3 rings, the warping angles rapidly increase to a value of 34“, then they slowly drop to 0°, at the 17th ring; the last 3 rings are, again, disturbed at w = 10°. The shape cannot be determined, because of the SP ambiguity (at = 0°). 174

15. NGC 628

We were unable to obtain a satisfactory model of NGC 628, a galaxy that has been observed by Briggs (1982). We include NGC 628 In our sample as an example of a spiral galaxy where large-scale, oval distortions (see

Bosma 1981b; Briggs 1982) dominate the entire disk, and for which no circularly symmetric, tilted-ring model can produce good results. A model that allows for elliptical orbits may explain the kinematical properties of NGC 628.

V. DISCUSSION

Tables 3 through 6 tabulate the parameters of the tilted-ring models that have been discussed in §§111 and IV, above. Table 3 presents all the parameters that were held fixed, in almost all cases, during our present modeling. These include the position angle y of the blue-shifted side of the major axis, the inclination angle iQ of the central (and reference) disk to the line of sight, the orientation of the southern edge of the galaxy, and the values Vc and Rc/R0 used for the rotation curve.

Also tabulated are the direction of the observed spiral arms, the orientation of the upper hemisphere, and the direction measured cw or ccw on the sky of the twist of the observed velocity map, if any, relative to the direction of the spiral arms. Finally, the reference column lists the a rtic le s from which these fixed parameters were taken.

In Table 4, we translate the parameters used In other authors' models into our own terminology. These include the position angle tQ at which the positive warp starts and the corresponding. Implied shape; the extent of the central disk; and the direction the rings twist for each model. Since most authors use varying increments for the twisting and 175

warping angles, we show in lable 4 only their values at the outer edges

(ring N) of the models. Table 5 is analogous to Table 4, but the

tabulated parameters have been derived from our own b e s t-fit models. It

additionally contains our incremented values of at and aw and the

predicted shape (based on the CT results) of the potential well of each galaxy.

In Table 6 we detail the structure of the warp that has been deduced for three galaxies in which constant twisting and warping

increments were not used. The structure given for NGC 3079 and NGC 3198

has been determined from our present modeling techniques; the structure given for NGC 3718 is Schwarz's (1985) best-fit model transformed to the

galaxy's frame.

The agreement between models and observations of the 15 studied

galaxies ranges from excellent to poor. Most of the above galaxies have

HI disks with a relatively simple structure and the models are sufficient

to analyze their kinematics. Others, like NGC ?805, 3079, and 6946, are complicated systems where large scale disturbances dominate, and their

kinematics and dynamics are noticeably different from what can be

predicted by any simple tilted-ring model. We also expect that many

spiral galaxies, similar to NGC 628, will be dominated by non-circular motions (oval disturbances) where any kinematical warp that might exist

becomes totally unimportant. In what follows, our discussion is divided

into five categories: a) HI radio observations; b) Structure and

properties of kinematical warps; c) Structure of kinematical twists and

shapes of the potentials of galaxies; d) Shape of the potential and Dark

Matter Hypothesis; and e) Other (edge-on or barred spiral, elliptical)

galax i e s . 176

a) HI radio observations

High resolution radio observations of galaxies cannot alone reveal

their true structure and kinematical and dynamical behavior, but detailed modeling is also needed, before conclusions are drawn. With respect to

kinematical modeling, more observations in the 21 cm line should be obtained especially in the outer regions of galaxies because a warped

structure is most likely to reveal Itself there. Perhaps even the outer regions of galaxies like NGC 5033, 5055, 2841 t 7331, and 6946 should be observed again with higher spatial resolution. Only then can the kinematical modeling of these galaxies be improved. This Is not a difficult task to undertake given that the kinematics of these systems appear to be relatively uncomplicated. High resolution observations will also provide information that is needed to c ritic a lly analyze future theoretical studies in which the time-dependent settling of warped, HI disks is modeled dynamically in a self-consistent manner.

b) Structure and properties of warps

Detailed radio observations and kinematical modeling, performed over the last 15 years, have clearly established that neutral hydrogen is not distributed in flat, disk-like structures in normal spiral galaxies.

Warps are evident in the outer regions of the disks of most galaxies.

Tilted-ring models of several galaxies reveal, in addition, that the warps are complicated, twisted structures and, in some cases, disturbed. Our tilte d -rin g modeling shows that complicated systems--!ike NGC 3079-- probably have complicated warped structures, while others--11ke most galaxies in our sample--show smoothly warped structures that monotonically 111

vary in orientation from the center of the disk to Its outer edge. On the other hand, we have constrained the variation of tw ists to be smooth in order to construct models that are consistent with the results of the

"preferred orientation" theory.

Lxamination of Table 5 reveals several trends exhibited by the warps of our models:

a) The radii of the unwarped central disks in our modeled galaxies range anywhere from 10/£ to 7bt of the entire disk. Almost half or the modeled galaxies appear to have quite extended central disks -- being composed here of from 12 to lb rings (out of 20) -- suggesting that the gas settling process has been substantially completed in many disk systems. This fraction is probably exaggerated, however, because most of the galaxies listed as having large central disks are either observed at low resolution, disturbed, or not very strongly warped.

b) Twisting increments range from 0" up to 11H5 and warping increments range from 0:'b up to 5^3. Large values of the warping and twisting increments tend to be associated with small central disks, but the converse is not necessarily true.

c) If the maximum twisting angles lie above - 30', then the twists are strong enough to resolve any ambiguity in the shape of the galaxies, unless the disk has high inclination relative to the plane of the sky. It should be noted that large twisting angles are not necessarily associated with large warping angles.

d) The twisting angles derived for the last five galaxies listed

In Table 6 are either zero, within the accuracy of our modeling, or very small. It is reasonable to suggest that the disks in these galaxies are 178

v e r y close to achieving a steady-state orientation. Possible exceptions

are NGC 3079, because it is disturbed, and NGC 6946, because it was observed at low spatial resolution.

Guided by the above discussion, we classify the warps of galaxy disks into three distinct types according to the magnitude of the maximum

twisting angles they exhibit:

1. strongly twisted warps (t io } : Their main property is FTluX, >~ that they are strong enough for the SP ambiguity to be resolved.

Therefore, the gross geometric shape of the underlying potential well can be determined. For the galaxies of our sample which belong to th is type, we find that the maximum warping angles range from 10“ to 40“. For galaxies that have maximum warping angles more than ~ 20", we also observe

that they exhibit exotic features in their surface density diagrams (HI spiral arms, bridges, or wings).

2. Moderately twisted warps (10' < t < 30"): These warps are max - certainly real. In most cases, however, they are not strong enough for the SP ambiguity to be resolved. For the galaxies in our sample which belong to this type we find that the maximum warping angles range from 4° up to 44".

3. weakly twisted warps (t . < JO"): These warps are very weak 3 ma x ~ and may not be real 1n some cases. If they are real, the small twisting angles suggest that they are almost in steady-state. The shape of the potential well cannot be determined. For these galaxies in our sample, a

typical value of the maximum warping angle is - 10".

The largest warps that we have found do not seem to exceed about

40" relative to the orientation of the central disk. A purely kinematical model cannot be expected to explain this behavior, but the behavior might 179

have been expected from dynamical arguments. Qualitatively speaking, for a highly inclined, gaseous disk under the influence of a spheroidal, non

tumbling po ten tial, the settling times between the inner and outer regions

should differ by several orders of magnitude. In addition, for disks at high inclinations significant inflow of matter is observed in dynamical calculations (Habe and Ikeuchi 1985; Steiman-Cameron and Durisen 1988;

Christodoulou 1989). One might expect, therefore, that a disk that is settling toward its preferred orientation from a high inclination will not develop a smooth warp.

c) Structure of twists and shapes of the potentials of galaxies

Tables 5 and 6 indicate that most of the warps of the studied galaxies are transient, either oblate or prolate (their definitions were given in §IIc), but a few galaxies seem to possess almost steady-state warps, i.e., the warped regions are not significantly twisted. The twists of the transient warps, in most cases, allowed us to determine the shape of the overall potentials of the galaxies.

Some twists were found to be very strong, extending as much as 180" over the radius of the disk. For those strong twists, it is possible to distinguish between an oblate and a prolate shape of the galaxy potential.

Other twists were found to be weak, spreading out for just a few degrees over the radius of the disk. In these cases, determination of the shape of the potential was not possible, and two equally good models were found,

(one oblate and another prolate) due to the SP ambiguity. Table 5 shows that the two strongest twists are those of NGC 5055 (tmax = 170") and of M

83 (tmax - 172"5). Although the studied sample of galaxies is not statistically complete, our results suggest that tmax = 180" is the 180

maximum value that one should expect to find for the twists of the warps of normal spiral galaxies.

If a warped disk is found to be moderately or strongly twisted (as is the case for most galaxies in this study), then the presently observed warp is probably a transient phenomenon that develops as the disk settles toward its preferred orientation with differentia) precession still at work. If the warp is smoothly varying and weakly twisted (as that of NGC

6946 under the present quality of observation or that of NGC 3718), then it is probably close to a steady-state orientation. A steady-state warp such as this can be maintained if the halo of the galaxy is tumbling

(Tohline and Durisen 1982; Simonson and Tohline 1983; Steiman-Cameron and burisen 1984; Habe and Ikeuchi 1988), or if its symmetry axis is tilte d relative to the rotation axis of the disk (Dekel and Shlosman 1983; Toomre

1983; Sparke 1984). If the warp is asymmetric and untwisted (e .g ., NGC

3079), then it probably is the result of a very recent interaction between the galaxy and a nearby companion, or of a recent internal, large-scale disturbance.

We confirm the hypothesis, laid out by CT, that the direction of the twists that appear in the isovelocity contour diagram of an HI disk provides a clue as to the true shape of the underlying galaxy potential.

For example, if the twist turns in the same direction as the optical spiral pattern, then the gravitational potential is prolate-1ike. This conclusion, however, is only valid under the following conditions: 1) the spiral arms are trailing relative to the direction of rotation of the galaxy and 2) the twist is sufficiently strong for the SP ambiguity to be resolved. 181

Finally, it is Important to note that models can be constructed in

which rings twist in one direction while the isovelocity contours twist in

the opposite direction (see, for example, the oblate models for NGC 5033,

5055 in CTS). However, for all the galaxies for which we have been able

to unambiguously determine the best model (either oblate or prolate), this model's twisting angles always are in the direction of the observed

twist. Thus the best models for NGC 2841 or M83 implied oblate shapes, while that the best models for NGC 5033 and 5055 Implied prolate shapes.

With this in mind, we predict that the shapes of all the SP ambiguous models will ultim ately be resolved in such a way that they correspond exactly with the "predicted" shapes listed in Table 5. That is, as better observations become available, we will find that NGC 300 and NGC 7331 have oblate halos while IC 342, NGC 3198, and NGC 628 have prolate halos.

From the 12 galaxies in our sample for which the shape of the halo can be predicted, about 40% show the signature of a prolate halo. This result seems to question the common belief that dark halos must be oblate spheroidal structures, although we do not presently know whether the rough equality between oblate and prolate shapes is only restricted among the galaxies of our sample or is s ti l l valid in universal scales. An

interesting question, deserving further investigation, is: What mechanisms can be responsible for shaping the dark matter in galaxies?

d) Shape of the potential and Dark Matter Hypothesis

Under the Dark Matter Hypothesis, determination of the shape of a galaxy's overall potential reveals the gross geometric shape of the dark halo in which the luminous components are embedded. Under any theory of gravity with a non-Newtonian dynamical law (see all references in §1), 18?

this subject presents renewed interest, since the modified theories of

gravity implicitly carry the assumption that there is no more matter than

the luminous matter in galaxies. Our results suggest that nature allows

for prolate mass distributions of galaxies, and support the Idea that dark

halos really exist. On the other hand, theories of modified gravity

(besides the theoretical problems they face, see Felten 1984) are not

flex ib le enough to allow for prolate mass d istrib u tio n s, as the luminous matter of spiral galaxies is distributed in a disk-like, definitely oblate

structure. For that reason, the results of this work argue that theories of modified dynamics should be ruled out. (This was also the principal

conclusion of CTS.)

It is very important to emphasize that, although our conclusion is derived under the assumption that the warps of spiral galaxies are

transient features in settling disks, it is not invalidated if one decides

to adopt Sparke's (1984) idea and assume that warps are instead long-11ved modes of oscillation of the disks. In this case, as Sparke points out, discrete modes exist or are important only if a galaxy disk is surrounded by a flattened, massive halo (see also Hunter and Toomre 1969; Toomre

1983; Sparke and Casertano 1988). Similarly, another alternative mechanism, proposed by Petrou (1980), that results 1n constant precession over the entire galaxy disk, requires the existence of a dark halo (see

CTS for details).

e) Edge-on or barred spiral and elliptical galaxies

A tilte d -rin g approach has also been used to study the edge-on spiral galaxies NGC 4565, 4631, and 5907, the e llip tic a l galaxies NGC 4278 and Cen A (NGC 5128), and the Seyfert-type, barred spiral NGC 5728. Our 183

models indicate that the observed surface density maps of the edge-on galaxies (see Sancisi 1976; Weliachew, Sancisi, and Guelin 1978) can be reproduced by using very strong warps. Byrd (19 78) has argued that these warps may be illusionary, resulting from projection effects. We have also constructed models that show illusionary warps, but they all seem to be washed out in the low-density regions and they are certainly not as sharply defined as the observed warps of edge-on galaxies. We argue, therefore, that the warps of edge-on spiral galaxies are real.

Our resu lts for NGC 4278 and NGC 5728 Indicate thct the HI disks in these systems cannot be represented very well by a circularly symmetric tilte d -rin g model. That was anticipated since the disks of these systems are dominated by non-circular motions (as in NGC 4278, see Raimond et a l.

1981 and Knapp 1982) or bar-like disturbances (as in NGC 5728, see Rubin

1980, Schortmer et al. 1988). We will present our details of kinematical modeling for these galaxies separately (Christodoulou 1989), but point out that fully dynamical calculations would be useful in analyzing the structure and dynamics of these galaxies.

Our modeling of Cen A (see Christodoulou 1989 for details) has produced an interesting and potentially controversial result. Based on observations of Bland et al. (1987), we find that the dust lane is composed of two orthogonal disks and that the outer disk is strongly warped (wmax = 23“) but not twisted. This result suggests that the outer disk of Cen A is 1n a warped steady-state orientation, contrary to the comnon belief that the dust lane is dominated by differential precession of orbits and is settling toward the preferred plane. 184

VI. SUMMARY

The main points of this work can be summarized, as follows:

1) A tilted-ring model (§IIa,b,d,f) can successfully reproduce the

observed characteristics of several spiral galaxies (§ IV) whose

disks possess relatively simple structure and kinematics.

2) The warp of a galaxy disk can be transient (twisted) or steady-

state (untwisted) -- §IIc.

3) The orientation of a galaxy's disk (§11d3) with respect to the

observer's line of sight must be known in order for the SSP

ambiguity to be resolved (§11e). We have demonstrated the SSP

ambiguity for M 31 (§IV). |If spiral arms are ultimately found

to be leading patterns in some galaxies, the halo shapes derived

in this Investigation will have to be altered.]

4) If the twisting in a model is weak, the results are still

ambiguous, because of the SP ambiguity (§IIe). Examples are

M31, NGC 300, 3079, 6946, 7331, and IC 342. We have

demonstrated the SP ambiguity for NGC 300, 7331, and IC 342

<§IV). b) The twisting of the warp of a galaxy may be camouflaged. I.e.,

even though the warp is strongly twisted, the observed velocity

map may just show a weak twist (§111, §IV).

6) We propose a new, compact terminology and system of measuring

and referencing all relevant parameters of the tilted-ring

model, derived from observation or modeling (§111).

7) Our results show that the kinematical structure of a warped disk

can often be determined, but dynamical information cannot always

be extracted from a tilted-ring model (§ 11e, §IV). 185

8) Three types of warps are identified (§V) according to the

magnitude of the maximum twisting angle: strongly twisted

(tmax > 30"), moderately twisted (10" < tmax ^ 30"), and weakly

twisted (tmax < 10") warps.

9) Our results suggest an upper limit of 40" for the warps

and - 180" for the twists of normal spiral galaxies (§V).

10) In relatio n to the dynamics of spiral galaxies, strongly twisted

warps are expected to signify the continuing evolution of the

disks toward a preferred orientation, while weakly twisted warps

signify rather the near end of such an evolution (§V).

11) Extremely disturbed systems cannot be modeled satisfactorily by

a circularly symmetric tilte d -rin g model (NGC 3079, 628, 6946)

— § IV.

12) In a number of cases, we were able to determine the shape of the

galaxy's potential well unambiguously (§IV). NGC 5033 and 5055

are galaxies with a prolate shape, while NGC 2805, 2841, 3718,

M33, and M83 have an oblate shape. For all galaxies with twists

in the observed velocity maps, we were able to predict -- but

not necessarily confirm -- the shape of the underlying potential

well using the hypothesis stated by CT (Table 5, §V).

13) The existence of prolate potentials in galaxies argues strongly

against theories that modify Newtonian dynamical laws (§V). By

the same token, it provides strong support for the idea that

dark halos exist and that they dominate the dynamics of the

outer regions of spiral galaxies.

14) Higher resolution radio observations are needed, especially of

the outer regions of galaxies, before kinematical modeling can 186

be improved and before detailed comparisons with resu lts from dynamical calculations can be made. Dynamical modeling is also

needed to resolve questions concerning the size and duration of

transient warps, the twisted structure of the warped layers,

stable polar orbits, and the importance of self-gravity of the

luminous matter (§V). 187

ACKNOWLEDGEMENTS

We are pleased to acknowledge the generosity and assistance of many researchers who gave us permission to use their original results,

including permission to reprint observational data and to reproduce figures showing contour maps from their own modeling effo rts.

Specifically, we thank Drs. J. Bland, A. Bosma, N. Caldwell, D. Emerson,

J. Irwin, G. Knapp, K. Newton, M. Roberts, I). Rogstad, and E. Seaquist.

We also thank Dr. Alan Sandage for providing most of the optical photographs of the galaxies that appear in Fig. 5 of this paper. This work was supported in part by grant AST 87-01503 from the National Science foundation. 188

A PP E N D IX A

NOMENCLATURE a) Symbols

Rq - total radius of the disk.

Vc - maximum rotational velocity.

Rc - radius at which Vc is reached.

N = number of concentric, circular rings. a R = R0/N = ring width, n - number of a ring, 1 * n ■; N.

Rn - outer-edge radius of ring "n" = naR.

- angular momentum vector of the reference disk.

Jn - angular momentum vector of the nth ring. wn = warping angle of ring "n", measured between J n and Jj. m = ring number at which the warp s ta rts , 2 -- m • N.

S(R) = surface density profile of the unwarped disk.

SQ =- surface density of the reference disk. t n = w - t = twisting angle of ring "n" measured from PAw.

1Q - inclination of the central disk to the plane of the sky = angle

measured between Jj and the observer's line of sight, 0U < i0 < 90“.

«d = twisting angle, measured from the projected major axis. y - position angle on the sky, measured from North, due East to the blue-

shlfted side of the major axis of the galaxy's central disk. tQ = position angle of the warp (PAw) = W(n. q = angle at which the nth ring reaches greatest linear height in the upper hemisphere = 90“ + u . a = projected major axis of the central disk. 189

b = projected minor axis of the central disk. s.e. - southern edge of the galaxy.

i = inclination of a ring to the line of sight. u - twisting angle of a ring in the plane of the sky.

b) Definitions

Standard model: an unwarped disk, composed of N coplanar rings.

Reference disk: the innermost ring segment of the standard model.

Upper hemisphere: the hemisphere into which points.

Warping angle wn: the angle between the vectors Jj and Jn .

Central disk: an extended reference disk composed of the innermost

coplanar rings.

Warp node: located in the central disk, the ascending line of nodes of

the first warped ring (ring "m").

PAw , Position angle of the warp, tQ: the twisting angle wm of the first

warped ring (ring number m).

Twisting angle tn: the twisting angle of the n*h ring referenced to

the position angle of the warp.

Twisting angle wn: the angle on the reference disk that locates the

ascending line of nodes of the n^^1 ring measured from the blue-

shifted side of the projected major axis.

Positive warp: the side of the warp that is located in the upper

hemisphere.

SSP ambiguity = Southern edge-shape-position angle of the warp ambiguity:

A situation that arises when two models of the same galaxy have. 190

simultaneously, a different southern edge, shape, and tQ, yet

they produce exactly the same projected images on the sky.

SP ambiguity = Shape-position angle of the warp ambiguity: A situation

that arises when two models of the same galaxies are indistinguish

able from one another when projected on the sky despite the fact

that the models have different "shape" and "PAw" parameters. 191

TABU 1

Determination of 1Q

ORIENTATION OF S.E.: Near Far

BLUE-SHIFTED East ( 0 ^ y < 180“) i ‘ 180“ - 1' EDGE: West (180“< Y < 360") 180“ - 1' r

Orientation of the Southern Edge

DIRECTION OF SPIRAL ARMS:

CW CCW

BLUE-SHIFTED East (0“ < y < 180“) near far EDGE: West (180“ < ? < 360“) far near

NOTES: a) Near or far refer to the southern edge as seen by the observer relative to the center of the disk.

b) Blue-shifted edge = negative radial velocities of the superimposed contour map on the optical photo graph of the galaxy.

c) cw = clockwise, ccw - counterclockwise. 192

TABLE 2

Bosma's (1981a) Models Transformed to our Terminology

NGC 5033

Bosnia: t Q = 328.7 (21U3) This paper: t Q = 315”

1(“) w(") tf“) w n t r

4 62.0 0 0 0 0 0 8 63.0 0 1.0 0 0 0 12 63.0 0 1.0 0 0 0 16 63.2 0 1.2 0 0 0 20 65.0 -2.0 3.5 0 1.1 0 24 66.5 -4.0 5.8 -8.3 3.3 -14.0 28 66.5 -4.0 5.8 -8.3 5.6 -27.3 32 67.0 -7.5 8.4 -23.9 7.8 -40.7 36 67.0 -12.0 11.9 -36.6 10.0 -54.0

NGC 5055

Bosma: tQ = 268115 Ihis paper: t Q - 260

kp in «(“) « (“> t ( “) w(') tr

12 55.0 0 0 0 4.1 -27.1 14 55.0 -5 0 4.1 0 5.1 -36.7 20 58.0 -3.0 3.9 -47.8 8.3 -65.2 26 52.0 +5.0 5.0 -140.3 11.4 -93.8 30 55.0 + 11.0 9.0 -175.3 13.5 -112.9 42 55.0 + 16.0 13.1 -173.9 19.8 -170.0 193

NGC 2841

Bosma: t Q = 90?8 This paper: t Q = 40°

R(kpc) i{ “) «<“) w(°) tr; wiU t n

8 68.2 0 0 0 0 0 10 68.2 2.0 1.9 0 0 0 12 70.0 0 1.8 89.2 1.7 0 19 74.0 2.0 6.1 70.8 5.6 22.8 24 76.0 6.0 9.7 52.0 8.3 36.6 30 78.2 10.0 13.8 43.9 11.7 53.4 32 78.2 12.8 15.8 36.4 12.8 58.9 36 81.8 12.8 18.4 45.1 15.0 70.0

NGC 7331

Bosma: t Q = 89;6 (90^4) This paper: t Q = 300° (240°)

Rfkpc) i ( u) a ( ’) w(“) t r * n t in ___

8 75.0 0 0 0 0 0 12 75.0 3.0 2.9 0 0 0 18 75.0 3.0 2.9 0 0 0 21 76.0 3.0 3.1 -18.2 1.0 -5.0 25 77.5 1.5 2.9 -59.2 2.3 -18.3 30 80.0 0 5.0 -89.6 4.0 -35.0

NOTES: a) The t0 values in parentheses represent the position angles of the warp for the oblate models of NGC 5033 and 7331. For the oblate models the sign of the t values must be changed to plus. b) For all radii between those listed the variation of all angles is smooth and the different values can be obtained by linear Interpolation. TABLE 3

Fixed Parameters

. Spiral Arms: Direction Galaxy *C> V c} s.e. Vc(kms_1) Rc/Ro uPPer Hemisphere9 of Twist Reference

NGC 5033 173 62 near 220 0.10 cw: close cw: same Bosma (1981a) NGC 5055 279 55 near 213 0.05 ccw: away ccw: same Bosma (1981a) NGC 300 108 45 near 94 0.20 cw: close ccw: opposite RCC NGC 7331 347 75 near 250 0.15 ccw: away cw: opposite Bosma (1981a) NGC 2841 328 68 far 300 0.15 cw: close ccw: opposite Bosma (1981a) M 83 45 24 far 180 0.15 ccw: away cw: opposite RLW M 33 21 55 far 107 0.10 ccw: away cw: opposite** RWL M 31 218 77 far 230 0.25 cw: close Roberts & Whitehurst (1975) IC 342 219 25 far 191 0.15 cw: close cw: same Newton {1980atb) NGC 2805 110 41c near 60c 0.20c cw: close ccw: opposite Bosma et al. (1980) NG1 3198 36 70 near 150 0.15 cw: close cw: same Bosma (1981) NGC 6946 62 30 far 208 0.15 ccw: away ------RSR NGC 3718 337 90 far 255 0.15 ccw: away cw: opposite Schwarz (1985) NGC 3079 345 75 near 195 0.30 ccw: away Irwin (1986) NGC 628 195 15 near 200 0.30e ccw: away ccw: same Briggs (1982)

NOTES: a) The location of the upper hemisphere is given relative to the observer. b) The direction of the observed twist in the isovelocity map is given relativeto the direction of the spiral arms. c) Determined values for NGC 2805 (this paper). d) No twist according to RWL, but cw twist from observations of Reakes and Newton (1978). e) Estimated value for NGC 628 (this paper). TABLE 4

Other Authors' Models

Direction Galaxy t 0(°): shape Central Disk/R0 of Twisting3 t ^ O wnC) Reference

NGC 5033 328.7: prolate 0.10 cw: same -36.6 11.9 Bosma (1981a) 211.3: oblate ccw: opposite +36.6

NGC 5055 268.5: prolate 0.25 ccw: same -173.9 13.1 Bosma (1981a)

NGC 300 60: oblate 0.45 ccw: same +38.0 30.0 RCC

NGC 7331 89.6: prolate 0.25 ccw: opposite -89.6 5.0 Bosma (1981a) 90.4: oblate cw: same +89.6

NGC 2841 90.8: oblate 0.20 ccw: same +45.1 18.4 Bosma (1981a)

M 83 0: oblate 0.15 cw: same +141.0 37.0 RLW

M 33 180: oblate 0.45 cw: same +45.0 40.0 RWL 273: oblate 0.50 cw: same +30.2 53.0 RN

M 31 35.4: prolate 0.75 cw: ------29.7 4.0 Roberts & Whitehurst (1975) 324.6: oblate ccw: ------+29.7

1C 342 294.8: prolate 0.60 cw: same -22.1 9.4 Newton (1980a) 245.2: oblate ccw: opposite +22.1

NGC 3198 1.00 0 0 Bosma (1981a)

NGC 3718 125.2: oblate 0.35 cw: same + 14.5 42.4 Schwarz (1985)

NOTE: a) The direction of twisting of the rings projected onto the sky is also given relative

to the observed velocity contour twist. L 56 TABLE 5

Determined Parameters

Shape

Galaxy t 0(c) Central Disk/RQ M ' ) t Nn wn ( ) Predicted9 Determined

NGC 5033 315 0.50 -6 1.0 -54 10.0 prolate prolate NGC 5055 260 0.10 -10 1.1 -170 19.8 prolate prolate NGC 300 80/120 0.40 +2.5 2.0 +27.5 24.0 oblate N0b NGC 7331 240/300 0.60 +4/-5 0.5 +28/-3S 4.0 oblate N0b NGC 2841 40 0.25 +5 1.0 +70 15.0 oblate oblate M 83 30 0.20 + 11.5 1.6 +172.5 25.6 oblate oblate M 33 180 0.45 +4.5 3.6 +45 40.0 oblate oblate 270c 0.50 + 3.3 5.3 +30.2 53.0 oblate oblate

M 31 330/20 0.75 +6 4.0 +24 20.0 ------N0b IC 342 275/320 0.60 + 3 1.5 121 12.0 prolate N0b NGC 2805 60 0.60 13 3.0 +21 24.0 oblate oblate

NGC 3198 60/75 0.60 —— +6 18.0 prolate N0b NGC 6946 50 0.75 0 2.0 0 10.0 ------N0b NGC 3718 125 0.35 1.1 3.4 +13.2 44.2 oblate oblate

NGC 3079 120 0.15 0 — 0 10.0 ------N0b

NGC 628 ------— ---- — prolate N0d

NOTES: a) The prediction 1s based on the direction of the observed twist in the 1soveloc1ty map, If any; see column (8) in Table 3. b) Determination of shape 1s not possible due to the SP ambiguity. c) Model based on observations of Reakes andNewton (1978). d) No good model was found for NGC 628. 197

TABLE 6

Structure of Warp for NGC 3079, 3198, and 3718

NGC 3079a NGC 3198 NGC 3718c

Number w(°) w(") t ( ‘ ) w D t r )

1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 4 0 0 0 0 5 9 0 0 0 0 6 15 0 0 0 0 7 24 0 0 0 0 8 34 0 0 7.4 0 9 30 0 0 14.8 +0.4 10 25 0 0 21.9 +0.8 11 18 0 0 28.2 + 1.7 12 15 0 0 33.4 + 3.0 13 12 1 0 38.8 +4.3 14 9 2 ±2 41.7 +6.1 15 6 4 + 4 42.6 +8.5 16 3 6 ±6 43.4 + 11.1 17 0 9 il2 b 44.0 + 13.6

18 10 12 16 43.3 + 14.2 19 10 15 ±6 42.4 + 14.5 20 10 18 16 42.4 + 14.5

NOTES: a) For NGC 3079, t = 0" for all rings. b) For NGC 3198, the twisting of the rings is not smooth in order to generate a local disturbance seen at the outer velocity contours. c) The tabulated values for NGC 3718 are calculated from Schwarz's (1985) published data. 198

REFERENCES

Arp, H. 1966, Ap. J . Supp1., 14, 1.

Bekenstein, J . , and MiIgrom, M. 1984, Ap. J ., 286, 7.

Bland, J ., Taylor, K., and Atherton, P. D. 1987, M.N.R.A.S., 228, 595.

Bosma, A. 1981a, Astron. J ., 86, 1791.

______. 1981b, Astron. J . , 86, 1825.

. 1983, in 1AU Symposium 100, Internal Kinematics and Dynamics of

Galaxies, ed. A. Athanassoula (Dordrecht: Reidel), p. 11.

Bosma, A., Casini, C., Heidmann, J ., van der Hulst, J. M., and van Woerden, H.

1980, Astron. Astrophys.. 89, 345.

Briggs, 1. H. 1982. Ap. J .. 259, 544.

Br inks, E. 1984, Ph.D. Thes is , Sterrewacht Le iden.

Byrd, G. G. 1978, Ap. J ., 222, 815.

Christodoulou, D. M. 1989, Ph. D. Thesi s, Louisiana State University.

Christodoulou, D. M., and Tohline, J. E. 1986, Ap. J . , 307, 449 (paper CT).

Christodoulou, D. M., Tohline, J. E., and Steiman-Cameron, T. Y. 1988,

Astron. J ., 96, 1307 (paper CTS).

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286, 53.

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p. 187.

Durisen, R. H., Tohline, J. E., Burns, J. A., and Dobrovo!skis, A. R. 1983,

Ap. J ., 264, 392.

Emerson, 0. T. 1976, M.N.R.A.S. , 176, 321.

Felten, J. E. 1984, Ap. J ., 286. 3.

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. 1988, Ap. J .. 326, 84.

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Kuhn, J. R., and Kruglayk, L. 1987, Ap. J ., 313, 1.

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Mitgrom, M. 1983a, Ap. J . , 270, 365.

______. 1983b, Ap. J ., 270, 371.

______1983c, Ap. J ., 270, 384.

______. 1984, Ap. J ., 287, 571.

______. 1986a, Ap. J .. 302, 617.

______. 1986b, Ap. J .. 306, 9.

Newton, K. 1980a, M.N.R.A.S. , 191, 169.

______. 1980b, M.N.R.A.S., 191, 615.

Newton, K., and Emerson, D. T. 1977, M.N.R.A.S. . 181, 573.

Petrou, M. 1980, M.N.R.A.S.. 191, 767.

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Ap. J . . 246, 708.

Reakes, M. L., and Newton, K. 1978, M.N.R.A.S. , 185, 277 (paper RN).

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Roberts, M. S., and Whitehurst, R. N. 1975, Ap. J ., 201, 327. 200

Rogstad, D. H., Crutcher, R. M.. and Chu, K. 19/9, A ^ J. , 229, 509

(paper RCC).

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193, 309 (paper RLW).

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22, 111.

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703 (paper RWL).

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Rubin, V. C ., Ford, W. K., J r ., and Thonnard, N. 1980, Ap. J ., 238, 471.

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Ap. J . , 261, 439.

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Galaxy, ed. W. B. Burton (Dordrecht: Reidel), p. 513.

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______. 1981, in The Structure and Evolution of Normal Galaxies, ed.

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______. 1988, ftp. J ., 325, 26.

______. 1989a, In Dynamics of

Astrophyslcal Gas Disks, ed. J. Sellwood (Cambridge: Cambridge

University Press), in press.

______. 1989b, ., submitted.

Tohline, J. E. 1983, in 1AU Symposium 100, Internal Kinematics and Dynamics

of Galaxies, ed. E. Athanassoula (Dordrecht: Reidel), p. 205.

Tohline, J. E., and Durisen, R. H. 1982, Ap. J . . 257, 94.

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Toomre, A. 1983, in IAU Symposium 100, Internal Kinematics and Dynamics of

Galaxies. ed. E. Athanassoula (Dordrecht: Reidel), p. 177.

lubbs, A. D., and Sanders, R. H. 1979, Ap. J . , 230, 736.

van Woerden, H. 1979, in IAU Symposium 84. The Large-Scale C haracteristics

of the Galaxy, ed. W. B. Burton (Dordrecht: Reidel), p. 501.

Varnas, S. R. 1986a, Ph.D. Thesis, Monash University.

______. 1986b, Proc. Astr. Soc. Australia, 6, 458.

Walterbos, R. A. M. 1986, Ph.D. Thesis, Sterrewacht Leiden.

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65, 37. 202

FIGURE CAPTIONS

Figure 1 - - The relationship between the two coordinate systems that we

use in the frame of the modei-galactic disk. All the

different quantities and axes are explained in the text.

Figure 2 -- The relationship between the two coordinate systems that we

use to project the model-galaxy onto the plane of the sky.

All the different quantities and axes are explained 1n the

te x t.

Figure 3 -- Schematic diagram for the determination of the orientation of

the southern edge of a galaxy. N is north, E is east, as in

the optical photographs of galaxies, and the negative velocity

(-V) indicates the blue-shifted side of the galaxy.

Figure 4 -- This simple combination of Figures 1 and 2 is used to indicate

the relationship between the parameters (i, a) in the plane of

the sky and (w.u) in the frame of the disk.

Figure 5 -- Optical photographs of galaxies: (a) NGC 5033, (b) NGC 5055,

(c) NGC 300, (d) NGC 7331, (e) NGC 2841, (f) M83, (g) M33, (h)

M3I, and (i) NGC 3079. North is at the top and east is on the

left of the page, except Figures 5b and 5c {North is on the

right and east is at the top) and Figure 5h (North is at the

top but east is on the right).

Figure 6 -- NGC 5033. (a,b) Observed (Bosma 1981a) isovelocity and

surface density maps. (c,d) Our results, for the best

(prolate) tilte d -rin g model, (e) Bosma's (1981a) result from

mode 11ng.

Figure 7 -- NGC 5055. (a-e) As in Figure 6. (f,g) Our reproduction of

the velocity field of Bosma's model for the galaxy. 2 0 3

Figure 8 -- NGC 300. (a.b) Observed (RCC) isovelocity and surface density

maps. (c,d) Our results for the best prolate model. (e,f)

The best oblate model, (g) RCC's result from modeling.

Demonstration of the SP ambiguity.

Figure 9 -- NGC 7331. (a-g) As in Figure 8, but from Bosma*s (1981a)

observations. The best prolate model (c,d) is not at all

different from the best oblate mode) (e.f); demonstration of

the SP ambiguity.

F i gure 10 - NGC 2841. (a-b) Observed (Bosma 1981a) isovelocity and

surface density maps. (c.d) Our results for the best (oblate)

model. (e) Bosma’s (1981a) result from modeliny.

F i gu re 11 - M 83. (a,b) RLW’s observed maps. (c.d) Our best (oblate)

model's contour maps. (e.f) Our model with small warping that

does not show HI spiral arms. (g,h) RlW's results from

modeling.

Figure 12 - M 33. (a) Observed (RWL) velocity map. (b) RWL's model

density map. (c,d) Our best (oblate) model.

Figure 13 - H 31. (a) Observed (Emerson 1976) velocity map. (b) Observed

(Roberts and Whitehurst 1975) density map. (c,d) Our best

prolate model (SP amgibuity appears for M 31).

Figure 14 - 1C 342. (a-g) As in Figure 9 but from Newton's (1980a,b)

observations and modeling. Demonstration of the SP

ambigu ity .

Figure lb - NGC 2805. (a.b) Observed maps (Bosma et al. 1980). (c.d) Our

best (oblate) model, that was based on the Southern side of

the galaxy. (Cover up the upper half of each map for a d irect

compari son.) ‘201

Letter 1 01-734 ^307

ROYAL ASTRONOMICAL SOCIETY

■ ORLINOTON h o u s e LONDON. WW ONL

1987 September 3

D.M. Christodoulou, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, U.S.A.

Dear Mr Christodoulou,

Thank you for your enquiry about reproduction of material from our publi cations, as follows:-

M.N.R.A.S., U 6 , 1976,p. 321. EMERSON, D.T.

M.N.R.A.S., 191, 19B0. p. 169. NEWTON, K.

M.N.R.A.S., 191, 19B0, p. 615. NEWTON, K.

The Society is happy for material to be reprinted from its publications, on the following conditions:- a) that the written permission of the authors has been obtained.

b) that a full reference is given to the source of the material, either in caption or in a list of references; if the latter, the words Royal Astronomical Society should be included in the picture caption.

Permission is therefore granted, subject to the other conditions being complied with.

The current addresses of Drs. Newton and Emerson are as follows:-

Dr K. Newton, Dr D. Emerson, 5 Cherwell Close, Department of Astronomy, Abingdon. University of Edinburgh, Oxon. 0X143 37D, Royal Observatory, England Edinburgh EH9 3HJ, Scot land

Yours sincerely. I

Mary Chibnall, Assistant Librarian Letter 2

Propgfuon LtbOfAIOry Ca - a ^ j1* 3** *3?'

May 8, 19S7 Refer to: 335.7-87-96

Dim itris CSiristodoulou Louisiana State University □epartrrent of Physics and Astronomy Baton Rouge, Louisiana 70803-4001

Dear S ir , I received your letter dated April 8, 1987, and I grant you permission to publish ccsrparisona between your models and the observations, by reproducing velocity and surface density diagrans in your paper co-authored with J. E. Tohline and T. Y. Steiman-Cameron in The Astrophysical Jcurnal.

Specifically, yrxi are permitted to use Figures 2, 3, 5, 7, 9, and 11 from our paper published in The Astrophys ical Journal (Bogstad, D. H., Lockhart, I. A., and Wright, M. C. H., 1974, 193, 309), Figures 2, 4, 15, and 16 frem our paper published in The Astrophysical Journal (ftogstad, D. H., Wright, M. C. H., and Lockhart, I. A., 1976, 1204, 7b3), Figures 1, 2, 4, 5 and 7 from our paper published in The Astrophysical Journal (Bogstad, D. H., Crutcher, R. M., and Chu, K., 1979, 229, 5011, and Figures 2, 3, 4, and 5 from our paper published in Astroncnr/ and Astrophysics (togotad, D. H., Shostak, G. S., and Rots, A. H., 1973, 2i, 111).

I was unable to find any copies of the original papers, so I will not be able to send them to you as requested. S in c e re ly

D. H. Rogstad S up erv iso r Concurrent Systems and Applications ■JU-;

L etter 3

National Research Council Consail national da rechetches Canada Canada

Hent>erfl institute Institul Herrberg oi Astrophysics dastrophysique

K1A0A6C a n a d a

June 20. 1989

Dr. Dimitris Chrlstodoulou Dept, of Physics Louisiana State University Baton Rouge Louisiana, 70803 U.S.ft.

Dear Dr. Chr1stodoulou:

The photograph of the galaxy, NGC 3079, **i1ch you received from me was taken from a Palomar Observatory Sky Survey Plate. Please feel free to use 1t In any publications you may require 1t for.

S1nee rely

Dr. Jud1 th Irw1 n

Jl/ck

Tele* 063-37 IS Tile* 053-3715 Letter 4

NATIONAL RADIO ASTRONOMY OBSERVATORY

S48 NODTH C M tnnr tVENUE CAMPUS BUILCKNC es TuCSON AHI/0N« S57JI . 0 6 S ! TELEPHONE 002 BBS 8?SD

Oc tober 13, 1987

Hr. Dloltrl* Chriatodoulou Department of Physics end Astronomy Louisiana Stat* University Baton Rouge, Louisiana 70603-4001

D*ar Mr. Chrlatodoulou,

Thank you for your letter of Scptambar 28th, concerning permission to reproduce diagrams from my paper in M.N.R.A.S. I hereby give you permission to uaa any of my diagrams. In particular Figures 1, 2 and 5 from M.N.R.A.S. 176, 321. 1976. I regret that I no longer have the originals of these figures.

Although I am not sure, I believe that the R.A.S. retains copyright over articles published in M.N.R.A.S. Please check with the editor of Monthly Notice*. If you have not already done so. In ease you need additional permission from M.N.R.A.S. In order to reproduce these figure*.

Your* sincerely

DTE:mt

□P1HATID •> ASSOC'* T »□ UNlVCBBlTlCa INC UfvDiff CONTRACT W IT H t H k P4JRTl O S A L f D U N O * f lO N L etter 5

~ 7 ? UNIVERSITY OF CAMBRIDGE )r£y> INSTITUTE OF ASTRONOMY The Obtcrvaionei. Madinfley Road. Cambridge CB3 OH A. England Telephone 0223-3J7J.. Telen *17297 ASTRON G Enquinet 0223-337J48 Telegram) Observer Cambridge UK J 7

hM Clujl't* & u /»**-

/'Ut< —

p /c -u 'Tic*.. Clt*4l4»

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Letter G

HARWELL UK ATOMIC ENERGY AUTHORITY Building 521 Harwell Laboratory United Kingdom Atomc Enetgy Authority Oxlordshire OX 11 ORA Telex 83135 Telephone Abingdon (0235) 24i«i Ext 5030

25 September 1987

Mr D Chrletodoulou Department of Physics end Astromony L o u is ia n a State University BATON ROUGE Louisiana 70803-4001 USA

Dear Mr Christodoulou

Permission to reproduce M.N.R.A.S figures

1 am writing, as you requested, to give you permission to reproduce the folovlng figures in your publicatlons :

Figures 6, 10 and Plate 1 from 'Neutral Hydrogen In IC342-1. The Large Scale Structure’ (Newton, K. , M.N.R.A.S. 1980, 191 , 196)

Figure 1 and Plates I, 2, 4 from 'N eutral Hydrogen In IC342-2. The Detailed Structure' (Newton, K-, M.N.R.A.S. 1980, 191 , 615).

I have located the line drawings from Paper 1 (copies attached). I think the plates should copy quite well from the M.N.R.A.S. publications, but if you w ant to t r y to locate the originals, you could write to:

Dr J E Baldwin Mullard Radio Astronomy Observatory Cavendish Laboratory Madlngley Road Cambridge CB3 0HE

Good luck with your project

Yours sincerely

K Newton 210

Letter 7

OBSERVATOIKE OE MARSEILLE a A l u i L t - I 1 1 4 * C I D E I 4 IM /H /tf

Mr. Dimitris Chr istafloulau Louisiana State University Department of Phy*ic* and Astronomy Baton Rouge, LI 70803-A00 I

Dear Mr, ChriitodOulOu,

Think you for your letter of April g regarding your wort on warps and the request for the figures. My response is as follows:

1. I hereby give you permission to reproduce the following figures : a' from the Astronomical Journal t B o S m a , 1 9 B 1 . A.J. 06, 1791) : Figures A, 3, 6 , 7, B , lit IB, 13t 1A , 15. lit 17, and IB b I from Astronomy and Astrophysics (Bosma et al. 1900 , A E> A B9i 3 ( * 5 * : Figures 2, 7 and 0.

5. I doubt that reproduction frcm the published figures will render copies which are by themselves publishable. I therefore decided to give you on loan my own copies of nearly all the figures Concerned. Vou may use them for inclusion in your paper and vour thesis, but after that I would 1|i» them bac h . Copies of these figures are very rare, and some of them are the unique copy left. I trust you will respect my wish and return the figures eventually.

3. Pe NGC 2805, 1 do not ha/e a copy of Figure 2 of that paper. I suggest you reproduce it from the journal directly. Since it is a line drawing that should not be that difficult. LJe did not give i he rotation curve because we could ..at derive it : the galaxy was considered too distorted. Even now, I wondi-r why you want to include it in your study ! any model will be very schematic. I would guess it has a rising rotation curve, with a relatively rapid rise out to 1 arcmin, and a slower rise thereafter. The inclination is very uncertain but should be of order E0-30 deg. a. Lia Athanassoula amd myself are very interested in y o u r models. In fact, Lia has been asked to discuss dynamical evidence for dark matter at the I .A .u . Symposium 130 in B a 1 atcnfured , Hungary, in J u 1n this year. She and I would appreciate very much to receive a preprint of yflur paper, if possible before Juin. Uarp* are phenomena which tell us something on dark matter, and we would like to know what you are up to in this respect.

Good luck w, i,h your paper and your thesis work .

Sincerely yours.

Albert Bosma reference disk 2 ] ’J

Figure 2

reference disk

plane of the sky North

projected major axis Figure 3

North -V East

s.e.

South 2 ] 1

Figure 4

reference disk

-plane of the orbit

plane of the sky X

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(d) DOCTORAL EXAMINATION AND DISSERTATION REPORT

Candidate: Dimitris Michael Christodou1ou

M ajor Field: Phys ics

Title of Dissertation Using Tilted-Ring Models and Numerical Hydrodynamics to Study the Structure, Kinematics and Dynamics of HI Disks in Galaxies

Approved

jor Professor and Chairman

Dean of the Graduate Sc:

EXAMINING COMMITTEE

v i OjctAls/

' 7

Date of Examination:

July 11. 1989