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
longdistance 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 twentyeight 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
11 GLOBAL PROPERTIES OF NGC 3359. . .. .. .. ... 14
21 CALIBRATION PROPERTIES .. .. .. .. .. .. .. 46
22 CONTINUUM SOURCES IN THE NGC 3359 FIELD. . .. ... 58
23 SINGLE CHANNEL SIGNAL AND NOISE CHARACTERISTICS. .. 90
31 LOGARITHMIC SPIRAL FIT TO FIGURE 33 .. .. ... 109
32 SUMMARY OF NEUTRAL HYDROGEN OBSERVATIONS . .. ... 138
41 BAR PROJECTION PARAMETERS. .. .. .. .. .. .. 168
42 EVALUATION OF INTEGRALS FROM GRADSHTEYN AND RYZHIK
(1980). .. .. ... .. .. .. .. .. .. 158
51 BEAM PROPERTIES, ONEDIM4ENSIONlAL CASE. .. .. .. 198
LIST OF FIGURES
Figure Page
11 The barred spiral galaxy NGC 3359, from a print of the
National Geographic SocietyPalomnar Sky Survey
blue plate. .. .. .... . .. ... .. . 13
21 Geometry of the twoelement interferometer .. .. .. 22
22 The (u,v) coordinate system. .. .. .. .. . ... 30
23 The tracks swept out in the (u,v) plane by a single
interferometer pair . .... ... .. ... 35
24 Singlechannel clean maps of the neutral hydrogen in
NGC 3359. .. .. . ... .. .... .. .. 62
25 Singlechannel dirty map of the H I in NlGC 3359, at the
center velocity of 1016 km/s. . ... .. .. .. 83
26 Singlechannel clean maps of neutral hydrogen in NGC
3359, showing a larger area than in Figure 24. .. 86
31 Contours of neutral hydrogen surface density in
NGC 3359. .. ... .. .. .. .. .. .. .. 102
32 The same integrated H I map as in Figure 31, shown
here in grey tone format. . ... .. .. .. .. 104
33 Distributions of H I and H II in NGC 3359. .. ... 107
34 Mean deprojected H I surface density in NGC 3359, as a
function of radius. ... .. .. .. . .... 114
35 Continuum emission in NGC 3359 .. ... .. .. .. 118
36 Contours of heliocentric, lineofsight velocity in
NGC 3359. . ... .. .. .. ... ... .. .. 122
37 Angleaveraged rotation curve of NGC 3359. .. ... 127
38 Rotation curve determined independently from velocities
along the two halves of the major axis. .. .. .. 128
FigurePage
39 Rotational velocities of the firstorder mass models
used in this chapter to estimate the mass of
N~GC 3359. .. .. .. ... .. .. ... .. 134
310 The observed global spectrum of neutral hydrogen in
NGC 3359. .. ... .. ... ... .. .. 137
311 Contours of integrated H I column density in the area
surrounding the isolated companion to NGC 3359. .. 140
312 Relative locations of NGC 3359 and its satellite . .. 142
313 The observed global spectrum of H I in the companion
galaxy to NGC 3359. . .... ... .. .. .. 144
41 The I passband image of NIGC 3359 .. ... .. .. 153
42 Geometry of a triaxial bar in a disk galaxy. . .. .. 157
43 Contour plot representation of Elmegreen's near
infrared plate of NGC 3359. ... .. .. .. .. 162
44 Near infrared brightness profiles, perpendicular
to the major axis of the disk of NGC 3359 .. .. 171
45 Brightness profiles along the bar of NGC 3359. .. .. 178
51 Gas response in the first of the purely triaxial
models of NGC 3359. .. .. ... .. .. .. .. 212
52 Gas response in the second of the purely triaxial
models. .. .. . ... .. ... ... . .. 216
53 Gas response in the pure oval distortion model .. .. 228
54 Gas response in the first of the hybrid oval distortion
and triaxial models .. . ... ... .. ... 235
55 Gas response in the second of the hybrid oval distortion
and triaxial models ... .. .. ... .. .. 239
56 Gas response in the third of the hybrid oval distortion
and triaxial models .. .... .. .. .. .. 243
57 Gas response in the final hybrid oval distortion and
triaxial model. ... ... .. ... . ... 247
58 The rotation curve derived from the model illustrated
in Figure 57 .. .. .. .. .. ... .. .. 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 singlechannel 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
wellknown "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 masstoluminosity 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 largescale 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 bestknown approaches are
the LinShu 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), LyndenBell 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,
quasistationary perturbation in an otherwise symmetric disk can, at
least under some conditions, be selfsustaining. 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, selfpropagating 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 highmass 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 selfpropagating 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
(11)
the equation of motion:
(pV)+ev(pv2+P)F = 0;
(12)
and the energy equation:
a(E)+o*[v(E+P)]Q= = .
at
(13)
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:
(14)
V2 e= nGp.
Since equations (11), (12), and (13) are nonlinear, the problems of
galaxian dynamics, contained in equations (11) through (15), 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 longlived 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 Nbody 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 selfgravity 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 smallscale 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 selfconsistent 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 selfattraction 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 diskbar 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 selfconsistent, 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 11 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
5m~eter in Sandage (1961); yellow and hydrogenalpha images from the
Palomar Schmidt telescope, in Hodge (1969); and the nearinfrared plate
of Elmegreen (1981), also using the Palomar Schmidt, which will be used
in Chapter IV of this study. Table 11 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, twoarmed 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 11. The barred spiral galaxy NGC 3359, from a print of the
National Geographic SocietyPalomar 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
SocietyPalomar Sky Survey. Reproduced by pennission of the California
Institute of Technology.
2
TABLE 11
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, twoarmed 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 firstorder 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 theprogram galaxy would suffice. Although the codes
used are gasdynamical, 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, earlytype 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 cm3) whose primary constituent
is atomic hydrogen in the ground state. This ground state has hyperfine
splitting into two levels separated by only 6x106 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 deexcitation is
much shorter than that for radiative deexcitation, 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.
(21)
If the velocity v is in km/s and TB in K, equation (21) gives NH in
atoms cm2. 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
(22)
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 twodimensional 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
millimeterwave 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 twoelement interferometer, as shown in Figure 21. 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
(23)
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).
(24)
TC
ANTENNA 1 ANTENNA 2
Figure 21. Geometry of the twoelement 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 lowpass 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 21cm 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 1ast 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
(25)
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. 215):
R(t) = coshgr 0 t)l
(26)
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* (ssg)+ c5(t ) k(t) *
27)
11l of the quantities on the righthand 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)
(28)
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 (27) in complexvariable notation:
Vjk(t)=Aj [t)Ak(t)exp~il 38a B* (ssg)+Jj(t) kit)l )*
(29)
By writing equation (29) 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 
(210)
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 (210). Observations of strong point sources, of wellknown
flux density and position, are used for this purpose. For a source of
this kind, equation (210) 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 (210) is
simply
Vjk(t)=Ajk(t)exp~ihjk~tt)
(211)
If this visibility were properly calibrated, it would be equal to
the flux density S of the calibrator. Thus, one can rewrite equation
(211) in terms of an effective complex gain for the jk baseline,
Gjk:
Vjk(t)=Gjk(t)S.
(212)
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)
(213)
where the asterisk denotes the complex conjugate. Combining the
expressions (28), (211), (212), and (213), we obtain the results
Aj(t)Aklt) A t
(214a)
(214b)
At each time for which calibration data are available, one has this
pair of equations for each of the N(N1)/2 baselines, so that the system
of equations for the N antennabased complex gains is highly over
determined. Therefore, a leastsquares 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 3045 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 online 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 twoelement 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 singledish 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
(215)
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 (29) 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
(2a0, 660). (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 (29), recalling the definition of T in equation (23), 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(aag,669)d x
x exp {i[ SD (B~r)+ mj(t)mkit)l )
(216)
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 eastwest in the earth's latitudelongitude system, as
illustrated in Figure 22. From the viewpoint of our source, it is much
more useful to describe this separation in the socalled (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 22. The (u,v) coordinate system. As viewed from a
celestial object, an interferometer which is oriented eastwest 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 22 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 earthrotation aperture
synthesis.
Figure 22 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 22 and in Figure 21. In the latter,
it is obvious that the vector baseline B may be written
B =b+rcs0
(217)
Therefore,
B~r = b~r+rcs0'E
Since ED and r are mutally orthogonal vectors,
B~r = br
(213)
and so equation (216) becomes
dVjk(a, 6,t)=Aj(t)Ak(t)I(a ,6 )PO(aa0** 0r)dn x
x exp {i[ D(b~r)mt)+@j(t) ki)3
(219)
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(aaD1660) exp~i( %g~br)]dn
(220)
This expression gives the response of each twoelement 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 (220) 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 singledish 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 twodimensional Fourier inte
gral (Fomalont and Wright, 1974, p. 261):
V'(u,v)= H I' (x,y)exp~i2a(ux+vy)]~ dx dy.
(221)
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 (221) 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. 101104), 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 largescale 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 23. 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 23. 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 22. T coverage is typical of an
a northerly source observed by an eastwest interferomneter.
be recovered from I' if the latter is available. However, I' must be
calculated by inverting the Fourier transform of equation (221):
I'(x,y)=if V' (u,v)exp[i~n(ux+vy )] du dv.
(222)
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 highresolution 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, narrowband 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 narrowpass 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 (221) to allow for its presence:
V'(u,v,r1)=i II '(x,y)exp fi~arl+2n(ux+vy)]} dx dy.
(223)
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 ,
(224)
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. 237). 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 ,
(225)
where the righthand side must now be evaluated at one of the narrowband
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 (225); 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 earthrotation 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 Yshaped 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 = abm1 hS
(226)
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 (221)
and (222) 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
socalled C and D arrays, were used. Since narrowband 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 signaltonoise ratio for any of the maps
is about 20. The next largest or "B" array synthesizes a beam of only
onetenth 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 lowamplitude, largescale 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 finescale 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 "crosstalk," 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 crosstalk. 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(N1)/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 signalfree 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 thirtyminute 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 online 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
multichannel 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 "pseudocontinuum" 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 continuumplusline 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 wellknown 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 arrayatmosphere system during
the observing run. The fluxes of all calibrators, and the total
onsource integration time for each, are listed for both observing runs
in Table 21. 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 21
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 spectralline 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
largescale 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)
(227)
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 twodimensional "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. 1210).
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 signaltonoise 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 signaltonoise 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
(228)
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 observationsthe hardwarerelated bandpass
glitch in the C array data, and the solar interference in the 0 array
databehave in the former way, for the most part. In both cases, the
most seriously affected of the singlechannel maps had readily visible
artifacts. These consisted of fairly highfrequency, concentric rings in
the first instance, and of lowfrequency, 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 lowamplitude 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
singlechannel 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 wellknown "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(N1)TiAv1/ mJy per beam.
(229)
Here N is the number of antennas used, Ti is the total onsource
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 fractionthe socalled loop gainof 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 crosstalk 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 22.
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 22
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 linefree
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 threeelement 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 (226):
1222.2
hTg aS
v2g1 2
(230)
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 23%, depending on the particular
background area measured.
Figures 24 (a) to 24 (u) are contour representations of the
singlechannel clean maps. The dirty map of the 1016 km/s channel is
shown, for comparison, as Figure 25. 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 23 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 singlechannel 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 24. The channel maps at velocities slightly lower
than the band center velocity of 1016 km/s exhibit more spatially
Figure 24. Singlechannel 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 lefthand corner. The synthesized beam (full width, half
maximum) is shown to scale in the lower left of Figure 24 (a).
Figure 24 consists of Parts (a) through (u).
62
809
C
O1
Right Ascension
Figure 24 (Part a)
830
Ri gt Ascension
Figure 24 Continued (Part b)
3
651
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Figure 24 Continued (Part c)
871
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Figure 24 Continued (Part d)
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Right Ascension
Figure 24 Continued (Part e)
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Figure 24 Continued (Part
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Figure 24 Continued (Part h)
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ae 42 33
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Figure 24 Continued (Part i)
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R33ight Ascension
24 Continued (Part j)
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Figure 24 Continued (Part k)
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Figure
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24 Continued (Part 1)
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Figure 24 Continued (Part m)
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Right Ascension
Figure 24 Continued (Part n)
f
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Figure 24 Continued (Part o)
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Figure
43 '_e 02. 42 30
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24 Continued (Part p)
78
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12 cr 02 43 30 02e 2 23
Right Ascension
Figure 24 Continued (Part q)
79
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e I
o
104 0 4 0 0 4 0D
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Fiue24 Cniud (atr
12 44 DB 4 020 4 00
Right Ascension
Figure 24 Continued (Part s)
12 44 2a 43 30 23 42 20 22
Right Ascension
Figure 24 Continued (Part t)
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2
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m
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Figure 24 Continued (Part u)
i I I I It
ICIE
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C3R1 ht Aiscensi:on
Figure 25. The singlechannel 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 24 (k). The map is labeled with
epoch .1950.0 coordinates. The contour levels are the same as those
used in Figure 24.
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, highsensitivity 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, lowmass 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 26 (a) and (b), which are
simply the relevant clean channel maps with a greater area of sky shown
than in Figures 24. (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 26, so that the weaker signal from the isolated cloud
can be seen. Finally, we include as Figures 26 (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 26. Singlechannel clean maps of neutral hydrogen in NGC
3359, showing a larger area than in Figure 24. Each map is labeled
with its heliocentric velocity in km/s. Contour levels are identical
to those of Figures 24 and 25. 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 26 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 26 Continued. Part (c).
