Title: Observations and modeling of the gas dynamics of the barred spiral galaxy NGC 3359 /
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Title: Observations and modeling of the gas dynamics of the barred spiral galaxy NGC 3359 /
Physical Description: xii, 283 leaves : ill. ; 28 cm.
Language: English
Creator: Ball, John Roger, 1956-
Publication Date: 1984
Copyright Date: 1984
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Subject: Spiral galaxies   ( lcsh )
Spiral galaxies -- Mathematical models   ( lcsh )
Astronomy thesis Ph. D
Dissertations, Academic -- Astronomy -- UF
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non-fiction   ( marcgt )
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Thesis: Thesis (Ph. D.)--University of Florida, 1984.
Bibliography: Bibliography: leaves 275-282.
Statement of Responsibility: by John Roger Ball.
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
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Resource Identifier: alephbibnum - 000479234
oclc - 11802247
notis - ACP5958

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OBSERVATIONS AND MODELING OF THE GAS DYNAMICS
OF THE BARRED SPIRAL GALAXY NGC 3359








BY

JOHN~ ROGER BALL


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1984





























for M~ary














ACKNOWLEDGMENTS

I must first offer my sincere thanks to my advisors, Drs. James

Hunter and Steve Gottesman. They have been deeply involved with the

progress of this dissertation, from its conception to the dotting of the

last i and crossing of the last t. They have been equally involved in my

development as an astronomer, and from each of them I have learned much

more than I could detail here. Their guidance and their support are very

warmly appreciated.

I would also like to express my gratitude to Dr. Jim Huntley of

Bell Laboratories, who has been a member of my supervisory committee in

all but name. Besides making available his hydrodynamical code, Jim has

provided many insights into its results, and into the dynamics of barred

spiral galaxies. Above all, he has lent a sympathetic ear to many a

long-distance call of bewilderment, for which I thank him profusely.

I thank the remaining members of mny committee, Drs. Haywood Smith

and Gary Ihas, for their interest and comments on this work. Addition-

ally, I thank Dr. T. D. Carr for his reading of Chapter II, and Dr. R. B.

Piercey for reviewing the manuscript and attending my defense on short

notice.

I acknowledge with pleasure the financial support, during my wrrit-

ing of this dissertation, of the National Science Foundation, under grant

AST 81 16312, and of the Graduate School of the University of Florida.

I am greatly indebted to Dr. Debra Elmegreen, of IBM~ Watson

Research Center, for the use of the near infrared data which forms the

basis of Chapter IV.










The original observations described in this dissertation were ob-

tained with the Very Large Array of the National Radio Astronomy Observa-

tory. The National Radio Astronomy Observatory is operated by Associated

Universities, Inc., under contract with the National Science Foundation.

I can hardly exaggerate the helpfulness of the staff, as well as many of

the visitors, of the NRAO, who patiently bore the thousand questions of

the neophyte observer struggling to achieve competence. Although there

are many who deserve mention, I will single out only the inimitable Drs.

Jacqueline van Gorkom and Rick Perley, to whom go my most heartfelt

thanks.

For expert assistance with the preparation of various figures, I am

grateful to Paul Gombola and Hans Schrader.

The physical quality of the volume which the reader has in his

hands is largely the product of the labors of Irma Snith, who has truly

gone the extra mile in helping to prepare this dissertation. Without her

masterful touch, putting together a satisfactory copy would have been

next to impossible. Therefore, I thank Irma for helping to preserve mny

sanity, and hope that she has not done so at the cost of her own.

There are also several people whose support demands a word of

thanks, although their contributions to this dissertation are less direct.

It is not only a pleasure, but an honor, to mention two gentlemen whose

tremendous abilities and dedication to the art of teaching have a great

deal to do with my decision to attempt a career in the physical sciences:

Mr. Howard Slaten, of Durant, Oklahoma, High School, and Dr. Frank Hart,

of the University of the South.

No one understands the travails of graduate school like another

graduate student, and what limited capacity I still have for normal










behavior owes a good deal to the understanding and friendship of the

astronomy graduate students who have been here at the University of

Florida during my stay, from 1977 to 1984. I also, needless to say,

learned a lot of astronomy from, and with, them. I thank all of them,

for innumerable reasons, and, especially, I thank Chris St. Cyr, Joe

Pollock, Greg Fitzgibbons, and Glenn Schneider.

The encouragement of my parents, Lee and K(ate Ball, through the

past twenty-eight years is a gift which I value more than I can hope to

express.

My deepest thanks of all go to my wife, Sandy, who has been with me

constantly, however many miles have been between us. Although the

hardships incurred by my pursuit of this degree *:ave fallen on her as

well as on me, she has given me only encouragement:, comfort, inspiration,

and hope. Mfy gratitude to her for sharing the L;,rdens of these years

goes far beyond what can be expressed here; let ? simply say that I

thank her for all that she has done, and for sil that she is: she is,

above all, the point.















TABLE OF CONTENTS


Page




viii

ix

xi


ACKNOWLEDGMENTS ....................

LIST OF TABLES. . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . .

ABSTRACT. . . . . . . . . . . . .

Chapter
I. INTRODUCTION. . . . . . . . .

An Overview of Disk Galaxy Dynamics....
Dynamics of Barred Spirals.........
The Barred Spiral Galaxy NGC 3359... ..

II. RADIO OBSERVATIONS..... .........

Neutral Hydrogen as a Kinematical Tracer ..
Elementary Aperture Synthesis Theory....
VLA Observations of NGC 3359. ......

III. DISTRIBUTION ANID KINEMATICS OF NEUTRAL HYDROGEN

Techniques of Spectrum Integration .....
Neutral Hydrogen Distribution in NGC 3359..
Kinematics of the Neutral Hydrogen ...
The Companion Galaxy to NGC 3359......

IV. THE STELLAR BAR OF NGC 3359... ......

Surface Photometry: Introduction......
Projection Effects in Triaxial Figures..
Brightness Distributions .
The Attraction of a Triaxial Homoeoid....

V. HYDRODYNAMICAL MODELS.............

The Beam Scheme...............
Models Using Triaxial Bars.........
Models Using Oval Distortions.... ...

VI. SUMMARY OF CONCLUSIONS.............

Observational Results............
Hydrodynamical Models............
















Page

APPENDIX A. .. .. .. .. .. .. .. ... . .. ... 265

APPENDIX B. .... .. .. .. .. .. .. .. .. .. 269

REFERENCES. ... .. .. .. .. .. ... . .. .. .. 275

BIOGRAPHICAL SKETCH .. .. .. .. .. ... .. .. . .. 235















LIST OF TABLES


Table Page

1-1 GLOBAL PROPERTIES OF NGC 3359. . .. .. .. ... 14

2-1 CALIBRATION PROPERTIES .. .. .. .. .. .. .. 46

2-2 CONTINUUM SOURCES IN THE NGC 3359 FIELD. . .. ... 58

2-3 SINGLE CHANNEL SIGNAL AND NOISE CHARACTERISTICS. .. 90

3-1 LOGARITHMIC SPIRAL FIT TO FIGURE 3-3 .. .. ... 109

3-2 SUMMARY OF NEUTRAL HYDROGEN OBSERVATIONS . .. ... 138

4-1 BAR PROJECTION PARAMETERS. .. .. .. .. .. .. 168

4-2 EVALUATION OF INTEGRALS FROM GRADSHTEYN AND RYZHIK

(1980). .. .. ... .. .. .. .. .. .. 158

5-1 BEAM PROPERTIES, ONE-DIM4ENSIONlAL CASE. .. .. .. 198















LIST OF FIGURES


Figure Page


1-1 The barred spiral galaxy NGC 3359, from a print of the
National Geographic Society--Palomnar Sky Survey
blue plate. .. .. .... . .. ... .. . 13

2-1 Geometry of the two-element interferometer .. .. .. 22

2-2 The (u,v) coordinate system. .. .. .. .. . ... 30

2-3 The tracks swept out in the (u,v) plane by a single
interferometer pair . .... ... .. ... 35

2-4 Single-channel clean maps of the neutral hydrogen in
NGC 3359. .. .. . ... .. .... .. .. 62

2-5 Single-channel dirty map of the H I in NlGC 3359, at the
center velocity of 1016 km/s. . ... .. .. .. 83

2-6 Single-channel clean maps of neutral hydrogen in NGC
3359, showing a larger area than in Figure 2-4. .. 86

3-1 Contours of neutral hydrogen surface density in
NGC 3359. .. ... .. .. .. .. .. .. .. 102

3-2 The same integrated H I map as in Figure 3-1, shown
here in grey tone format. . ... .. .. .. .. 104

3-3 Distributions of H I and H II in NGC 3359. .. ... 107

3-4 Mean deprojected H I surface density in NGC 3359, as a
function of radius. ... .. .. .. . .... 114

3-5 Continuum emission in NGC 3359 .. ... .. .. .. 118

3-6 Contours of heliocentric, line-of-sight velocity in
NGC 3359. . ... .. .. .. ... ... .. .. 122

3-7 Angle-averaged rotation curve of NGC 3359. .. ... 127

3-8 Rotation curve determined independently from velocities
along the two halves of the major axis. .. .. .. 128












FigurePage

3-9 Rotational velocities of the first-order mass models
used in this chapter to estimate the mass of
N~GC 3359. .. .. .. ... .. .. ... .. 134

3-10 The observed global spectrum of neutral hydrogen in
NGC 3359. .. ... .. ... ... .. .. 137

3-11 Contours of integrated H I column density in the area
surrounding the isolated companion to NGC 3359. .. 140

3-12 Relative locations of NGC 3359 and its satellite . .. 142

3-13 The observed global spectrum of H I in the companion
galaxy to NGC 3359. . .... ... .. .. .. 144

4-1 The I passband image of NIGC 3359 .. ... .. .. 153

4-2 Geometry of a triaxial bar in a disk galaxy. . .. .. 157

4-3 Contour plot representation of Elmegreen's near
infrared plate of NGC 3359. ... .. .. .. .. 162

4-4 Near infrared brightness profiles, perpendicular
to the major axis of the disk of NGC 3359 .. .. 171

4-5 Brightness profiles along the bar of NGC 3359. .. .. 178

5-1 Gas response in the first of the purely triaxial
models of NGC 3359. .. .. ... .. .. .. .. 212

5-2 Gas response in the second of the purely triaxial
models. .. .. . ... .. ... ... . .. 216

5-3 Gas response in the pure oval distortion model .. .. 228

5-4 Gas response in the first of the hybrid oval distortion
and triaxial models .. . ... ... .. ... 235

5-5 Gas response in the second of the hybrid oval distortion
and triaxial models ... .. .. ... .. .. 239

5-6 Gas response in the third of the hybrid oval distortion
and triaxial models .. .... .. .. .. .. 243

5-7 Gas response in the final hybrid oval distortion and
triaxial model. ... ... .. ... . ... 247

5-8 The rotation curve derived from the model illustrated
in Figure 5-7 .. .. .. .. .. ... .. .. 254















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


OBSERVATIONS AND MODELING OF THE GAS DYNAMICS
OF THE BARRED SPIRAL GALAXY NlGC 3359



John Roger Ball

August 1984

Chairman: James H. Hunter, Jr.
Major Department: Astronomy

We have conducted a detailed study of the dynamics of the neutral

hydrogen gas in the bright northern barred spiral galaxy NGC 3359.

Observations of the 21 cmn line at the Very Large Array have been reduced

to give single-channel maps w~ith a spatial resolution of 18", and a vel-

ocity resolution of 25 km/s. The acquisition, calibration, and reduc-

tion of the data are discussed in some detail.

Maps of the integrated column density and mean velocity of the

atomic hydrogen, derived from the channel maps, provide the principal

data for an investigation of the dynamics associated with the spiral

structure of the galaxy. On scales comparable to the resolution of this

survey, approximately 1 kpc at the distance of NGC 3359, the gas is

broken up into a somewhat chaotic distribution of local maxima and mini-

ma. However, on larger scales the column density shows a smooth, grand

design spiral pattern with two principal spiral arms. The extent and

density of these two arms are roughly equal in the 21 cm map, unlike the












optical image. These neutral hydrogen arms are very well correlated

with the positions of H II regions.

The observed velocity field of NGC 3359 shows large kinematical

deviations from simple rotation. The locations of the disturbances are

closely correlated with the optical and 21 cm spiral structure, and

indicate that significant spiral streaming motions are present in the

arms.

We have performed numerical modeling of the galaxy, using the

well-known "beam scheme" hydrodynamical code, to attempt to reproduce

the observed kinemnatics. Two types of model have been employed. In the

first, the spiral structure is driven by an inhomogeneous, triaxial bar,

whose density distribution is determined from near infrared surface

photometry. The alternative is an oval distortion of the underlying

disk surface density. In models which depend solely on the triaxial

bar, the kinematical .perturbations in the galaxy's outer regions are far

too weak, relative to those near the bar. The oval distortion models

fit the observed velocities much more accurately. We conclude that the

spiral structure in NGC 3359 cannot be driven entirely by the nonaxisym-

metric force produced by the observed optical bar.















CHAPTER I
INTRODUCTION

An Overview of Disk Galaxy Dynamics

This dissertation is concerned with the application of certain

hydrodynamical models to the spiral structure in the barred spiral

galaxy NGC 3359. In this introductory chapter, we hope to explain the

importance of our chosen subject, and to place it in its astrophysical

context. Accordingly, we first will review, briefly, the topic of spiral

structure in disk galaxies, pointing out the chief obstacles that arise

in an analytical approach to the general problem. We will see that

restricting our attention to the barred spiral galaxies eliminates one of

the central, physical difficulties of the analytical theory, and, at the

same time, dictates the use of numerical techniques. We then introduce

the reader to the particular barred spiral, N~GC 3359, which will be the

subject of our investigation into the details of barred spiral dynamics.

We close the chapter with an outline of the remainder of the present

work.

The problem of spiral structure in galaxies is one of the major

challenges currently facing dynamical astronomy. It seems probable that

spiral galaxies make up quite a large proportion of the luminous mass of

the universe, especially outside of rich clusters of galaxies, though per-

haps not as large a numerical fraction of galaxies. For example, in our

Local Group of galaxies, van den Bergh (1981) finds only three true spir-

als out of 29 known members, but these are estimated to contribute some

90% of the total luminosity contained in all 29 galaxies. By comparison










with the other principal classes of galaxies, namely, the elliptical,

lenticular, and irregular galaxies (Sandage, 1961), spirals pose intrigu-

ing problems for dynamical study. Their structures are, to all appear-

ances, much more complex than those of elliptical and lenticular gal-

axies, and yet remain ordered and fairly symmetrical, unlike the irregu-

lars. The dynamics governing the spirals should, therefore, be both

interesting and ultimately tractable, if difficult. Furthermore, there

are several techniques available for measuring radial velocities in these

galaxies, so their internal motions can be explored observationally.

However, the dynamics of spiral galaxies are not yet well under-

stood. While the light distribution of a typical spiral is dominated by

a fairly flat, thin, rotating disk, with the addition, in some types, of

an inner spheroidall" or "bulge" component, the mass distribution is

often inferred to be markedly different. It is common to postulate a

"dark halo," more or less spherical in shape, of some kind of matter

which has a high mass-to-luminosity ratio. The visible disk is pictured

as lying in the equatorial plane of this halo, though often the halo is

believed to extend to considerably greater radius than the disk. Halos

have been discussed extensively in the recent literature, and work on

them is reviewed by van den Bergh (1978) and by Haud and Einasto (1983).

iWe will have relatively little to say about them in this dissertation.

In light of the uncertainties in our knowledge of the masses and

other large-scale properties of spiral galaxies, perhaps it is not

surprising that there is not yet general agreement on a specific

mechanism for the most striking dynamical features of these objects,










the spiral patterns themselves. All the promising theories of spiral

structure advanced to date depend on the existence of shear in the gal-

axies' disks, due to differential rotation. It is well established that

spirals do rotate in that way. However, the problem of deriving a self-

consistent treatment of the disk dynamics, which explains the spiral

patterns, remains an open one. Perhaps the best-known approaches are

the Lin-Shu density wave theory, and the scenario of stochastic, self-

propagating star formation (SSPSF). These two hypotheses are radically
different.

The density wave theory is an analytical treatment, based on a

linearization of the relevant equations (see below). Some elements of

this treatment are present in the work of Lindblad (e.g., Lindblad,

1963), but the modern form of the theory derives primarily from Lin and

Shu (1964), Toomre (1964), and Kalnajs (1965). Subsequent papers of

special importance include, among others, those by Lin and Shu (1966),

Lin, Yuan, and Shu (1969), Toomnre (1969), Shu (1970), Lynden-Bell and

Kalnajs (1972), Mark (1974, 1976), and Lau and Bertin (1978). Lin and

Lau (1979) give a thorough mathematical presentation and summary of dens-

ity wave theory. As Lin and Shu (1967) emphasize, this theory is an

attempt to explain the grand design of spiral features over the whole

face of a disk galaxy. It consists of a demonstration that a small,

quasi-stationary perturbation in an otherwise symmetric disk can, at

least under some conditions, be self-sustaining. The spiral arms are

seen as manifestations of a "density wave," which is, in essence, a

tightly wound, rotating, spiral potential minimum, through which the

material of the galaxy flows. That is, the matter which one sees in the

spiral arms at any given moment is not permanently bound to the arms. In











its original form, the theory made no effort to explain the origin of the

spiral perturbation, and to do so remains one of its great unsolved prob-

lems. It has also grown increasingly clear that even the maintenance of

the waves over many rotations is not adequately dealt with by this theory

as it now stands. Toomre (1977) gives an excellent review of the the-

oryls progress and its difficulties. Despite the persistence of the lat-

ter, the density wave theory is probably the most significant advance, to

date, in our understanding of the basic dynamics involved. Also, it

seems to succeed quite well, for some galaxies, in matching the observed

morphology (Roberts, Roberts, and Shu, 1975).

The stochastic, self-propagating star formation theory takes a very

different tack. It explains thle global pattern as an effect of local

interactions in the gas. Also, it is a numerical scheme by its very

nature. The idea first appeared in a paper of Mluller and Arnett (1976),

but the theory in its present form has been developed, primarily, by

Gerola and Seiden (1978), Seiden, Schulman, and Gerola (1979), Seiden and

Gerola (1979), and Seiden (1983). The thermodynamical interpretation

which underlies the seemingly heuristic approach of SSPSF is brought out

in a very interesting pair of papers by Shulman and Seiden (1978, 1982).

As Gerola and Seiden (1979) point out, it is not a dynamical theory, but a

statistical model, which works in the following way. One begins with a

small number of high-mass stars, randomly distributed about a galactic

disk, which experience supernova events. For each of these, there is a

probability Pst that the shock wave engendered by the supernova

triggers the formation of a similarly massive star in a nearby gas cloud.

The physical justification for this step has been studied extensively, by,

among others, Elmegreen and Lada (1977), Elmegreen and Elmegreen (1978),











Lada, Blitz, and Elmegreen (1978), and Welter (1982). As this process

slowly percolates through the available gas of the galaxy, clumps of OB

stars are produced. These groups are then pulled out into trailing,

spiral features by differential rotation. Not surprisingly, the

appearance of the model galaxy depends sensitively on its rotation curve

(through its shear), but the general tendency is toward rather fragmented

spiral structure with many disconnected, or imperfectly connected,

branches. The Sc galaxy M 101 is an example of one whose overall

morphology is strikingly similar to models of this kind.

Although the density wave and self-propagating star formation

theories differ in method, results, and, most importantly, in their

assessment of what is fundamental to the physics of the problem, they

need not be mutually exclusive. As discussed by Gerola and Seiden

(1979), there is no conceptual barrier to the simultaneous operation of

both processes in a single galaxy (Jensen, Talbot, and Dufour, 1981).

Perhaps a more common view is that these two analyses may correspond to

the subjective, but useful, division of spiral galaxies into grand design

and flocculentt" types (e.g., Elmegreen, 1981).

The techniques to be used in the present study differ in spirit

from both of those just discussed. As in the density wave theory, we

will seek an explanation of spiral structure as a global mode in the

galaxy under investigation. However, our approach is strictly through

numerical experimentation. This methodology is forced upon us by the

desire to explore dynamics which are inaccessible to the density wave,

and related, theories. The fundamental difficulty arises from the

nonlinearity of the problem, which is contained fully in the following

set of equations. We employ hydrodynamical equations here to represent










the behavior of the constituent matter of a galaxy, as we will be

concerned primarily with the gaseous component in this work. It can be

shown that somewhat similar expressions hold for the case of stellar

dynamics in the gravitational field of a galaxy (Freenan, 1975; Lin and

Lau, 1979). Let us denote the gas density by p, the velocity vector by

v, the pressure by P, and the kinetic energy density by E. Neglecting

the effects of magnetic fields, we have the equation of mass continuity

for a compressible fluid:



--(p)+p*(py) = 0;
at


(1-1)

the equation of motion:



-(pV)+ev(pv2+P)-F = 0;


(1-2)

and the energy equation:



-a-(E)+o*[v(E+P)]-Q= = .
at


(1-3)

Here Q represents explicitly dissipative terms, such as viscosity and

heat conduction. Finally, the force, F, consists only of gravitational

and viscous terms. We will not usually write the latter explicitly in

this study (though they may not be negligible), because the numerical

calculation with which we will model these equations contains the










effects of viscosity implicitly. If we omit the viscous force, then F

is solely gravitational, and so is specified by Poisson's equation:





(1-4)

V2 e= nGp.



Since equations (1-1), (1-2), and (1-3) are nonlinear, the problems of

galaxian dynamics, contained in equations (1-1) through (1-5), is not

susceptible to linear analysis, although one can sometimes develop a set

of linear equations which approximates this system for disturbances of

small amplitude. The density wave theory is an example of that approach,

in which the approximation is valid as the perturbation density and

force, which are involved in the spiral pattern, go to zero. Unfortun-

ately, there is no reason -: believe that these perturbations should be

small, in general. Eve- it the difficulties currently confronting the

density wave theory are overcome, this theory can only be regarded as the

precursor to a less restrictive theory.

Dynamics of Barred Spirals

A large subclass of spirals is that of the barred spiral galaxies,

so named because of elongated concentrations of light at their centers

(e.g., Sandage, 1961). These bars are generally interpreted as non-

axisymmetric mass concentrations, with the form of the bar rotating as a

solid body, though the individual stars which compose it may well have

more complicated orbits. Such nonaxisymmetric configurations have been

found to be long-lived an" robui' in a great many numerical simulations

of gravitationally interacting particles (Miller, 1971, 1976, 1978;










Hohl, 1971, 1972, 1976, 1978; Ostriker and Peebles, 1973; Sell wood, 1980;

and, especially, Miller and Smith, 1979). The attack by Kalnajs (1972)

upon the very difficult analytical problem of the stability of rotating

disk galaxies points toward similar conclusions. Initially, these bars

were viewed as aberrant instabilities in rotating disks. However, they

seen to form without fail when the disks are dynamically "cold," i.e.,

have small velocity dispersions. It is more to the point, therefore, to

say that these experiments demonstrate the stability of the equilibrium

figure in a flattened system whose internal motions are dominated by

rotation: namely, a bar. Indeed, one of the reasons for invoking dark

halos, in nonbarred or "normal" spirals, is that a halo a few times more

massive than its embedded disk helps to suppress bar development, as wias

first surmised by O~striker and Peebles (1973). The ubiquity of bars in

the N-body results immediately suggests a ready source of a strong, peri-

odic, nonaxisymmetric driving perturbation on the background disk potent-

ial in a spiral galaxy. If the bar is taken to be the cause of the non-

axisymmetric perturbation in the rest of the galaxy, one major aspect of

the problem of the origin and maintenance of spiral structure is resolved

at once. One the other hand, the nonlinearity of the problem can no

longer be evaded, as the "perturbation" force is far from negligible.

The theoretical study of barred spirals, then, is best carried out

by numerical techniques. In recent years, many numerical experiments of

this kind have been published. The first successful models were those of

Sanders and Huntley (1976) and of Sorensen, Matsuda, and Fujimoto (1976).

The former used the "beam scheme," a hydrodynamical code originally

developed by Sanders and Prendergast (1974), to follow the gas response.











(The beam scheme is discussed in some detail in Chapter V of this disser-

tation.) Shortly thereafter, Huntley (1977), Huntley, Sanders, and

Roberts (1978), and Sanders and Tubbs (1980) explored fairly thoroughly

the barred spirals which can be generated by varying the dynamical param-

eters in the beam scheme. Huntley (1980) investigated the inclusion of

the self-gravity of the gas. The beam scheme also served as the basis

for the barred spiral models of Schempp (1982). In the meantime, other

hydrodynamical codes were used for this purpose by Sanders (1977), by

Berman, Pollard, and Hockney (1979), by Roberts, Huntley, and van Albada

(1979), by Matsuda and Isaka (1980), and by van Albada and Roberts

(1981). These codes give slightly different results from one another,

especially in calculating the strengths and locations of shocks. How-

ever, these are differences of detail, arising largely from differences

in the effective spatial resolution from one scheme to another. Both the

essential physics, and the observable consequences, of the models are

quite similar, for similar input potentials. Therefore, while it should

be borne in mind that small-scale structure in the models depends some-

what on the particular code employed, this choice is less important in

examining the global response of the gas. It is unlikely that the dif-

ferences between the various computer programs will be very significant,

for most galaxies, at the relatively coarse resolutions which typify

radio observations, such as wre report in this dissertation.

It is now clear that any of several hydrodynamical computer codes

is capable of producing stable, trailing spiral density patterns in

response to a rotating, barred potential. The goal of the present work

is to examine the validity of these codes, by comparing in detail the

predicted gas distribution and kinematics with observations of a real











galaxy. The code we use is a version of the beam scheme developed by Dr.
J. M. Huntley, who has kindly made it available for this study.

The limitations of this work are quite fundamental, and we wish to

alert the reader to them at once. Neither the present study, nor any

other yet undertaken, provides a fully self-consistent treatment of the

dynamical behavior of a complete galaxy. The discussion above should

make it clear why that is a goal beyond our abilities at this stage of

our understanding of galaxies. W~e will simply assume that an underlying,

stellar disk has been perturbed by the formation of a bar into some

stable configuration, and that the potential of this system remains con-

stant, except for its rotation, during the time followed by our numerical

computation. This is further assumed to be the only potential present in

thle model galaxy. That is, we ignore the self-attraction of the gas

which is modeled. The justification for this assumption is that the

density of gas, over most of a typical galaxian disk, is small compared

to that in the disk-bar system. We also assume a gaseous component which

is in a single, continuous, isothermal phase. Further, it is character-

ized by a viscous length scale equal to our computational cell size, and

has no sources and only one sink, namely, outflow beyond our numerical

grid. The intricate .interactions of the interstellar medium with the

galactic ecology are thus omitted entirely for simplicity's sake. We

will elaborate on these questions in Chapter V. At a deeper level, the

developing spiral pattern is not allowed to influence the imposed

potentials of the disk and bar, so that there is no dynamical feedback in

the models. (One consequence of this approach is that, if a model accur-

ately reproduces the desired rotation curve, no further information is

necessary about the axisymmetric component of the mass distribution










responsible for it. Thus, the presence or absence of a halo need not be

considered explicitly.) In these ways does our approach fail to be

strictly self-consistent, and to include all of the nonlinear behavior of

the problem. Obviously, our investigation, even if successful in its

stated aims, is simply one additional step toward a full understanding of

spiral galaxy dynamics.

The Barred Spiral Galaxy NGC 3359

The galaxy whose dynamics we will attempt to model is NGC 3359, a

system classified as S(B)clI by van den Bergh (1960), as SBc(rs) by

Sandage (1961), as SBT5 by de Vaucouleurs, de Vaucouleurs, and Corwin

(1976), and as Sac(s)I.8pec by Sandage and Tammann (1981). Figure 1-1 is

an enlargement from the blue plate of the Palomar Observatory Sky Survey,

showing the galaxy. The reader may also find the following published

photographs of NGC 3359 of interest: a blue exposure with the Palomar

5-m~eter in Sandage (1961); yellow and hydrogen-alpha images from the

Palomar Schmidt telescope, in Hodge (1969); and the near-infrared plate

of Elmegreen (1981), also using the Palomar Schmidt, which will be used

in Chapter IV of this study. Table 1-1 lists properties of NGC 3359,

compiled from a variety of sources, which we will not attempt to verify

independently in the present work.

The galaxy is dominated by a strong central bar, of length approxi-

mately 1.7 arc minutes, from which a fairly prominent, two-armed spiral

pattern emerges. The spiral pattern is, however, rather asymmetric, with
the ann that begins at the northern end of the bar being significantly

more prominent than its counterpart. The asymmetry is equally clear in

the map of NGC 3359 in H e by Hodge (1969), where a string of H II

regions delineates the stronger arm much more sharply than the other.































Figure 1-1. The barred spiral galaxy NGC 3359, from a print of the
National Geographic Society--Palomar Sky Survey blue plate. North is at
the top, and east to the left. Several foreground stars are marked with
crosses, the positions of these stars are used as fiducial points in
figures throughout the present work.@ 1960, National Geographic
Society--Palomar Sky Survey. Reproduced by pennission of the California
Institute of Technology.













































2












TABLE 1-1
GLOBAL PROPERTIES OF NGC 3359


Right Ascensiona 10h43m20s 7
Declinationa +63029'12"

Morphological Typeb SBTS
Distance 11 Mpc

Photometric Diameterc at 25 mag/(arc sec)2 6!3

20.2 kpc

Corrected Blue Magnitudec 10.60

Corrected Blue Luminosityd 1.08 x 1010 Lg

aGallou~t, Heidmann, and Dampierre (1973).
bde Vaucouleurs, de Vaucouleurs, and Corwin (1976).
cde Vaucouleurs (1979)
dCalculated using above values of distance and magnitude, and with
MB ()) =+ 5.48 (Allen, 1973).












Nevertheless, the open, two-armed spiral structure is quite similar to

typical results of the hydrodynamical barred spiral models, so that one

may reasonably hope to be able to model at least the first-order behavior

of the system. This structure was one factor in the selection of NGC

3359 as our program object. The galaxy is also at a high, northerly

declination, and is known to be bright in the 21 cm line of neutral hydro-

gen, the medium from which we will derive our kinematical data. Single

dish observations of the system have been published by Roberts (1968), by

Rots (1980), and by Fisher and Tully (1981). Early aperture synthesis

maps of thle hydrogen were made by Siefert, Gottesman, and W~right (1975)

and by Gottesman (1982). These investigations produced images with

resolutions of 72" and 36", respectively. As will be seen in Chapter II,

the observations reported here improve on the latter figure by another

factor of two. This was possible because of the exceptional sensitivity

and resolution of the N'ational Radio Astronomy Observatory's Very Large

Array, which has only recently become available for 21 cmn work. The

resolution achieved here gives information on the kinematics and structure

of NGC 3359 on a scale which is a small fraction of the galaxy's diameter.

Therefore, an excellent opportunity is at hand to obtain data of suffic-

ient resolution to test the predictions of the hydrodynamical models.

That is our fundamental goal in the present work.

The remaining chapters of this dissertation form the body of our

work on NIGC 3359. Chapter II presents the observational data, with a

brief introduction to the techniques used in obtaining them. In Chapter

III, we will derive the distribution and kinematics of the neutral,

atomic gas in the galaxy from these data. We will also determine certain











global properties of dynamical importance, and comment briefly on the

relation of our results to other published data on the galaxy. Chapter

IV introduces some further data, namely, near infrared surface

photometry, which wi11 be helpful in determining the driving potential 1n

NIGC 3359. A theoretical framework for using these data is also

developed. The hydrodynamical modeling and its results are covered in

Chapter V. Finally, Chapter VI gives a brief summary of the major
conclusions of this work.
















CHAPTER II
RADIO OBSERVATIONS

Neutral Hydrogen as a Kinematical Tracer

The first step in understanding the dynamics of the gas in a par-

ticular barred spiral galaxy must be to determine its present kinematical

state. In this study of the barred spiral NGC 3359, we use observations,

at radio wavelengths, of the columnn density and velocity distributions of

neutral hydrogen for this purpose.

For the purposes of dynamical modeling, observations of any of sev-

eral components of the-program galaxy would suffice. Although the codes

used are gas-dynamical, any component whose kinematics are closely cor-

related with those of the gas could be used as a tracer. Thus, optical

observations of the hydrogen alpha line could be used, for example,

because the H II regions whose velocities are measured are associated

with young, early-type stars. The dynamics of these stars as a group

will be quite similar to those of the gas from which they have only

recently formed. The choice of neutral, atomic hydrogen gas, however,

offers several important advantages. These result from the nature of the

radiation fromn neutral hydrogen in the interstellar medium (ISM).

Interstellar space, in a disk galaxy, is filled by a tenuous medium

having a number of characteristic phases, of varying density and tempera-

ture (McKee and Ostriker, 1977; Spitzer, 1978). One of the dominant

phases consists of cool, intermediate density gas (approximate tempera-

ture and density: 100 K, 20 atoms cm-3) whose primary constituent











is atomic hydrogen in the ground state. This ground state has hyperfine

splitting into two levels separated by only 6x10-6 eV. The upper of

these levels is occupied when the spin angular momentum vectors of the

proton and the electron in the atom are aligned parallel to one another;

in the lower state, they are anti parallel. The energy difference corre-

sponds to a temperature of 0.07 K, far below the ambient temperature of

the medium. Also, the radiative transition between the two levels is

forbidden. Consequently, the time scale for collisional de-excitation is

much shorter than that for radiative de-excitation, even in the low dens-

ities typical of the ISM. However, the large column density of hydrogen

atoms along a typical line of sight through a galaxy makes the radiation

observable. This line, at a wavelength of 21.1 cm, was predicted by van

de Hulst (1945) and confirmed a few years later (Ewen and Purcell, 1951;

Muller and 00rt, 1951; Christiansen and Hindman, 1952). Observations of

this line have since seen wide application in studies of our own and

external galaxies. The latter work has been reviewed b~y Roberts (1975),

by Sancisi (1981), and by Bosma (1981a, b; 1983), among many others.

The low temperature of the gas in the ISM causes the thermal

Doppler width of the 21 cm line to be very small, although the line is

broadened, by macroscopic turbulence, to velocity widths of several km/s

(Kerr and Westerhout, 1965). The observed total line widths for external

galaxies, therefore, are controlled by their rotational velocities.

Because of this rotation, the optical depth in the line at any given vel-

ocity (characteristic of the bulk rotation of the galaxy at a given

point) remains low for most lines of sight (M~ihalas and Binney, 1981, p.

488).











Let us assume that the H I is optically thin, in this sense.

Should this assumption be false, the observed brightness temperature TB

will approach the physical, spin temperature of the gas, which is

typically about 100 K, as we have said. For our observations of

NGC 3359, the highest brightness temperature recorded was 42 K, averaged

over the beam. This implies an optical depth of about 0.55, which would

lead to an underestimate in the column density of about 20%, at this

isolated peak. The underestimate would be less serious elsewhere. The

assumption of small optical depth is nonetheless retained, since the

optical depth structure, or "clumpiness," of the medium is not known.

Then a simple integration, over velocity, of the observed brightness of

the image suffices to determine the column density of the gas at any

point (e.g., Mihalas and Binney, p. 489):



NH(x,y)=1.8226 x 1018 m" TB(x.y) dv.

(2-1)

If the velocity v is in km/s and TB in K, equation (2-1) gives NH in

atoms cm-2. Similarly, the first moment with respect to velocity

gives a measure of the mean velocity at the point:



v(x,y) = -2" vTR(x,y) dv
+m
/ TB(x,y) dv

(2-2)

The simplicity of these results is the first great advantage of neutral

hydrogen observations for work of this kind. The other accrues from the

way in which the gas is distributed in space. First, especially if one

excludes regions furthest from the galactic centers, it is observed to be












among the fl attest and thinnest of the disk components of both our own

(Jackson and Kellman, 1974) and other galaxies (van der Kruit, 1981;

also, compare Sancisi and Allen, 1979, to van der Kruit and Searle,

1981). Therefore, one may determine, with a reasonable degree of con-

fidence, the two-dimensional position in the plane of the galaxy of any

observed emission. Secondly, this phase of the intersellar mnediumn is

pervasive enough that the emission is seen by radio telescopes as being

continuously distributed, even for relatively nearby galaxies. The main

alternative source of kinematical data is optical spectroscopy, but in

practice only small regions of ionized gas near the hottest stars can be

detected by that technique (though the resolution is very good). The

ubiquity of H I across the disk component allows one to assign a mean

surface density and velocity to each point in the disk, whereas the

distribution of the optical spectra is necessarily patchy.

The only other Population I component which gives rise to prominent

spectral features is molecular gas, which may have mass comparable to, or

even greater than, the H I. Unfortunately, the molecular hydrogen, which

is presumably its dominant constituent, is very difficult to detect. The

most easily detected transitions, those of carbon monoxide, fall in the

millimeter-wave region, where observational difficulties are greater than

at longer wavelengths, and where the first aperture synthesis arrays (see

below) are only now becoming operational. Also, molecular gas is less

broadly distributed in galactocentric radius than H I in most galaxies.

It is usually far more concentrated to the inner disk and, often, even to

the nucleus (Morris and Rickard, 1982).










Elementary Aperture Synthesis Theory

NGC 3359 was observed in the 21 cm transition using the Very Large

Array (VLA) radio telescope of the National Radio Astronomy Observatory.

The VLA is the largest and most sensitive of the radio arrays which oper-

ate by the principle of aperture synthesis, a technique for obtaining

good sensitivity and excellent resolution in this wavelength regime. We

will give a brief description of aperture synthesis below. Fuller dis-

cussions may be found in Fomalont and Wright (1974), Hjellming and Basart

(1982), and Thompson and D'Addario (1982).

The fundamental building block of an aperture synthesis telescope

is the two-element interferometer, as shown in Figure 2-1. Consider the

case where each dish of this interferometer tracks a celestial source,

whose direction is given by the unit vector i0. Let their separation

in meters be written as B. Then, for wavefronts from this source, there

is a delay r in arrival time at antenna 1 relative to antenna 2:



c -B10

(2-3)
where c is the speed of light. Therefore, for monochromatic radiation of

angular frequency W the signal received at antenna 2 is proportional to

cos w t, and that at antenna 1 is proportional to cos a (t + T). After

detection and amplification, the signals from the two are multiplied

together. In the simple scheme discussed so far, the result of this

multiplication would be


R(t) cos(2wt WT) + cos(WT).


(2-4)






























TC


ANTENNA 1 ANTENNA 2

Figure 2-1. Geometry of the two-element interferometer. The
unit vector 10 points toward the source being observed. Vector b is
the component of the mutual separation B which is perpendicular To
10. Note that b and 10 need not lie in the plane of the paper.
The additional propagation length to antenna 1 causes a relative delay
for wavefronts traveling at velocity c.










When this signal is sent through a low-pass filter, only the second term

remains. This is because the first term varies at a frequency twice that

of the emission being observed, the second only at the rate of the

sidereal change in the projected baseline B*sO (in wavelengths),

giving a period of order 1 second for 21-cm emission and typical VLA

spacings.

In practice, the instrument must be made more sophisticated in a
number of ways. The most fundamental is caused by the fact that

astronomical radiation is not monochromatic, and neither are most

receivers, so that a band of frequencies as is always observed. The

geometrical delay -I then, corresponds to a different number of wavelengths

for each frequency in the band, and the unavoidable integration over all

these frequencies in the multiplier will cause the signal to be severely

depressed in amplitude. The solution is to introduce a delay T' into the

propagation path from antenna 2 which almost cancels r, at 1-ast at the

band center wc, enabling coherence over broader bandwidths. This

procedure is called delay tracking. Suppose the band contains signals

between frequencies q and wu = Y + Am. Let rg be the net
difference in delay between paths 1 and 2: r, = r -c'. The change in

phase, for frequency oi, over this delay is TO wi, and therefore the

phase difference over the whole band at the end of the path is To dw
If this difference amounts to 2n, all coherence is lost; the condition

for satisfactory coherence is evidently



0T ~ 271


(2-5)










Of course, the effects of a finite bandwidth must still be taken into

account by integrating the output signal over 6

Secondly, before multiplication the signals must be reduced to a

lower frequency (usually called the intermediate frequency) than that

received from the source. This is achieved by a heterodyne system, in

which both antenna outputs are mixed with the signal from a monochromatic

local oscillator (LO). A single sideband, of the two created by this

mixing, is selected by appropriate filtering. This step is nonnally

implemented before delay tracking. These considerations complicate the

intermediate calculations considerably. However, if thle band passes are

stable, symmetrical, and identical in the two channels, and if delay

tracking is acceptably good, one still obtains the remarkably simple

expression (Hjellming and Basart, 1982, p. 2-15):


R(t) = coshgr 0 t)l

(2-6)

Here WO is the frequency of the local oscillator, and 4 (t) the

phase difference between the LO signal as received at the two mixers of

the heterodyne system.

The formal extension of this result to a system of many antennas is

straightforward. One simply replaces the local oscillator by a network

of slaved local oscillators, one at each antenna, under the control of a

master LO and rigorously synchronized. (The electronics system in use at

the VLA actually uses a number of LOs in each signal path to optimize the

frequency characteristics of each component in the system. However,

there is no conceptual difference between this scheme and the use of a

single LO.) Also, delay tracking is implemented for all antennas










relative to a single point (for example, the geometrical center of the

array). In order to avoid the introduction of errors into the data by

incorrect tracking, the positions of all of the individual telescopes

must be known very accurately. This calibration is ordinarily performed

when the configuration of the array is changed. It is carried out by

observing point sources at a variety of hour angles and declinations, to

obtain many measurements of B ~g for different orientations of

sp. B can then be estimated quite accurately for each antenna pair.
Of course, the delay tracking is strictly accurate only for a

source at the pointing center s0. Suppose emission is detected from a

source at some other position, s. Let us write the instantaneous phase

of the signal from the jth antenna, relative to the master LO, as e3.
This includes the phase shift associated with the delay tracking.

Because JD is the reference point for delay tracking, the correlated

signal from the antenna pair j and k becomes



Rj k(t ) = Aj (t)Ak (t)co s[ 5 7 B* (s-sg)+ c5(t ) k(t) *


2-7)
11l of the quantities on the right-hand side of this equation should be

quite slowly varying. The quantities Aj and g are called the
instrumental amplitude and phase, respectively, of the jth antenna, and

the expression



Gj = Aj exp (i @j)

(2-8)
is often referred to as the complex gain.









It is common to speak of the correlated signal obtained in this way

as the "complex visibility function" Vjk (t). This is simply the

representation of equation (2-7) in complex-variable notation:



Vjk(t)=Aj [t)Ak(t)exp~il 38a B* (s-sg)+Jj(t)- kit)l )*


(2-9)
By writing equation (2-9) as the product of independent terms for

the two antennas, we have implicitly assumed that "antenna based" cali-

bration is appropriate, as explained below. However, the quantities

which are actually measurable are those specified by the equation



Vjk(t)=Ajk(t)exp Ci[ AD 4 (1."0)* Ojklt)l -


(2-10)
The calibration of the data base consists of determining the complex gain

of each antenna, as a function of time, from the observed quantities of

equation (2-10). Observations of strong point sources, of well-known

flux density and position, are used for this purpose. For a source of

this kind, equation (2-10) becomes particularly simple. The assumption

one normally makes is that s = sO for these calibrator observations; if
this is not true, a phase gradient across the array will be introduced.

Clearly, it is important that the positions of all calibrators be known

as accurately as possible. Wlith this assumption, equation (2-10) is

simply



Vjk(t)=Ajk(t)exp~ihjk~tt)
(2-11)











If this visibility were properly calibrated, it would be equal to

the flux density S of the calibrator. Thus, one can rewrite equation

(2-11) in terms of an effective complex gain for the jk baseline,

Gjk:


Vjk(t)=Gjk(t)S.
(2-12)

As we have stated, the usual assumption when dealing with VLA data, which

is almost always valid (Hjellming, 1982), is that the complex gains of

the antennas are separable. Then



Gjk(t)=Gj(t)Gk (t)
(2-13)

where the asterisk denotes the complex conjugate. Combining the

expressions (2-8), (2-11), (2-12), and (2-13), we obtain the results



Aj(t)Aklt) A t


(2-14a)


(2-14b)

At each time for which calibration data are available, one has this

pair of equations for each of the N(N-1)/2 baselines, so that the system

of equations for the N antenna-based complex gains is highly over-
determined. Therefore, a least-squares solution is appropriate. The

residuals of Aij(t) and ej(t) in this solution are called thle closure
errors in amplitude and phase, respectively, for antenna j. Obviously,










the calibrator observations must be spaced closely enough in time so that

the time dependence of Gj(t) is known. One can then interpolate to

find its value during the observations of the program object. For the

VLA, the system is ordinarily stable enough so that a calibrator scan

once every 30-45 minutes is adequate.

Before leaving the topic of calibration, we should mention the

calibration of the antenna pointing. Like the telescope position

calibration, this function is performed at regular intervals by the

observatory staff. The errors in pointing for each antenna, as a

function of altitude and azimuth, are determined by point source

observations, combined with the known primary antenna power patterns.

Corrections for these errors are then applied, in the on-line observing

computers, to data taken subsequently for scientific programs. This

is necessary for observing programs where highly accurate positioned

information is sought, but is of little importance in the present work.

Now let us consider the response of the two-element system to an

extended source of emission. As before, ID gives the direction of the

point on the sky which is tracked by the antennas; that is, the maximum

of the single-dish power pattern PO is kept aligned with the changing

position of iD. Let the vector s give the position of some emitting

point within the field of view, not located at s.0. Since the primary

beam of each antenna will restrict this field to a relatively small solid

angle about sp, s may be written as




(2-15)

where r is a vector perpendicular to, and much smaller than, the unit










vector S0. The complex visibility associated with this patch of emnis-

sion will differ from that given in equation (2-9) for two reasons.

First, we must now include explicitly the product of the source intensity

distribution I(a, 6) with the normalized, primary power patter PO

(2-a0, 6-60). (Here the equatorial coordinates (a0> 60) give the

position of ID.) This ensures that integration, over the solid angle
of the source, of the apparent intensity yields the apparent flux

density. The second, and more important, effect is that the delay r is

now a function of the vector s, not sO. if we let s play this role in

equation (2-9), recalling the definition of T in equation (2-3), we

obtain for the contribution from a small emitting patch of solid angle

dn:


dVjk(,6,t)=Aj(t)Ak(t)I(a,6)PO(a-ag,6-69)d x

x exp {i[ SD (B~r)+ mj(t)-mkit)l )


(2-16)

Before proceeding to simplify this expression, let us consider the

geometry of the interferomneter pair as viewed from the object under

observation. Suppose, for simplicity, that the orientation of the

baseline B is east-west in the earth's latitude-longitude system, as

illustrated in Figure 2-2. From the viewpoint of our source, it is much

more useful to describe this separation in the so-called (u,v) plane.

This coordinate system has one axis (the v axis) parallel to the earth's

rotational axis, while the other (u) is perpendicular to it and is

broadside to the source, as shown. Clearly, if the object were located

at the zenith as seen from the interferometer, then one could identify

















POLE


EQUATOR


Figure 2-2. The (u,v) coordinate system. As viewed from a
celestial object, an interferometer which is oriented east-west on the
rotating earth has the projected baselines b1= b2, and b3 at
three different times.


NORTH
I









the u and v directions with east and north, respectively, on the earth's

surface. In general, however, the transformation between the two systems

depends on the interferometer's location on the rotating earth as seen by

the source, and is therefore a function of time. It is apparent from

Figure 2-2 that both the orientation and the length of the projected

baseline seen by the source will change as the earth rotates, for a fixed

pair of antennas. Thus, without the need to move either dish, one

samples the visibility function with a wide variety of effective

spacings. This is the basic principle of earth-rotation aperture

synthesis.

Figure 2-2 suggests that it may be helpful to recast the observing

geometry in terms of b, the projection of B in the (u,v) plane. This

quantity is shown both in Figure 2-2 and in Figure 2-1. In the latter,

it is obvious that the vector baseline B may be written



B =b+rcs0

(2-17)
Therefore,



B~r = b~r+rcs0'E


Since ED and r are mutally orthogonal vectors,


B~r = br

(2-13)
and so equation (2-16) becomes











dVjk(a, 6,t)=Aj(t)Ak(t)I(a ,6 )PO(a-a0** 0r)dn x

x exp {i[ D(b~r)mt)+@j(t)- ki)3

(2-19)

Integration over the solid angle of the source gives


VIjk(t)=Aj (t)Ak(t)exp { i C j(t)- 0k(t)] ) x

x I(a,6)PO(a-aD16-60) exp~i( -%g~br)]dn

(2-20)

This expression gives the response of each two-element interferometer in

the array to an extended source of emission. Of course, in an actual

observation a great deal of noise, generated both by the atmosphere and

in the instrumental system, also will be detected, and this has been

neglected in our analysis. However, this noise should not be correlated
between antennas, and so, over a long enough period of time, it should

average to zero for each antenna pair. In practice, statistical

deviations from this ideal, together with the variations in the amplitude

and phase gain of each antenna, will generally cause a small, but

nonzero, correlation of this noise component for each baseline. If these

effects are more or less equal for all the visibility records, as they

should be, the noise in the final map will be spatially random. Noise

which does not satisfy this condition can create map artifacts, and

should be edited out of the data base, if possible. An example is

interference, although the rejection of uncorrelated signals gives









aperture synthesis data some degree of protection from many types of

interference.

The factors outside the integral in equation (2-20) are not of

astronomical interest, as they contain only calibration factors in ampli-

tude and phase. The integral itself is accordingly sometimes referred to

as the visibility function; let us denote it by V'. By inspection, V' is

the Fourier transform of I'(a,6), where I' is the product of the true

brightness distribution I and the single-dish power pattern PO. If the

source brightness is specified in a coordinate system (x,y), which is

centered on (a0000) and parallel to the (a,6) system at the source,

V' can be written in the formn of a standard two-dimensional Fourier inte-

gral (Fomalont and Wright, 1974, p. 261):


V'(u,v)= H I' (x,y)exp~i2a(ux+vy)]~ dx dy.
(2-21)

This equation assumes that the field of the view is small enough that the

curvature of the sky plane can be neglected.

The quantities u and v in this expression can be identified with

the components of b in the (u,v) plane, provided that the latter are

measured in wavelengths of the emitted radiation so that they are dimen-

stonless. Equation (2-21) implies another interpretation of u and v:

they play the role of "frequencies" in the spatial Fourier "spectrum" V'

of I'. Consequently, they are usually called spatial frequencies. In

light of this interpretation, we see that a single measurement of the

complex visibility function of two antennas at a given time, correspond-

ing to a particular projected baseline and hence a particular (u,v),

gives a single Fourier component of I'. Because of the similarity

theorem of Fourier transforms (Bracewell, 1965, pp. 101-104), features

which have large extent in the (x,y) domain will have small extent in











the (u,v) domain, and vice versa. Therefore, in order to obtain high

angular resolution, one must have wide spacings present in the array.

Conversely, the detection of large-scale structure requires the inclusion

of low spatial frequencies (i.e., small spacings are required).

Suppose we define one antenna of our pair as the reference position

and plot the relative position of the other in the (u,v) plane. As time

goes by, this point will sweep out an arc whose shape depends on the

orientation of the baseline B on the earth and the declination of the

source. Since the choice of the reference antenna is arbitrary, the arc

which is diametrically opposite will also be traced out. In effect, two

Fourier components are measured at once. However, no more information is

gained because I' is a real function, and therefore its Fourier transform

is Hennitian (Bracewell, p. 16). Hence it is completely defined by its

values over half the plane. An example of the arcs produced for a

particularly simple geometry is shown in Figure 2-3. When many antennas

are present in an array, and all their relative separations are plotted

in this way, the entire (u,v) plane becomes covered with these tracks,

indicating that very many of the Fourier components of the source

brightness have been measured. The array has then approximately

synthesized the Fourier sampling of a single filled aperture equal to the

greatest separation in the array. This is the origin of the term

"aperture synthesis." However, spacings less than the physical diameter

of the dishes are missing from the synthesis. The importance of this

point is discussed below.

Intensive coverage of the (u,v) plane is necessary, because the

quantity of interest is actually, of course, the intensity I in the sky

plane. The primary beam PO is usually known well enough so that I can






































Figure 2-3. The tracks swept out in the (u,v) plane, over the course of several
hours, by a single interferometer pair. The points labeled hib 2, b3 correspond
to the three interferometer positions shown in Figure 2-2. T coverage is typical of an
a northerly source observed by an east-west interferomneter.










be recovered from I' if the latter is available. However, I' must be

calculated by inverting the Fourier transform of equation (2-21):



I'(x,y)=if V' (u,v)exp[-i~n(ux+vy )] du dv.

(2-22)
Since V' is measured only at a finite number of points, and each

measurement includes observational noise, I' cannot be determined

uniquely, nor without error. We will return to a discussion of this

inversion later in the present chapter.

We have now described all of the fundamentals of this observing

technique which bear directly on the present research, except for the

acquisition of high-resolution spectral data. Good spectral resolution

of the signal received by an aperture synthesis instrument requires that

one of several possible methods be used to divide the emission into a

number of independent, narrow-band spectral channels which can be pro-

cessed separately. While all these methods are conceptually equivalent,

in practice the necessity of holding the number and complexity of elec-

tronic components to a manageable level has dictated the use of correla-

tion receivers. A thorough introduction to these devices, with some

attention to practical design considerations, is given by D'Addario

(1982). The mathematical analysis and the electronic implementation of

this type of system are quite complex, as Fourier transforms in both the

time/frequency and space/spatial frequency domains are encountered.- Here

we will only outline the basic principle of their operation, beginning

with a qualitative demonstration of the underlying reasoning.

In our discussion of delay tracking, the compensating delay served

the purpose of restoring the time coherence of the wavefront received











at the two antennas. Indeed, in the idealized case of monochromatic

incident radiation, if one had perfect, continuously adjustable delay

tracking, and no atmospheric or ionospheric distortions of the wavefront,

the delay r' would restore full coherence to the wavefront. Except for

the contribution due to background noise, the outputs of the two

receivers would then be perfectly correlated for a point source at the

tracking center 1.0. Now suppose that a relatively large additional

delay 1l is inserted, after T', in this blissful arrangement.
Obviously, the effect is to destroy the coherence of the two signals at

the inputs to the multiplier, resulting in zero signal detected after

correlation. The only exception to this will be for a very narrow range

of frequencies centered on some frequency wi, for which the shift

corresponds to a complete cycle of 2rr in phase. In effect, the

introduction of the lag T1 has acted as a narrow-pass filter,

isolating a single spectral channel from the band Am. The use of mnany

such lags allows one to divide the signal into many channels. The

frequency resolution of the system is controlled by the maximum lag. The

most common correlator scheme, which is used at the VLA, employs lags of

both positive and negative sign in equal numbers, and spaced at equal

intervals AT1. If there are a total of N1 of these, the frequency

resolution is (CNydT1)-1, where C, of order unity, depends

on the weighting used in the frequency Fourier transform (Baldwin et al.,

1971).

Mathematically, there is no distinction between T1 and the

other various delays which have been considered previously. We can

rewrite equation (2-21) to allow for its presence:










V'(u,v,r1)=i II '(x,y)exp fi~arl+2n(ux+vy)]} dx dy.

(2-23)
If we integrate over the bandwidth, we find the integrated visibility

VII :


VII(u,vT)= iiII'(x,y)F(w)exp{ iwarl+2x~ux+vy)] } dx dy dw ,

(2-24)
where F( ) is the bandpass function. Because of thle symmetry of the lag

spectrum, only the real part of V'g need be Fourier transformed
(Hjellmning and Basart, 1982, p. 2-37). When we take the Fourier trans-

form with respect to the lag 1,weotn


Re(VI')exp(iwry)dT1=iI I'(x,y)F(w)exp~i2n(ux+vy)] dx dy ,

(2-25)
where the right-hand side must now be evaluated at one of the narrow-band

frequences wi determined by the la9 spectrum. Therefore, the extension
of ordinary aperture synthesis techniques to spectral line work involves

three principal complications, the first two of which are resolved in the

design of the instrument and are not of immediate concern to the user:

(1) the need for a complicated correlating receiver with provision for a

large number of digital lags; (2) the need to Fourier transform the lag

spectrum once per integration period, as implied by equation (2-25); and

(3) the need to calibrate the band pass F(w). The latter is accomplished

by observing a strong continuum source, which is assumed to exhibit no

spectral variation over the quite narrow total bandpasses normally used

in spectral line observations.










VLA Observations of NGC 3359

The NRA0 Very Large Array was used to observe the 21 cm line in

NGC 3359. The VLA is an earth-rotation aperture synthesis instrument,

consisting of 27 fully steerable dishes of 25 m aperture apiece (Thompson

et al., 1980). Nine antennas are deployed along each of the three arms

of a Y-shaped pattern. The arms are oriented 1200 apart, approximately

in the north, southeast, and southwest directions. This arrangement

causes the set of baseline vectors at any given time to cover a wide

range in azimuth, providing good (u,v) coverage. The radial spacing

along an arm increases outward from the array center. This gives denser

sampling of the (u,v) plane near its origin, which results in better

sensitivity and sidelobe suppression. We will return to this point in

our discussion of tapering.

The spacings of the antennas can be changed to allow observations

at different resolutions, but remain fixed during a single observing run.

There are four standard sets of antenna positions. All have roughly the

same relative spacings, but the overall scale of the array is changed

from one to another. The choice among these configurations involves a

compromise between resolution and sensitivity requirements of the obsery-

ing program. The similarity theorem dictates that the highest resolu-

tions can only be achieved with large separations. On the other hand, it

also requires the inclusion of short spacings if broad structure is sus-

pected to be present. Even more importantly, high resolution observa-

tions have relatively poor brightness sensitivity. Although the minimum

detectable flux density hS depends only on the system temperature,

bandwidth, integration time, and total collecting area, for mapping a

resolved source it is the detectable brightness which is important.










This is related to AS by (e.g. Kraus, 1966, p. 102)



aB = abm-1 hS

(2-26)

where nbm is the solid angle of the "synthesized beam," in other

words, the power pattern of the array as a whole. The synthesized beam

is thus equal to the nonnalized brightness distribution observed for a

point source. The complementary relation expressed by equations (2-21)

and (2-22) indicates that the bean is smaller when larger (u,v) spacings

are used. Consequently, the sensitivity to extended structure is degrad-

ed whenever the resolution is improved, and vice versa.

For this project, the two smallest available configurations, the

so-called C and D arrays, were used. Since narrow-band spectroscopy

divides the available signal power from the galaxy into many channels,

sensitivity tends to be poor. The C configuration was expected to be the

largest array with adequate brightness sensitivity. This is demonstrated

by our results. The best peak signal-to-noise ratio for any of the maps

is about 20. The next largest or "B" array synthesizes a beam of only

one-tenth the solid angle of that of the C array. Thus, we could have

achieved a "two sigma" detection, at best, with the B array. The C con-

figuration contains spacings from a maximum of about 3 km, or roughly

15,000 wavelengths at 21 cm, down to approximately 100 m, but the inner

spacings are rather sparsely sampled. Data from the smaller D array were

desired as well, to ensure that no low-amplitude, large-scale emission

was missed. The separations in the 0 array vary from about 40 m up to

just under a kilometer, overlapping the inner C spacings fairly heavily.

The total range of available spatial frequencies, therefore,










gives sensitivity to structure on scales from about 14" to 17'. If

spatial wavelengths larger or smaller than this are present within a

single spectral channel, they will not be detected. Naturally, a real

object is not likely to be composed of a single spatial frequency. It

is, perhaps, more relevant that a source whose brightness distribution is

Gaussian will be at least 50% resolved, if its full width at half power

lies between about 7" and 8' (see Figure 10.A4b of Fomalont and Wright,

1974). Of course, it is certain that fine-scale structure is indeed

missed in this way. The lower limit, in practice, was increased from 14"

to about 18" because of the taper employed (see below). Since the (u,v)

coverage was fairly uniform at all azimuths, these limits should not

depend strongly on the shape of the source.

These spacings are unprojected, and are therefore upper limits.

The physical size of the antennas places a lower limit on the available

(u,v) spacings. When the projected separation of two antennas becomes

less than the dish diameter of 25 m, one of them is partially blocking

the other's view of the source. This "shadowing" effect can be corrected

in the calibration and editing of the data. However, a more insidious

result of this situation is "cross-talk," in which the shadowed antenna

detects signals from the electronics of its neighbor. In practice, it

has been found that in every case where even slight shadowing occurs,

there is a quite noticeable deterioration in signal quality, attributable

to cross-talk. All such data were simply discarded. The problem only

arises when the elevation of the source is low, so for NGC 3359

(6=+63"5) very few data are involved.

The observations of NGC 3359 were obtained on 27 January 1983, with

the C array, and on 21 June 1983, with the D array. In most spectral












line applications, it is not possible to use all the antennas because of

the limitations of the correlator. The correlator must multiply the sig-

nals from 2n delay lines, for each of N(N-1)/2 baselines, where n is the

desired number of channels and N the number of antennas. The details of

the particular correlator used, therefore, place an upper limit on the

product nN2. The choice of n is dictated by two conflicting astronom-

ical requirements. The larger n is, the better one's spectral (and hence

velocity) resolution, but the poorer the sensitivity. At a minimum, n

must be large enough to cover the entire velocity width of the 21 cm

line. For NGC 3359, past experience and inspection of previously pub-

li shed global 21 cmn profiles indicated a choice of approximately 20 km/s

for the single channel width. Rots (1980) had measured a width of 260

km/s, at the 25%, level, for the global spectrum. To be certain of

detecting any faint emission at extreme velocities, it is necessary to

allow a margin beyond this. It is also desirable to have a "baseline" of

a few signal-free channels. Since n is restricted to being a power of

two, so that the Fourier transform of the lag spectrum can be calculated

with Fast Fourier Transform (FFT) techniques, n=32 was chosen. During

the January observations, this necessitated restricting the number of

antennas used to 21. Because of the northerly declination of NlGC 3359,

the southeast and southwest arms are the most important for getting good

coverage of the (u,v) plane. Of the six antennas to be omitted, four

were selected from the northern arm, and one fron each of the others. By

June, the correlator had been upgraded, and only two antennas had to be

omitted. These were simply chosen on the basis of their recent malfunc-

tion history.










In each observing run, the data were calibrated in amplitude and

phase, as described in the previous section, using a somewhat complicated

calibration scheme. The purpose of this calibration is to correct the

slow variations in the system response to a point source of constant

flux. These variations are of two types: those arising somewhere in the

instrument, and those imposed by fluctuations of the atmosphere. The

latter are usually more severe at the VLA. Consequently, it is important

that the calibrator source be near the program source in the sky, and

especially that it be at nearly the same elevation. However, the number

of available calibrators is small, since few bright radio sources are

both unresolved and constant in flux. For NIGC 3359, it proved helpful to

bracket it in declination between two different calibrators. A third

calibrator had to be used to complete this scheme, on account of the dif-

fering hour angles of the calibrators and NGC 3359. Finally, the bright

source 3C 286 did double duty as the "primary" calibrator, i.e., the

fundamental standard used to determine the flux densities of the other or

"secondary" calibrators, and as the bandpass calibrator. In the C array

observations, 3C 386 was observed twice, once on either side of transit.

This made it possible to optimize the choice of switchover time between

secondary calibrators, while ensuring that all of them had primary cali-

bration. It also guarded against losing the primary calibration for any

antennas which happened to be experiencing technical malfunctions during

the time of primary calibration. Unfortunately, the hour angle made it

impossible to do this for the June observations. In both cases, the bas-

ic observing cycle alternated thirty-minute integration on NGC 3359 with

much shorter scans of the calibrators. The duration of the latter










depended on the flux density of the calibrator, varying from three

minutes for 3C 286 to eight for secondary calibrator 1031+567.

There are three relevant integration times for these data. The one

of ultimate interest is, of course, the total amount of time spent

observing the program source. The integration time of immediate concern

in the on-line computer system, however, is the time for which the output

signals from the multipliers are averaged, before the FFT of the lag

spectrum is taken. The latter step produces a single estimate of the

multi-channel spectrum of the source for each baseline. Subsequently,

this signal is itself averaged for a slightly longer time to reduce the

volume of data to manageable levels. These times were 20 and 40 seconds,

respectively, for both observing runs on NGC 3359.

The implementation of the calibration procedure outlined in the

previous section is conceptually straightforward, in the case where there

are no unusual problems in the data. Primarily for calibration purposes,

a "pseudo-continuum" channel is generated along with thle narrow band

channels. This channel is so called because it contains the broadband

signal fromn the central 75%, of the original intermediate frequency band-

pass; it would be more accurate to term it the continuum-plus-line chan-

nel. In the present case, this "channel 0" has a bandwidth of approxi-

mately 4.7 MHz, compared to the frequency separation of about 98 kHz for

the spectral line channels. Therefore, the sensitivity to the cali-

brators is almost seven times better in this channel than in the individ-

ual line channels. The calibration is determined for channel 0, and then

applied to the line channels, along with the bandpass calibration. The

detailed procedure is as follows.











First one forces the flux density of the primary calibrator to

assume its well-known value at the frequency of observation. A solution

for the amplitude and phase of each antenna in the array, as a function

of time, is then computed, using all the scans of the various calibrat-

ors. The flux densities of the secondary calibrators are used as free

parameters in the amplitude solution. When this has been done, baselines

with large closure errors in the amplitude or phase solutions can be

identified. For the observing mode being discussed here, one often has a

few such errors of greater than 10% in amplitude or 100 in phase. It is

best to delete such baselines from the data for the span of time in which

they are troublesome, and repeat the solution. After iterating this pro-

cedure until the data seem acceptable, one can use the flux density of

the primary calibrator to determine those of the secondary calibrators.

The latter are said to be bootstrappedd" fluxes. Their errors are

indicative of the overall stability of the array-atmosphere system during

the observing run. The fluxes of all calibrators, and the total

on-source integration time for each, are listed for both observing runs

in Table 2-1. The source names used for the secondary calibrators are

the IAU designations. The integration time is also given for NGC 3359

itself, for comparison.

Once this stage has been reached in the calibration process, the

observer must decide whether the particular solution for the antenna

gains, which he now has in hand, is satisfactory. Since the solution

uses a particular reference antenna to establish the phase reference for

the array, it is well worth one's while to try to select one which is

particularly stable. This is done by listing the phases and amplitudes

of the other antennas relative to the reference antenna, after the


















Integration
Data Base Object Typea Flux Dens ity, Jly Time, min


TABLE 2-1
CALIBRATION PROPERTIES


14.76b

4.172 +

1.883 +

2.529 +




14.755b

4.268 +

1.938 +

2.539 +


Jan. 1983,

C Array








June 1983,

0 Array


3C 286

0836 + 710

1031 + 567

1311 + 678

NIGC 3359


3C 286

0836 + 710

1031 + 567

1311 + 678

NIGC 3359


aPC = Primary Calibrator, SC = Secondary Calibrator, PO = Program Object.

bAssumed flux density, found by interpolating the results of Baars et al.
(1977) at the frequency of observation.










solution has been made. The hope is to find an antenna whose varia-

tions, compared to the ensemble of all the other antennas in the array,

are slow, as small as possible, and not monotonic functions of either

space or time. If the first antenna chosen does not meet these criteria,

one can usually improve the solution by choosing another reference anten-

na and repeating the entire process from the beginning.

When a good solution has been found, it is applied to the entire

data set, including the observations of the program source, by a simple

running mean, or "boxcar," interpolation of the computed amplitude and

phase gains of the individual telescopes. Inspection of the quality of

the solution at this point can identify more baselines which may merit

removal. Finally, the bandpass is calibrated by assuming a flat spec-

trum, over the small total spectral-line bandwidth, for the primary cali-

brator. Then the calibration is complete, and the data may be Fourier

inverted to give maps.

Unfortunately, both the C array and the 0 array observations of

NGC 3359 were affected by peculiarities which complicated the calibration

and editing considerably. The spectrum of the bandpass calibrator,

3C 286, in the C data was afflicted by a rather sharp drop, mimicking a

spectral line, of about two channels in width. The drop in flux was

about 2%, many times the mean noise across the band. Subsequently, the

variation was found to be caused by a hardware problem in the Fluke

synthesizer, which controls the precise frequency of observation. It was

not possible to isolate the bad data in the (u,v) plane and the effect

eventually had to be corrected in the mapping process.

The 0 array observations were rather adversely affected by the

presence of interference from the sun. The sun is a bright source at









21 cm, and its sidelobes can create difficulties, especially for the type

of observing program described here. The sidelobes will drift through

the field being observed, but normally when one is well away from the

position of the sun in the sky, they will be washed out by bandpass

smearing. For spectral line observations, of course, this helpful effect

of the bandwidth is greatly reduced. Also, since the sidelobes are

large-scale features, they are most deleterious at the shortest base-

lines, so that the D array is the most susceptible of the VLA configura-

tions. Although the sun was some 710 from NGC 3359 in the sky during the

D observations, strong interference was nevertheless found in the data.

Its most obvious manifestation was the presence of very large fluctua-

tions in the amplitudes and phases of the individual baselines after

applying the calibration solution. That noise ceased abruptly at sunset.
This conjecture as to the nature of the interference was also confirmed

by its being tied to specific baselines, rather than to antennas, and

primarily to the shorter baselines. It was decided, again, that the

problem was best dealt with in the map plane. The only solution to this

type of interference is to remove baselines from the data set, but if one

simply deletes all the short baselines from the run, the (u,v) coverage
is badly degraded. The approach must be to reject the worst offenders

until the effect is no longer visible in the maps. While still in the

calibration and editing process, however, the amplitudes and phases were

studied carefully, and a list of recurrent problem baselines was pre-

pared. These provided a starting point for flagging after the data were

transformed into the map plane. It was found that most of the baselines

of less than 60 m had at least sporadically large errors of this type.

The worst of these was the second shortest baseline in the array, at










40 m unprojected separation. For that antenna pair, the problem was so

pervasive that the baseline was removed in the editing mode to allow a

good calibration solution.

There are several decisions to be made in the Fourier inversion of

the visibility data, for each channel, to give the single channel maps.

These concern primarily the compromise between sensitivity and resolution

referred to in the previous section, and they arise in the assignment of

certain weights to the data before transforming. Obviously, the calibrat-

ed visibility data are very poorly distributed in space, for the purposes

of computing their Fourier transform. As the number of individual points

is on the order of 250,000 in the present instance, FFT techniques are a

necessity. These require that the data be arranged in an m x 1 rectangu-

lar grid, where m and 1 are both powers of two. In the present work, we

have used a square grid (m = 1), as the sampling is quite similar in the u

and v directions. The visibility data, on the other hand, lie on ellipti-

cal arcs, with spacings between measurements determined by the rate of

change of the projected (u,v) separation for each baseline. This situa-

tion is handled by computing an approximate representation of the data on

the desired square grid, where each tabulated value is found by averaging

nearby observed points in some way. To do the averaging, we convolve

these points with a function C(u,v) whose Fourier transform is rather flat

to some radius and then falls off rapidly. However, it should not have

high sidelobes beyond the map area, which disqualifies the simple "two-

dimensional boxcar," or "pill box," function. (This requirement helps

suppress aliasingg," onto the map, of sources which lie outside the field

of interest, Sramek, 1982.) In our maps, the convolving function used was

the product of an exponential and a since function, i.e.










C(u,v) = C'(u)C,'(v)

where


C'()=ep(-I u 2 ,sin(ru/1.556u)
2.52 Au (nu/1.55 Au)

(2-27)
and Au is the (u,v) "cell size," that is, the spacing of the square grid

to be formed. The values of the numerical constants have been chosen to

optimize the detailed shape of the Fourier transform of this function.

Sramek gives a thorough discussion of the factors affecting the choice of

C(u,v), as well as the weighting functions discussed below. After

convolution, the map is then sampled by the two-dimensional "shah"

function (Bracewell, 1965, p. 214) to obtain the gridded values, and the

FFT can be computed.

Before the transform is performed, one may wish to assign some

additional weights to the gridded data. In general, there are two kinds

of weighting employed, the first of which is global in nature, the second

local. The global weighting is called tapering, and consists of

multiplying the weights of all points by a factor which decreases at

greater distances from the origin of the (u,v) plane. The purpose of

tapering is to decrease the small scale sidelobes of the array by
reducing the importance of the measurements at large spacings, which

determine the high spatial frequency structure of the map. Tapering is

usually effective at reducing sidelobes because these outer portions of

the (u,v) plane are less densely filled with data, and hence less well
determined. Another way of viewing the situation is in terms of the

"dirty beam," which is the response of the array to a point source. For










extended observations with the VLA, the dirty beam usually consists of a

small, Gaussian core, with broader sidelobes at the level of one to ten

percent. The shape of these sidelobes is simply the Fourier transform of

the unsampled spacings in the (u,v) coverage (Ekers, 1982, p. 12-10).

This statement is strictly true if we include, in the unsampled spacings,

those at radii from the outer edge of the sampled aperture to infinity,

whose Fourier transform is the Airy diffraction disk. The low density of

observed points in the outer (u,v) plane is therefore the direct cause of

the strong inner sidelobes, which is why tapering to reduce the

importance of these measurements improves the sensitivity. However, it

is equally obvious that it does so at the expense of the resolution. A

simple Gaussian taper is normally chosen, if one is used at all.

One can also assign a weight to each cell, based upon some measure

of the expected signal-to-noise ratio within that cell. In one scheme,

the weight is simply proportional to the number of records in the cell.

This is called "natural weighting." It is useful for observations where

the signal-to-noise ratio needs improvement, but it tends to lead to a

significant loss of resolution because it weights the center of the (u,v)

plane very heavily. The other common approach, "uniform weighting,"

simply assigns equal weight to all nonempty cells. When this weighting

method is adopted, the beam characteristics are controlled by the taper

(Sramek, 1982).

Finally, one must choose the (u,v) cell size Au. From a

consideration of the gridding in the (u,v) plane, it is obviously

desirable to have the product of Au with the number of cells on a side,

m, be neither so large that the outer cells are all empty and the inner

ones heavily undersampled, nor so small that many points at large










spacings are thrown away. Furthermore, Au is related to the cell spacing
Ax in the map plane by





(2-28)
where Ax is in radians. For the VLA, an empirical relation which

produces good sampling is that the synthesized beam should be three to

four times hx.

After the transform has been calculated, producing a dirty beam and

its "dirty map," one can try to remove faulty data, such as those alluded

to in the discussion of calibration, by analyzing the image in the map

plane. As pointed out by Ekers (1982), many serious errors are difficult

or impossible to detect in the (u,v) plane. Conversely, many errors

which seem significant in the (u,v) plane produce no detectable features

in maps, and so need not be pursued further. The two most serious

difficulties in the NGC 3359 observations--the hardware-related bandpass

glitch in the C array data, and the solar interference in the 0 array

data--behave in the former way, for the most part. In both cases, the

most seriously affected of the single-channel maps had readily visible

artifacts. These consisted of fairly high-frequency, concentric rings in

the first instance, and of low-frequency, linear ripples in the second.

The solution in each case was to isolate a group of possible culprit

baselines, as found from the (x,y) wavelength of the image defects, and

to delete each of them in turn, checking whether the imaage was then

noticeably improved. This method worked especially well for the solar

interference. The removal of a second baseline solved thle problem











entirely. That no more drastic remedy was required, in view of the

apparent problems in the calibrated amplitudes and phases mentioned

earlier, probably can be attributed to the comparatively short duration

of most of those (u,v) errors. For the C array data, a similar procedure

was followed, and again only two baselines were deleted. In this case,

although the other affected channels were corrected quite well, the most

severe example, that at a central velocity of 1119.4 km/s, could not be

wholly restored, and some very low-amplitude ripple remains in the map.

However, the level of this spurious anission is well below the threshold

which would be detected as a real signal by the routines used to

calculate mean densities and velocities. Therefore, the contamination of

the final integrated maps by this feature should not be too important.

After one has transformed the visibility data into a set of

single-channel maps, these often suffer noticeably from contamination by

the sidelobes of the dirty beam. An equivalent way to state this is that

the map is flawed, owing to an unrealistic estimate of the brightness at

unsampled spatial frequencies (Hi~gbom, 1974, Cornwell, 1982). In

particular, the straightforward application of the FFT upon the

visibility records gives an estimate of zero brightness at all unmeasured

(u,v) positions, wherever they occur. In one sense, this estimate is the

truest to the available data, since it assumes absolutely nothing about

the source which is not measured. For this reason, it is called the

principal solution. On the other hand, it is obviously implausible, and

it would be very desirable to get a better estimate of the true

brightness distribution at the unmeasured positions. To put it another

way, we wish to remove the effects of the sidelobes of the dirty beam.










There are several ways to attempt this. In the case of NGC 3359,

we have used -the well-known "CLEAN" program, devised by Higbom (1974)

and modified by Clark (1980). This program, in the form used at the VLA,

has been discussed extensively by Cornwell (1982) and by Clark (1982).

The algorithm of CLEAN performs an explicit correction for the effects of

the dirty beam. In essence, CLEAN simply subtracts the entire dirty

beam, including side10bes, at the position of greatest brightness in the

map. Then, the remaining map is scanned for its brightest point, where

the procedure is repeated, and so forth. This process is continued until

it is decided that the remaining emission on the map is simply noise.

The best estimate of the true brightness is then considered to be the set

of points that CLEAN has selected, each with its proper brightness, but

without sidelobes. When CLEAN has been applied properly to a map, the

"blank" areas of that map should have no visible sidelobe structure, only

random noise. This criterion, judged fairly subjectively for a few

example maps, was used to choose an iteration limit for CLEAN and all

channels were then processed in a uniform manner. An empirical check on

the validity of this process is provided by measuring the ans noise of an

apparently blank region of sky. This value should be nearly the same in
each channel, and should not exceed by more than factor of about two the

following empirically predicted value (Rots, 1983), which holds for

spectral line maps mnade with natural weight:


op=a[N(N-1)TiAv-1/ mJy per beam.

(2-29)
Here N is the number of antennas used, Ti is the total on-source

integration time in hours, Av is the narrow bandwidth in kHz, and a is a

constant which is equal to 620 in the 21 cm band. Taking N = 23,












Ti=12.4, and 60=98, we have for our observations 09=0.8 mJy per beam
solid angle. The measured value of about 1.1 mJy per beam is quite

acceptable.

The CLEAN algorithm is complicated in several ways by the require-

ments of stability and accuracy of the final or "clean" image. First,

only a small fraction--the so-called loop gain--of the brightness at the

selected point is actually multiplied by the dirty beam, and subtracted,

at each step. Second, it is best to restrict the area which CLEAN

searches for flux to be cleaned, in order to restrict the number of

degrees of freedom available in fitting the data (Cornwell, 1982). The

most common practice is to set a few rectangular boxes, often consisting

of the inner quarter of the field, augmented by any other areas of obvi-

ous signal. CLEANI is allowed to find real flux only within these boxes.

Finally, the "clean beam" must be mentioned. As explained above, the end

result of CLEAN is that the observed flux density has been distributed

into a set of discrete spikes of varying intensity. This is not felt to

be a very good representation of the true brightness distribution in a

typical, extended astronomical source. Some sort of spatial smoothing of

these spikes must be performed. This is accomplished by convolving them

with a "clean beam," consisting of an elliptical Gaussian function, fit

to the inner portions of the dirty beam. The convolved images, hereafter

designated "clean maps," will form the basis for our discussion of the

neutral hydrogen characteristics of NGC 3359. Note that the resolution

of the clean map is controlled directly by the dimensions of the clean

beam (and so, ultimately, by those of the dirty beam).










Besides improving on the sidelobe characteristics of the principal

solution, there is a further advantage to the use of CLEAN. Most of the

gaps in the (u,v) coverage result from the vagaries of the detailed

sampling available, and their importance is slightly lessened by the

presence of observations at the same absolute separation, but different

azimuthal orientation. There is a significant exception, however. At

the very center of the (u,v) plane is a hole of some finite radius, which

cannot be filled in. For the larger array configurations, this radius is

set by the array scale, but when D observations are included, the size of

the hole is reduced until a physical limit is reached. That limit is at

least one antenna diameter, because of shadowing problems; the presence

of cross-talk will~ increase the limit slightly beyond this. The point of

greatest interest is that at the origin of the (u,v) plane. The

visibility at this point is simply the Fourier transform of the total

flux in the mapped field, but it cannot be measured. Clark (1982) has

discussed the resulting difficulties in detail. Their root cause is

that, in these circumstances, the dirty map will have an average value of

zero, despite the fact that all real flux is positive. This is achieved

by balancing the relatively small region of actual emission on the map by

a large area of slightly negative "brightness." Nonnally, one sees this

as a slight, negative "bowl" in which the source sits. Since CLEAN

provides an estimate of the brightness at points in the (u,v) plane not

actually sampled, it may be able to interpolate successfully across the

central hole, if it is not too large. As Clark mentions, the positive

mean value of the clean beam is instrumental in producing this effect.

This is one of the best justifications for its use. CLEAN succeeded

quite well in performing this task, in the case of NGC 3359. Of course,










these data do contain 0 array spacings, so that the case is a rather

favorable one.

One further application of the CLEAN program is quite useful in 21

cm line work. At that wavelength, there are many unresolved continuum

sources in a typical field which are bright enough to pose a problem of

possible confusion with the neutral hydrogen emission. The most satis-

factory way to eliminate such sources (van Gorkom, 1982) is to estimate

their contribution in the (u,v) plane and to subtract them there, before

inverting the visibility data to obtain dirty maps. This estimate can be

provided by CLEANing a continuum map of the field, with the algorithm

allowed to find flux only in a few very small regions centered on the

discrete continuum sources. Six continuum sources were removed from the

NGC 3359 field in this way. Their estimated positions and flux densities

are listed in Table 2-2.

The mapping parameters used for the combined C and D visibilities

for NGC 3359 were as follows. The data were gridded onto a 512 x 512

(u,v) plane, giving, of course, output maps 512 x 512 pixels in extent.

The pixel size in the (x,y) plane was 6 arc seconds. Uniform weighting

was selected, as the brightness sensitivity of the data was quite good.

A Gaussian taper was applied, having a scale, from the origin to the 30%

level, of 10,000 wavelengths. (This Gaussian function, therefore, has a

dispersion of 6450 wavelengths or 1.36 km.) These values were chosen by

experiment, using one channel map with a very strong signal and another

with only weak H I, in order to find an acceptable compromise between

sensitivity and resolution.

Since care was taken, in the choice of bandwidth parameters, to

leave a baseline of observed channels with no expected signal, the





_~~


Table 2-2
CONTINUUM SOURCES IN THE NGC 3359 FIELD


Flux
Density,
mJya

5.62

24.5

103.3

19.8

25.3

3.54


Source Right Ascension
(1950.0)

1 10h 44m 07519

2 10 45 08.88

3 10 44 48.60

4 10 42 43.65

5 10 41 27.87

6 10 42 27.90

WT~e~se ~vTiialu~ees have bee orete orte


Declination
(1950.0)

+63033'38'.'5

+63 28 52.9

+63 22 08.8

+63 22 41.7

+;13 11 06.2

+63 34 11.4

effects of the


primary beam.











highest and lowest velocity channels in the data base were strictly

continuum maps. For safety, the first three seemingly line-free

channels, on either side of the band center, were assumed to be possible

line maps. This left five maps on either side of the line emission which

were not processed further. Although the remaining channel maps are

often called "line maps," it should be remembered in this case that they

still contain any continuum signal from NGC 3359 itself.

The dirty line maps were subjected to the CLEAN algorithm, with a

loop gain of 0.2, and 2000 iterations were used. The clean bean was

18'.'04 x 17'.'64 in size (full width at half maximum or FWHM), with its

major axis at a position angle of -5324 (astronomical convention).

Thus, as expected for a source of such northerly declination, the beam is

nearly round. More importantly, the linear resolution achieved is more

than twice as fine as that reached for this galaxy by Gottesman (1982)

using the Green Bank three-element interferometer. It is also somewhat

better than the typical resolutions of about 25" previously achieved at

the VLA for H I in barred spirals by Gottesman, et al. (1984) for

NGC 3992 and NGC 4731, and by van der Hulst, et al. (1983) for NGC 1097

and NGC 1365. This is primarily because the H I in NGC 3359 is so bright

that, in choosing the mapping parameters, there is little need to

sacrifice resolution for greater sensitivity.

The mean rms noise of the dirty maps is 1.16 mJy per beam solid

angle, with very little scatter from one map to another. That of the

clean maps is 1.11 mJy per beam. This value can be converted to a

brightness temperature via the following conversion for the clean maps,

which is based on equation (2-26):










1222.2
hTg -aS
v2g1 2


(2-30)

or ATB = 2.13 K for the present observations, where we have specified

aS in mJy per beam, v in GHz (1.4156 for these data), and 91. 02

are the full widths at half maximum, in arc seconds, on the major and

minor axes of the elliptical, Gaussian clean beam. On a single map, the

value for the noise level varied by 2-3%, depending on the particular

background area measured.

Figures 2-4 (a) to 2-4 (u) are contour representations of the

single-channel clean maps. The dirty map of the 1016 km/s channel is

shown, for comparison, as Figure 2-5. As might be expected from their

respective noise statistics, there is almost no difference between the

dirty and clean maps. This is not too unusual when uniform weighting

has been used, as the sidelobes generated are ordinarily less extensive

than for natural weighting. Table 2-3 gives information on the signal

and noise properties of the dirty and thle clean maps for each channel.

We will, for the most part, defer our consideration of the astro-

nomical significance of these results to the next chapter. There, we

will discuss the integrated maps of density and velocity for NGC 3359,

prepared from the single-channel maps. However, twro points are worthy of

mention before passing on to the next stage of our analysis.

First, the asymmetry of the spiral arms noted in Chapter One is

reflected in Figures 2-4. The channel maps at velocities slightly lower

than the band center velocity of 1016 km/s exhibit more spatially






























Figure 2-4. Single-channel clean maps of the neutral hydrogen in
NGC 3359 made with a velocity separation of 20.7 km/s and an effective
velocity resolution of 25.2 km/s. Coordinates of epoch 1950.0 are
shown. The contour levels are multiples of the approximate rms noise
of 1.15 mJy/beam: the first contour is at three times this value, the
next at five times, and succeeding contours at increments of 2.5 times
the noise level. The mean heliocentric velocity of each map is given
in its upper left-hand corner. The synthesized beam (full width, half
maximum) is shown to scale in the lower left of Figure 2-4 (a).

Figure 2-4 consists of Parts (a) through (u).







62














809















C
O1
















Right Ascension

Figure 2-4 (Part a)




















830


Ri gt Ascension
Figure 2-4 Continued (Part b)























3

651


C-3 I~r


:2 sc 02 Rigt Ascensio~n ^ 0
Figure 2-4 Continued (Part c)




























871

L..











C,'

O
*R

*

u
a,


12 4t OB 4? 32 00 42 c> r
RIGhT ASC-NSION

Figure 2-4 Continued (Part d)
























692









~1 -e*












20~









12 :0 c3 j3 22 2 30 e0
Right Ascension

Figure 2-4 Continued (Part e)


















I IO


42 32 20

f)


i/' _-,
=~
Y
~;~3'
c~\ ~~
'5 v
';j
GO


20Li


12 44 g2 43" 2.
Right Ascension
Figure 2-4 Continued (Part







68












c33















o


134 o4 00 2S

Rih seso
Fiue24 otned (atg







69













954



















CS






:2 44 ct 4i3 3 20 42' 32 :3
Rght Ascension
Figure 2-4 Continued (Part h)






































~-?
c: :


975


10 44 23 C3 32


ae 42 33
Right Ascension


Figure 2-4 Continued (Part i)

























































43 3020 l a?3C
R33ight Ascension
2-4 Continued (Part j)


13 44 23

Figure


i


995


m

V C
j;

~:~c~:


20


22







72















1016








*r--














22







10 re 02 43 3a 00 42 33 Et2
Right Ascension
Figure 2-4 Continued (Part k)
























































12 44 22

Figure


i


43 30 00 4^ 32 20
Right Ascension
2-4 Continued (Part 1)


i-
3~t ''
-- ~~


V


20







74













on~



Cj 15 !























10 44 22 c3 33 00 42 33 02
Right Ascension
Figure 2-4 Continued (Part m)






































































I


3:


c 20 '3 30 00 c
Right Ascension
Figure 2-4 Continued (Part n)


f


20


o

*r-


~

i""~r'~
,,
~ ~,c --~-
^? ~i
.:j


12Z 4







76















0O99













c-















12 44 80 ;3 13 22 52 20 a2
Right Ascension
Figure 2-4 Continued (Part o)


























%1 ~5 r


I




~e -


r


O
~17~~ D
i~--
sr ~S"
~


12


I
20


Figure


43 '_e 02. 42 30
Right Ascens10n
2-4 Continued (Part p)








78















1140








C.)C
GJo


12 cr 02 43 30 02e 2 23
Right Ascension
Figure 2-4 Continued (Part q)









79





























"cc




e I
o











104 0 4 0 0 4 0D
Rih Acnso
Fiue24 Cniud (atr
































12 44 DB 4 020 4 00












Right Ascension
Figure 2-4 Continued (Part s)

















































































12 44 2a 43 30 23 42 20 22
Right Ascension

Figure 2-4 Continued (Part t)


;Z202


2


r
O
C)
m
c


u
ar
D ,, L























1223













o
















0* ~~~ I
12 ic 20ga 3 22 1 3" 039
Right Ascension
Figure 2-4 Continued (Part u)



























































i I I I It


ICIE


i
~t ;C


\;"


:2 cc as


C3R1 ht Aiscensi:on


Figure 2-5. The single-channel dirty map of the H I in NlGC 3359,
at the center velocity of 1016 km/s. This contour map should be
compared to the clean map of Figure 2-4 (k). The map is labeled with
epoch .1950.0 coordinates. The contour levels are the same as those
used in Figure 2-4.









extended emission than their counterparts at higher velocities. These

former maps are those which contain the signal from the southern half of

the galaxy. While the neutral hydrogen associated with the dominant arm

is also seen in these maps, the outlying emission is well beyond the rad-

ius of that arm. As will be discussed in the next chapter, in the inte-

grated density map this emission seems to form ragged, outer arms or a

partial ring. Here we simply point out that the hydrogen at this large

radius is asymmetric in the same sense as the optical arms.

Second, one major surprise emerged when the C and D observations

were concatenated to give a single, high-sensitivity data set for map-

ping. At a distance of some fifteen minutes of arc from the center of

NGC 3359, and at least seven minutes from any other detected H I, a

small, isolated 21 cm feature was found in two adjacent spectral chan-

nels. This object seems to be a previously unknown, low-mass satellite

galaxy to NGC 3359. Support for this interpretation will be given in the

next chapter. It is not listed in the Master List of Nonstellar Optical

Astronomical Objects (Dixon and Sonneborn, 1980).

The object is clearly visible in Figures 2-6 (a) and (b), which are

simply the relevant clean channel maps with a greater area of sky shown

than in Figures 2-4. (An area centered on this feature and about 12' on

a side was searched for emission in the CLEAN program, along with the in-

ner quarter of the field centered on NGC 3359.) The emission from the

main body of NGC 3359 has been "overexposed" by the choice of contour

level in Figures 2-6, so that the weaker signal from the isolated cloud

can be seen. Finally, we include as Figures 2-6 (c) and (d) the adjacent

map on either side of this pair, at the same scale. The satellite HI-

clearly decreases drastically in brightness at these latter velocities.

The velocity width of its spectrum is evidently quite small.






















Figure 2-6. Single-channel clean maps of neutral hydrogen in NGC
3359, showing a larger area than in Figure 2-4. Each map is labeled
with its heliocentric velocity in km/s. Contour levels are identical
to those of Figures 2-4 and 2-5. Coordinates are for 1950.0.

(a) and (b). Two channel maps containing anission from the newly
discovered satellite galaxy of NGC 3359.

(c) and (d). The two channels adjacent to these; no emission
from the satellite is detectable.






















s$o


00 43 30


I I .I ((I
00 42 30 00 41 30 00


~ I I Ill


83 3S


25 .


O


I I i


10 44 30


Right Ascension


Part (a)


. Il '

954













6975


30 |-


2S F


on


I I I I I II
10 44 30 00 43 30 00 42 30 00 41 30 00

Right Ascension

Figure 2-6 Continued. Part (b).


P m';K~Yo










933
83 35

















2S




-1 I I I
10 44 30 00 43 30 00 42 30 00 41 30 80
Right Ascension
Figure 2-6 Continued. Part (c).




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