KINEMATICS AND DYNAMICS OF BARRED SPIRAL GALAXIES
MARTIN NICHOLAS ENGLAND
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
"To the Reader Concerning the Hypothesis of this Work"
Andrew Osiander cl543 in
"De Revolutionibus Orbium Caelestium"
It is the job of the astronomer to use painstaking and
skilled observation in gathering together the history of the
celestial movements, and then, since he cannot by any line
of reasoning reach the true cause of these movements to
think up or construct whatever causes or hypotheses he
pleases such that, by the assumption of these causes, those
same movements can be calculated from the principles of
geometry for the past and for the future, too. And if
[mathematical astronomy] constructs and thinks up causes--
and it has certainly thought up a good many--nevertheless it
does not think them up in order to persuade anyone of their
truth but only in order that they may provide a correct
basis for calculation. And as far as hypotheses go, let no
one expect anything in the way of certainty from astronomy,
since astronomy can offer us nothing certain.
As being is to becoming, so is truth to belief. If
then, Socrates, amid the many opinions about the gods and
the generation of the universe, we are not able to give
notions which are altogether and in every respect exact and
consistent with one another, do not be surprised.
It is rather difficult to acknowledge all the people
who in one way or another have contributed to this document
without using the standard oft-repeated phrases so common in
these sections of dissertations.
Dr. Stephen Gottesman led me through the intricacies
of the VLA and extraglactic radio astronomy. This was no
trivial achievement as he was dealing with someone who was
initially a confirmed optical stellar spectroscopist. His
success in this can be measured by the results contained in
the next few hundred pages. Not only was he my dissertation
chairman but a person who was always willing to help in
other matters of general well-being, and above all, a
Dr. James Hunter, who continually challenged me with
his seemingly "straightforward" problems, acted as
cochairman for the dissertation. He is also responsible for
showing an observational astronomer that theoretical
astronomy is not the great insurmountable barrier that it
was first considered to be. He, probably more than anyone
else, taught me the virtue of sitting down with something,
as with his course work problems (generally unpleasant) and
pe rseve ring until it was done The s ati sfac ti on of
completing the problem was worth the effort.
The rest of my committee, Drs. Thomas Carr, Haywood
Smith and Gary Ihas, performed their duties competently and
allowed me the fre edom, within guidelines, to do as I
The 21cm observations utilized in this dissertation
were obtained at the Very Large Array of the National Radio
Astronomy Observatory. The National Radio Astronomy
Observatory is operated by Associated Universities, Inc.,
under contract with the National Science Foundation. My
thanks to all the staff, especially Drs. Jacqueline van
Gorkom and Patrick Palmer. They not only helped make a
competent spectral line observer out of me but introduced me
to the mountains of southwest New Mexico.
My thanks go also to Drs. Bruce and Debra Elmegreen who
made their surface photometry available and to Drs. C.
Telesco and I. Gatley who allowed me to use their 2.2um
The diagrams and photographs were produced by Paul
Gombola and Hans Schrader.
Computing was done using the facilities of the
Astronomy Department and the Northeast Regional Data Center
(NERDC). I thank the numerous people who provided free
consultation in the hallways and helped with problems as
diverse as image processing and dissertation printing.
Thanks especially to Virginia Hetrick and Jim Parkes. This
dissertation was produced using UFTHESIS on NERDC.
Irma Smith typed the equations and the "fiddley bits",
and provided typing services throughout my stay in Florida.
Finally, my heartfelt thanks must go to my parents,
Michael and Maureen England, who supported and actively
e nc our aged their "professional student ." Wi thout their
support none of the next few hundred pages would have been
written. I hope that I can repay them someday for their
sacrifices and dedication.
My wife, Shei la, has been a veritable "Rock of
Gibraltar" and has put up with more and had less than any
wife and woman should reasonably be expected to endure. It
is all over now and it is to her and my parents that this
volume is dedicated.
TABLE OF CONTENTS
LIST OF TABLES ....
LIST OF FIGURES ....
. . . . . . xiv
Selection Criteria... ....
NGC 1073 . . . . . .
NGC 1300 . . . . .
NGC 3359 . . . . . .
NGC 3992 . . . . .
II. RADIO OBSERVATIONS ...........
HI As A Kinematic Tracer .....
Aperture Synthesis Theory .....
Observing Strategy and Calibration
Map-Making and Image Processing ..
. . . 18
. . .
III. DETERMINATION OF THE NEUTRAL HYDROGEN PROPERTIES
Spectrum Integration Techniques ......
Neutral Hydrogen Distribution .......
Continuum . . . . . . . . .
Kinematics of the Neutral Hydrogen ....
Mass Models . . . . . . . .
IV. SURFACE PHOTOMETRY ...............
Calculation of the Volume Mass Distribution
Surface Photometry of NGC 1300 ......
Modeling the I Passband Features ....
Comparisons Between Different Passbands
Triaxial Ellipsoid ...........
V. MODELING .
. . . . . . 185
The Beam Scheme ..........
Hydrodynamical Modeling of NGC 1300
. . .
Triaxial Bar Models ....
Oval Distortion Models ...
Composite Models ......
Bulge Models ........
VI. RESULTS FROM OTHER GALAXIES ...
NGC 3359 . . . . .
Observational Results ..
Hydrodynamical Models ..
NGC 3992 . . . . .
Observational Results ...
Hydrodynamical Models ...
NGC 1073 . . . . . .
Observational Results ...
Hydrodynamical Models ...
VII. PROPERTIES OF BARRED SPIRAL GALAXIES
Observational Comparisons ...
Dynamical Properties .....
VIII. SUMMARY .. . . . . . .
Neutral Hydrogen Results for NGC 1300 .
Hydrodynamical Results ........
Dynamical Properties .........
A. DERIVATION OF VOLUME BRIGHTNESS DISTRIBUTIONS
B. OVAL DISTORTIONS FOR N=1 TYPE TOOMRE DISKS .
BIBLIOGRAPHY . . . . . . . . . .
BIOGRAPHICAL SKETCH .
LIST OF TABLES
1.1. Global Properties of Survey Galaxies .. .. .. 14
2.1. Properties of Survey Calibrators .. ..... 38
2.2. Image Signal and Noise Characteristics ...... 53
3.1. Signal Characteristics for Spectrum Integration . 85
3.2. Summary of Neutral Hydrogen Observations for
NGC 1300 .. .. . ..... . .. . 139
4.1. Bar Projection Parameters for NGC 1300 . ... 183
6.1. Summary of Integrated Properties of NGC 3359 ..254
6.2. Summary of Integrated Properties of NGC 3992 262
6.3. Summary of Integrated Properties of NGC 1073 ..270
8.1. Summary of Results for NGC 1300 .. .. .. 284
LIST OF FIGURES
1.1. Survey Galaxies .. .. .. .. ... .. .. 9
2.1. Spheroidal Convolving Function .,, 43
2.2. (u,v) Coverage .. .. .. ... .. .. .. 45
2.3. Spectral Line Channel Maps ., 55
2.4. Wide Field Map .,,, 76
2.5. Channel Zero ., ., 77
3.1. Neutral Hydrogen Distribution Contour Plot . 88
3.2. Neutral Hydrogen Distribution with the Optical
Image .. .. .. .. ... .. ... . 89
3.3. Neutral Hydrogen Distribution Gray Scale Image . 91
3.4. Neutral Hydrogen Distribution False Color
Image .. ... .. .... . .. .. . 92
3.5. Logarithmic Fit to Spiral Arms ., ., 96
3.6. Deprojected Azimuthal Profiles ., ., 97
3.7. HII Regions in NGC 1300 ,, ., 98
3.8. Deprojected HI Surface Density . 104
3.9. Profiles Through HI Surface Density
Distribution ... .. ... 107
3.10. Continuum Emission . .. .. .. 114
3.11. Velocity Contours ... . ... .. ... 116
3.12. False Color Representation of Velocities .. 117
3.13. Velocity Field Superimposed on Optical Object .119
Angle-Averaged Rotation Curve . 125
Wedge Rotation Curve . .. ......... 126
Optical and HI Rotation Curves .. .. 129
Rotation Curve to 6.4 arcmin . ... 131
Mass Models for NGC 1300 . .... 134
HI Observed Global Profile . .. .. .. 138
NGC 1300 Gray Scale I Passband ... 147
Contour Plot of I Plate ... ... 148
Convolved I Passband Image NGC 1300 ......151
Bar Brightness Profiles ... ... 153
Disk Surface Brightness .. ..... 155
Bulge Component Model ... ......... 158
Bulge Subtracted Disk Profile .... 159
Bulge Subtracted Contour Plot . . .. 160
I Band Model Isophotes . .. . .. .. 163
I Band Model Profiles . .. .. .. .. 164
Gray Scale of Blue Passband . .. 166
Contour Plot Blue Passband . .... 167
Minor-axis Profiles Blue and I Passbands 169
Profile Comparison . 171
Comparison of Blue and I Profiles . .. 172
Contour Plot 2.2um . ..... 176
False Color Plot 2.2um ... .... 177
Flux Profile 2.2um . .. .. .. .. 179
Comparison Between Different Wavelengths . 180
Dependence of Rotation Curve on Projection
Parameters . .... ........ 198
Gas Response for Disk and Triaxial .
Velocity Field for Bar and Triaxial
Model Rotation Curve ........
Supermassive Bar Rotation Curve ..
Oval Distortion Model Gas Response .
Oval Distortion Model Velocity Field
Composite Model Gray Scale .....
Composite Model Contour Plot ....
Composite Model Velocity Field ...
Composite Model Rotation Curve ...
Slow Pattern Speed Model Gray Scale
Velocity Field Vectors .......
Velocity Field in Perturbation Frame
Noncircular Velocities .......
Gas Response for Bulge Model ....
Density Compared with Observations .
Bulge Model Velocity Field .....
Velocity Compared with Observations
Comparison of Rotation Curves ...
Velocity Vectors for Bulge Model ..
Velocity Field in Perturbation Frame
Noncircular Velocities .......
HI Distribution NGC 3359 ......
Velocity Field NGC 3359 ......
Rotation Curve NGC 3359 ......
HI Distribution NGC 3992 ......
. . . 202
. .. .204
. . . 205
. . . 206
. . . 211
. . . 212
. . . 215
. . . 216
. . . 217
. . . 218
. . . 225
. . . 229
. . . 230
. . . 231
. . . 236
. . . 237
. . . 238
. . . 239
. . . 241
. .. .242
. . . 243
. . . 244
. . . 249
. . . 250
. . . 253
. . . 257
6.5. Velocity Field NGC 3992
6.6. Rotation Curve NGC 3992 .. .. ... .. 260
6.7. HI Distribution NGC 1073 . .........265
6.8. Velocity Field NGC 1073 .. ... .. .. 266
6.9. Rotation Curve NGC 1073 . ......... 268
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
KINEMATICS AND DYNAMICS OF BARRED SPIRAL GALAXIES
Martin Nicholas England
Chairman: S. T. Gottesman
Cochairman: J. H. Hunter, Jr.
Major Department: Astronomy
The kinematics and dynamics of a group of barred spiral
galaxies are analysed. Hydrodynamical models using the
"beam scheme" are calculated and provide a set of dynamical
properties for barred spiral galaxies.
Neutral hydrogen radio observations of NGC 1300 show
the galaxy to be an excellent example of a grand design
spiral system. The HI gas is confined almost entirely to the
spiral arms with very little interarm gas. These HI arms
correlate very well with the position of the optical arms.
The HI arms can be traced through about 310 degrees in
azimuth. The central region, the region occupied by the bar,
is deficient in gas.
The velocity field shows that circular motion is the
dominant component but that large non-circular motions,
mainly in the arms, are present. The rotation curve rises
to a maximum of 185km/sec at r=2.5' and then remains
essentially flat out to about r=3.2'.
Near infrared surface photometry is used to calculate a
triaxial ellipsoidal figure for the bar. Blue and 2.2um
photometry is analysed and compared with the I band and 21cm
Hydrodynamical models for NGC 1300 are partially
successful in reproducing the obse rved mo rpho logy and
kinematics of NGC 1300. Various combinations of parameters
are investigated and a composite "best model" presented.
This model consists of an n=1 Toomre disk, a triaxial bar,
an l=2 oval distortion and a halo. The pattern speed of
19.3km/sec/kpc places corotation just outside the end of the
Hydrodynamical models for NGC 1073, NGC 3359 and NGC
3992 are examined and compared with that for NGC 1300. This
results in a set of dynamical properties for barred spiral
Barred spiral galaxies present a very interesting but
very difficult problem for astronomers. De Vaucouleurs
(1963) in his sample of 994 spiral galaxies found that about
37% are pure barred galaxies, while another 28% are mixed
spirals. The remaining galaxies are pure spirals, and are
thus in the minority. Therefore, barred systems are common,
and a good understanding of the physical processes occurring
in them would give insight into the formation and
maintenance of the observed spiral structure.
Until recently the comparison of theoretical models and
observations of barred spiral galaxies has not been very
fruitful for a number of reasons. Hydrodynamical
calculations have indicated that the gas distribution and
kinematics in barred spiral galaxies are very sensitive
tracers of the underlying gravitational potential (Roberts,
Huntley and van Albada, 1979). This is due to the fact that
the gas may respond in a highly non-linear way to even small
deviations from axial symmetry (Sanders and Huntley, 1976).
Theory is capable of producing high resolution models of gas
kinematics and structure. However, the observations have
either not had sufficient spatial resolution for a good
comparison to be made, or their spatial coverage has been
poor. Traditionally, optical measurements of the kinematics
of spiral galaxies have provided high spatial resolution but
very poor spatial coverage as they have relied upon HII
regions. The distribution of these regions is patchy; Hodge
(1969) has distributions of HII regions in spiral galaxies.
These regions are not a good tracer of the kinematics of the
gas in barred spiral galaxies. These HII regions are mainly
in the inner regions and spiral arms. However, they would
provide an excellent complement for some other, more global,
Neutral hydrogen is known to be distributed over large
regions in spiral galaxies and could provide the global
kinematic tracer needed to test the theoretical models
re sul t s. However, until recently, obse rvati ons of the
neutral hydrogen in barred spiral galaxies have been made
using single dish radio telescopes. These observations do
not have the required spatial resolution needed to compare
with the theories, and even have difficulty in isolating the
bar from the underlying disk. The National Radio Astronomy
Observatory Very Large Array (NRAO VLA) is a sensitive
instrument of high resolution, on the order of 20" for the
HI emission from barred spiral systems. The VLA is sensitive
enough that a complete two-dimensional map of the velocity
field can be completed in a reasonable amount of observing
time, even though there are no bright, nearby, easily-
observed large barred spirals. Thus, observations of the
gas kinematics and structure can now be made that can
confront theoretical models in a quantitative fashion.
High resolution mapping of the gas kinematics in barred
systems, combined with two-dimensional hydrodynamical
modeling, could address the following questions (Teuben et
1. What is the radial mass di stribution in barred
2. Are principal resonances present in barred spiral
galaxies? Sanders and Huntley (1976) have shown that
the gas flow changes character at the resonances,
consistent with the dominant periodic orbits. Within
the inner Lindblad resonance, gas flow is on
elliptical streamlines oriented perpendicular to the
bar maj or- axi s. Determining the location of the
resonances, combined with the radial mass
di stribution, would allow an estimation of the
pattern speed of the bar.
3. What is the character of the gas motions? Elliptical
streaming is recogni zab le as a skewing of the
velocity contours along the major-axis (Bosma, 1981).
The degree of skewing is related to the bar strength
(Sanders and Tubbs, 1980).
4. What is the nature of the parallel, straight dust
l ane s in barred spiral galaxies of type SBb? In
hydrodynamical calculations, such structures arise
naturally as shocks. These dust lanes often lie along
the leading edge of the rotating bar, for example NGC
1300 and NGC 1365, and as yet there have been no
unambiguous, kinematic verifications that they are
actually associated with shocks. Ondrechen and van
der Hulst (1983) have shown that for NGC 1097 the
radio continuum emission is enhanced along the dust
lanes, which is to be expected from compression in
Asymmetries in the mass distribution, such as barlike
configurations or oval distortions, play an important role
in the dynamics of galaxies. Various numerical simulations
have indicated that barlike configurations are robust and
long- lived and may be a preferred configuration for
gravitationally interacting particles (Miller, 1971, 1976,
1978; Ostriker and Peebles, 1973; Hohl, 1978; Miller and
Smi th, 1979 ). Evidence, both theoretical (Sanders and
Huntley, 1976) and experimental (Sanders and Huntley, 1976;
Huntley, Sanders and Roberts, 1978; Sanders and Tubbs, 1980)
has been presented that supports the origin of spiral arms
as being the dynamical response of a gaseous disk to a
rotating stellar bar. On the other hand, it may be the case
that spiral arms could result from either the dynamical
response of the gas to a rotating barred-spiral potential
(Liebovitch, 1978; Roberts, Huntley and van Albada, 1979) or
from the effects of self-gravity in a bar-driven disk of gas
(Huntley, 1980). Evidence that the gas in barred galaxies
does "sense" the presence of a stellar bar is concluded from
a morphological study of barred systems (Kormendy, 1979).
Other observational features which may have significant
implications for the modeling process are
1. The sharp bending of the bar into spiral arms.
2. The presence of luminous, giant HII regions which
often distinguish the spiral arms from the bar in the
region where the arms break from the bar.
The basic aim of this study is to observe a variety of
barred spiral galaxies and to calculate theoretical models
for each galaxy, using some of the observational parameters
as input quantities for the modeling procedure. The VLA was
used to provide detailed, high resolution observations of
the properties of the atomic hydrogen in each of the
galaxies at the highest possible si gnal- to-no ise ratio.
These observations provide an estimate of the rotation curve
for each galaxy and allow the mapping of the galaxian
velocity field and neutral hydrogen gas distribution.
Observations in the near infrared region (1 =82501) are
reduced to provide data for an "observed" bar, after some
M/L assumptions. This bar is used as an input parameter for
the modeling procedure. Optical observations of the gas
kinematics, where available, are used to complement the
neutral hydrogen kinematical information.
The modeling procedure consists of a hydrodynamical
computer code, the "beam scheme" of Sanders and Prendergast
(1974), kindly provided by Dr. J. M. Hunt ley. This code
calculates the response of a gaseous disk to an imposed
perturbati on, for ex amp le a bar figure or an oval
distortion. The results from these models are compared with
the observations of the kinematics and distribution of the
neutral hydrogen gas.
The sample of barred spiral galaxies used for this
study was selected using several criteria;
1. The galaxy should be large, with an optical diameter
of at least 5'.
2. The bar should be prominent and large in comparison
with the 15"-30" beam synthesized at the VLA.
3. The HI surface brightness should be reasonably high
to allow observations with good signal-to-noise
4. The object should not be too far south.
5. The system should be symmetrical.
6.The inclination of the disk of the galaxy, with
respect to the sky plane, should not be too high.
7. Surface photometry, especially in the near infrared,
I passband (A =82501) should be available.
8. A variety of types of barred spiral galaxy should be
The first four criteria are used to ensure that the
observations are feasible in a reasonable amount of
observing time, and that the signal-to-noise ratio is
optimal. The size of the object and the bar allow good, high
resolution observations to be made and the declination
requirement is imposed to obtain as circular a synthesized
beam as possible. Criterion 5 is used to facilitate the
modeling procedure. If the galaxies are not symmetric-al, the
complexity of the modeling procedure is increased greatly.
Criterion 6 avoids the problems associated with observing a
galaxy with a line-of- sight through a di sk of finite
thickness. The availability of surface photometry,
especially near infrared (Criterion 7) allows an approximate
determination of the underlying distribution of non-gaseous
lumi nou s matter (stars) in the gal axy Near infrared
photometry gives valuable information on the distribution of
the bar mass as it can penetrate, to some extent, the dust
lanes. This in turn provides constraints on the non-
axisymmetric bar component of the gravitational potential
which is required as input data for the modeling procedure.
Lastly, a variety of galaxies is needed, spanning a range of
galaxy types. This will allow some general conclusions to be
drawn about barred spiral galaxies as a class of object.
The galaxies NCC 1073, NGC 1300, NGC 3359, NGC 3992
satisfy most of the selection criteria.
The four galaxies used in this study are shown in
Figure 1-1 (a-d). These photographs are taken from various
sources. Other photographs of these galaxies which may be of
interest are near infrared exposures in Elmegreen (1981),
yellow and hydrogen alpha images in Hodge (1969), and for
NGC 1073, NGC 1300 and NGC 3359 blue exposures from the
Palomar 200" in Sandage (1961). Table 1-1 lists some global
properties of these galaxies compiled from a variety of
sources. No independent effort has been made to verify
these parameters. As can be seen from Figure 1-1 these
galaxies all have rather different morphologies and each
should present different problems for the modeling
procedure. Thu s, a wide range of morphological types is
represented by this sample and should allow some general
conclusions to be drawn.
This galaxy, shown in Figure 1-1 (a), is classified as
an SBT5 by de Vaucouleurs, de Vaucouleurs and Corwin (1976)
and as an SBc(sr) by Sandage (1961). The two prominent
spiral arms do not begin at the ends of the bar, but at 300
from the ends. The bar has a bright, central, elliptical
Figure 1-1. Survey Galaxies. Optical photographs
of the four galaxies used in this survey.
A. NGC 1073 (Arp and Sulentic, 1979).
B. NGC 1300 (National Geographic--Palomar Sky Survey).
C. NGC 3359 (N~ational Geographic--Palomar Sky Survey).
D. NGC 3992 (National Geographic--Palomar Sky Survey).
In all photographs north is to the top and east is
to the left except for NGC 3359 where north is to the top
and west is to the left.
. : .- ..
Figure 1-1 cont.
Figure 1-1 cont.
Figure 1-1 cont. (Part C).
Figure 1-1 cont.
Parameter NGC 1300 NGC 1073
Right Ascensiorit 3 17 25.2 2 41 09.0
Dec linatiord -19 35 29.0 1 09 54.0
Morphological TypdP SBT4 SBT5
Distance (Mpc)c 17.1 1.
Photometric Diameter D25 (arcmin) 6.5 4.9
Photometric Diameter (kpc) 32.3 19.4
Dimensions of Optical Bar (arcmin) 2.3x0.5 1.2x0.2
Corrected Blue Luminosityd~(1101 Lo) 2.39 0.93
Corrected Blue Magnitude 10.7 11.2
Parameter NGC 3359 NGC 3992
Right Ascensiod*~ 10 43 20.7 11 55 01.0
DeclinatiorP 63 29 12.0 53 39 13.0
Morphological Typeb SBT5 SBT4
Distance (Mpc)c 11.0 14.2
Photometric Diamete D25(aramin) 6.3 7.6
Photometric Diameter (kpc) 20.2 31.4
Dimensions of Optical Bar (arcmin) 1.7xO.6 1.7x0.5
Corrected Blue Luminosityd(1010 Lo) 1.08 2.40
Corrected Blue Magnitude 10.6 10.22
a Gallouet, Heidmann and Dampierre (1973).
b De Vaucouleurs, de Vaucouleurs and Corwin (1976).
c De Vaucouleurs and Peters (1981).
d Calculated using above values of distance and magnitude,
and using I(o)=+5.48 (Allen, 1973).
Global Properties of Survey Galaxies
region, decreasing in brightness noticeably before meeting
the arms. The ring is not complete and there are no straight
absorption lanes. Both the arms and the bar can be resolved
into many knots. The west arm appears to bifurcate at about
the end of the bar. Arp and Sulentic (1979) identified three
quasars in the field of NGC 1073, namely objects 1,2 and 3
in Figure 1-1 (a).
NGC 1300, Figure 1-1 (b), is described by Sandage
(1961) as the prototype of the pure SBb(s) system. It is
classified as an SBT4 by de Vaucouleurs, de Vaucouleurs and
Corwin (19'76). The bar is very prominent, di stinct and
smooth in texture, with two straight dust lanes emerging at
an angle from the nucleus and following the bar to its ends
and turning sharply and following the inside of the spiral
arms. The two arms start abruptly at the ends of the bar
each forming almost complete ellipses with the nucleus and
the other end of the bar being the approximate faci. They
can be traced through almost 3400
Thi s galaxy, Figure 1-1 (c), described by Sandage
(1961) as being a broken ring galaxy, is classified an
SBc(rs), and as an SBT5 by de Vaucouleurs, de Vaucouleurs
and Corwin (1976). A fairly prominent two-armed pattern
emerges from a strong central bar. The arms are asymmetric,
with the arm beginning at the southern end of the bar being
far less structured than the other. This arm appears to
break up into two or more segments whereas the other arm
more closely follows a "grand design" spiral pattern. There
is a high degree of resolution of both bar and arms into
Significant spiral structure (two bifurcated arms or
possibly even a three-arm patte rn ) eme rge s from an
incomplete ring surrounding the bar in NGC 3992 (Figure 1-1
(d)). De Vaucouleurs, de Vaucouleurs and Corwin (1976)
classify this galaxy as an SBT4. Two absorption lanes are
visible emerging from a bright, central nuclear region. The
bar is smooth in texture but the arms can be resolved easily
This dissertation will describe in detail, the
obse rvati ons reduction and analysis, and hydrodynamical
modeling of NGC 1300. Data for NGC 1073, NGC 3359 (Ball,
1984, 1986) and NGC 3992 (Hunter et al., 1986) are published
elsewhere and only the conclusions are utilized here. The
neutral hydrogen data collection, reduction and analysis are
described in Chapters 2 and 3. Surface photometry in the
blue, near infrared and 2.2um passbands is discussed in
Chapter 4, with the results from Chapters 3 and 4 being used
in the hydrodynamical mode ling in Chapter 5. The
observational and modeling results for NGC 1073, NGC 3359
and NGC 3992 are summarized in Chapter 6 and comparisons
between these galaxies are made in Chapter 7. A summary of
all the results is presented in Chapter 8.
HI As A Kinematic Tracer
To successfully model and understand the dynamics of a
barred spiral galaxy, some sort of tracer of the dynamics of
the system is needed. Any tracer which is closely associated
with the gas may be used. Several components are available
for use as this tracer. Observations of the optical Hydrogen
alpha line provide velocities for HII regions, which are
associated with hot young stars which have recently formed
from the gas. Observations of the other Population I
component, molecular gas clouds, also could provide the
kinematical information needed. However, both these
measurements have serious drawbacks. The HII observations
have high spatial resolution but generally very incomplete
coverage. This is due to the clumpiness of these regions
which means that only velocities near the hottest stars can
be measured. Molecular hydrogen, which presumably makes up a
significant portion of the molecular clouds, is difficult to
detect. Carbon monoxide, CO, the second most abundant
interstellar molecule, coexists with molecular hydrogen and
can be used to map the molecular regions in galaxies and
elucidate the varying rates of star formation (Black, 1985;
Dalgano, 1985). CO is usually far more concentrated in the
inner disk (Morris and Rickard, 1982), although it does
appear to follow the intensity distribution of the blue
light (Young et al., 1984; Young, 1985). The CO transitions
are fairly easy to excite and lie in the milIlimeter
A dominant component of the gas of the interstellar
medium consists of neutral hydrogen, HI, in its ground
state. It is well-distributed spatially and is relatively
easy to detect. This gas has a spin temperature, T'S, of
approximately 100K (Mihalas and Binney, 1981 p485). The
ground state is split into two hyperfine levels separated by
6x176 eV. This energy difference is extremely small; it
corresponds to a temperature T=0.07K (through E=ktT), well
below the ambient temperature of the surrounding medium,
and, consequently, much of the gas is in the upper level.
The upper level, or ortho-state, has the dipole moments of
the electron and nucleus parallel and the lower level, the
para- state has the dipole moments anti-parallel. The
probability of the forbidden ortho-para radiative
transition, the F=1 to F=0 spin-flip transition, is so low
that the mean lifetime of the excited level is 1.1x107yrs.
In contrast, the collisional de-excitation timescale is much
shorter, 400yrs at N =20atoms/cm3, than the radiative de-
excitation timescale, even in the low densities typical of
the interstellar medium. This implies that collisions can
establish equilibrium populations in the two levels, which
means that there will be nearly three atoms in the upper
level (which is threefold degenerate) to every one in the
Because the collisional excitation and de-excitation
rates are so much faster than the rate of radiative decay,
the atomic populations n1 and n2 in the two levels will be
essentially the same as those expected in thermodynamic
n2/n1 (2 91) exp (-hv/kTS) (2-1)
where g2 l1=3 is the ratio of the degeneracies of the two
levels. In a typical cloud TS=100K, so (h v/kTS =6. 8x10- and
exp (-h v/kTSWO.9993, giving,
n2/n1 ~ 2 1 = 3* (2-2)
In terms of probability coefficients,
nlC12 = n2(C21+A21) = n2C21(1+A21/C21) (2-3)
where Cl2 and C21 are collisional probabilities and A21 is
the Einstein probability coefficient for spontaneous
radiative decay from level 2 to level 1. As A21 is small,
nlC12 = n2C21 (2-4)
and approximate equilibrium is established. Although A21 is
small ~2.868x10-15 sec-1, radiative decay is the observable
transition mechanism. The large column densities along a
typical line-o f- sight in a galaxy make thi s radi ative
transition detectable. This transition is observable at a
frequency of 1420.40575MHz (ho=21.105cm). Its observation
was predicted by van de Hulst (1945) and first measured by
Ewen and Purcell (1951). Muller and Oort (1951) and
Christiansen and Hindman (1952) confirmed the measurement.
Neutral hydrogen generally covers a region larger than
the observed optical object and thus provides good coverage
of the whole disk of the galaxy and not just selected
regions, as do Hydrogen alpha observations. If the outermost
regions are excluded, then HI is among the flattest and
thinnest of the disk components of ours and other galaxies
(Jackson and Kellerman, 1974). This allows the
determination, with a reasonable degree of confidence, of the
two-dimensional location of any observed emission. This gas
is pervasive enough that the emission recorded by radio
telescopes appears to be continuously distributed.
If the neutral hydrogen gas is assumed to be optically
thin, a simple integration of the brightness temperature,
TB' over velocity, V, determines the column density, Nh, of
the gas at that point (Mihalas and Binney, 1981 p489):
Nhx~) 1826 118 B(xy) dV, (2-5)
where V is in km/s, T is in Kelvin~and, Nh in atoms/cm2
The mean temperature-weighted velocity at a point is
given by the first moment with respect to velocity,
= B(x'y) V(x,y)dv
If the neutral hydrogen gas is not optically thin this
will lead to an underestimate of the surface density. In
this case the observed brightness temperature, Tf'B would
approach the physical, spin temperature of the gas, TS.
TB (-e) (2-7)
and, for an optically thin gas, t<<1, TB:6b, while for an
optically thick gas, TB=S(iaa n iny 1981 p487).
The highest observed brightness temperature for NGC
1300, averaged over the beam, was 16.95K. Assuming a mean
temperature for the gas of 100K (McKee and Ostriker, 1977;
Spitzer, 1978) gives an approximate optical depth of t=0.19,
thereby justifying the optically thin assumption. Although
this leads to an underestimate of the column density the
effect is <15% at the peak emission and will be less at
other points. As the "optical depth structure" of the medium
is not known, the assumption of an optically thin medium
will be retained.
Thus, in summary, neutral hydrogen provides a good
tracer for the kinematics of the gas in a galaxy;
1. It is well distributed spatially.
2. It is relatively easy to observe.
3. As sumi ng it is optically thin, the above simple
expressions hold for the column density of the gas
and the mean velocity of the gas at an observed
location, equations 2-5 and 2-6.
Aperture Synthesis Theory
The neutral hydrogen content of NGC 1300 was observed
using the Very Large Array (VLA) of the National Radio
Astronomy Observatory (NRAO). The VLA is the largest and
most sensitive radio telescope which exploits the principle
of earth- rotati on ape rture synthe si s. The array is a
multiple-interferometer instrument using a' maximum of 27
antennae. As the basic theory of interferometry and earth-
rotation aperture synthesis is well covered in Fomalont and
Wright (1974), Hjellming and Basart (1982), Thompson (1985),
D' Addario (1985), Clark (1985) and, from an electrical
engineer's perspective in Swenson and Mathur (1968), only a
brief discussion will be given here and some fundamental
The basic process of interferometry is the cross-
correlation of signals from two antennae observing the same
source. The resulting signal is analogous to the
interference pattern in the classical optical double slit
experiment. The cross-correlation of these two signals
produces information on both the intensities of sources in
the beam of the antennae and on their positions relative to
the pointing position of the antennae. Any distribution of
radio emission in the beam of an antenna can be considered
as a superposition of a large number of components of
different sizes, loc nations and orientations. As the
relationship between intensity distributions and the
components can be described in terms of a Fourier integral,
it follows that an interferometer pair, at any instant,
measures a single Fourier component of the angular
distribution of sources in the beam pattern. The essential
goal in radio aperture synthesis observations is to measure
a large number of these Fourier components. This procedure
allows the reconstruction of an image of the spatial
intensity distribution of sources in the beam. The VLA
achi eve s the measurement of a large number of Fourier
c omponent s by using multiple interferometer pairs and
allowing their geometric relationships with the sources in
the sky to change by utilizing the rotation of the earth,
hence the term earth- rotati on aperture synthes i s. For
multiple interferometer pairs, N antennae, there are
N (N- 1 )/2 different baselines, or s amp le s of the Fourier
components, at any one instant. The VLA has a maximum of 27
antennae or 351 samples of the Fourier components. These
samples are not all unique as there is redundancy in the
The output from a two element interferometer can be
shown to be
V'(u,v) = I(x,y) exp[-i2n(ux+vy)]dx dy (2-8)
where I' (x,y) is the observed brightness distribution,
V' (u,v) is the observed complex visibility, and u,v are
projected spacings in east and north directions
re spec ti vel1y sometimes called spatial f requ enc ie s
(Hjellming and Basart, 1982).
Thi s shows that a single measurement of the complex
vi sibili ty, y*I, corre spending to a particular projected
baseline, or particular (u,v) point, gives a single Fourier
component of I', the observed brightness distribution. The
similarity theorem of Fourier transforms (Bracewell, 1965)
shows that large extent in the (x,y) plane means small
extent in the (u,v) plane and vice versa. Thus, achievement
of high spatial resolution requires large spacings between
the antennae in an interferometer pair, and conversely,
large scale structure requires low spatial fr equenci es,
Equation 2-8 can be inverted to give the observed
brightness distribution I' as a function of the measured
complex visibilities, V' ,
I'(x,y) =lv V'(uv expl i~nuxivy)]du dv (2-9)
where I' is the product of the true brightness distribution
Io and the single antenna power pattern A,
I'(x,y) = A(x,y) I (x,y) (2-10)
These results have been calculated in the absence of
noise. Since all observations measure only a finite number
of (u,v) points and all contain noi se I' c anno t be
determined uniquely or without error. A later section deals
with the problem of missing complex visibilities and the
non-uniqueness of the solution of equation 2-9.
The extension of these results to spectral line
observations introduces several complications. The signal
has to be divided into a number of independent, narrow-band
spectral channels At the VLA thi s is achi eved by
introducing an additional delay, t., into the signal path.
This delay destroys the coherence of the received signals
except for those in a narrow frequency range centered on
some frequency v .. Changing this delay changes the frequency
v and allows the signal to be divided into a number of
independent, narrow-band channels. The integration of
equation 2-8 over bandwidth gives
V'(u,v,t) = ~lI'(x,y)Fiv)exp[-i~vt.+2n(ux~vy)] dxdy at
where F(v)is the frequency bandpass function.
Due to the symmetry of the delays introduced, only the
real part needs to be Fourier transformed, giving (Hjellming
and Basart, 1982)
I'~I(xy,y F(viexpli2ny(ux~vy)]dx dy. (2-12)
This is the Fourier transform at one of the frequencies
and contains all the visibility information necessary to map
the source at that frequency. Equivalently, as the number of
delays t. is finite, this procedure allows the mapping of
the narrow-band channels. The right hand side of equation
2-11 contains the bandpass function,F(v) which must be
calibrated. This is done by observing a strong continuum
source, which is assumed to exhibit no spectral variation
over the quite narrow total bandpass normally used for
spectral line work.
Observing Strategy and Calibration
In an interferometer, such as the VLA, high resolution
is achieved by using large separations of the antennae.
Conversely, broad structure requires relatively small
spacings; thus, both long and short spacings are required to
measure both the small scale and the extended structure in a
galaxy. However, the higher the resolution, the poorer the
brightness sensi tivi ty Thi s conflict demand s that a
compromise be made between sensitivity and resolution.
The minimum detectable flux density, AS .,depends
only upon system temperature, bandwidth, integration time
and effective collecting area, viz.,
miS a aT /Ae FEG (2-13)
where T is the system temperature in Kelvin, A is the
effective collecting area, av is the bandwidth in Hz, and, t
is the integration time in hours.
The effective collecting area A =lAT where A is the
total area and r, is the aperture efficiency.
However, for resolved sources the detectable brightness
temperature is the important quantity, and
TB =Amin BEAM (-4
where BBEAMV is the synthesized beam solid angle.
The synthesized beam is the power pattern of the array
as a whole, rather than the power pattern of an individual
antenna. Thus, for a point source, the synthesized beam is
the observed normal ised bri ghtne ss di stributi on .
Consequently, as resolution is improved the brightness
sensitivity is degraded, and vice versa.
If observing time were unlimited, the choice of arrays
would be an easy undertaking. The resolution required would
dictate the largest separation of the antennae, and the
required signal-to-noise ratio would dictate the amount of
integration time needed. However, as observing time is
limited, in order to determine which array configurations
were practical to use for this project required
consideration of both the resolution needed to observe the
structure and the sensitivity needed to ensure that the
majority of the gas was observed. Another factor which had
to be considered was that, as the VLA was used as a
spectrometer, the sensitivity in each narrow line channel is
relatively poor. With these considerations in mind, it soon
became evident that the two lowest resolution
configurations, the D and C arrays, would be the only two
practical configurations to use for a reasonable amount of
observing time. The D array would ensure that no low
amplitude large scale structure emission was missed, whereas
the C array would resolve the-smaller scale structure. Using
only these two arrays means that some small scale structure
below the resolution limit of the C array will be missed,
but will ensure that the majority of the emission was
observed. As the best peak signal-to-noise ratio observed in
any of the channels was 13.4 this would mean that the best
detection achievable with the next largest array, the B
array, would be, for the same amount of observing time, a
less than "two sigma" detection.
In spectral line observations the co rrel1ato r must
multiply the signals from 2n delay lines for each of
N(N-1)/2 baselines, where n is the number of spectral
channels and N is the number of antennae used. The
correlator thus has an upper limit for the product nN which
necessitates a compromise when choosing n and N. The larger
the value of n, the greater the spectral, and hence
velocity, resolution but the poorer the sensitivity. The
larger N is, the better the sensitivity as more antennae
contribute to the signal Ideally the largest values
possible for n and N are required. However, as n has to be
an integer power of two to allow the Fourier transform of
the lag spectrum to be calculated using Fast Fourier
Transform techniques (FFT), this also places some
restrictions on n.
The choice of n depends upon the velocity range of the
global profile of the galaxy under study and the velocity
resolution desired. Also, a few "line-free" channels on each
end of the spectrum are desirable to allow the continuum
emission to be mapped. Previous studies and single dish
results (Bottinelli et al., 1970) indicate that the global
profile for NGC 1300 has a velocity width (full width at a
l evel1 of 25% of the peak l evel1) of 2 90km/sec The se
observations, coupled with the other considerations above,
lead to a choice of n=32 with a single channel separation of
20.63km/sec, 97.656krHz. This choice of n allowed a maximum
of 25 antennae to be used. The discarded antennae were
chosen simply on the basis of their recent malfunction
During the observing run the central channel, channel
16, was centered on 1540km/sec, a value equal to the
approximate mean of other previous determinations of the
systemic velocity; Sandage and Tammann (1975) find
1535 +9km/sec; de Vauc ou leurs de Vaucouleurs and Corwin
(1976) find 1502+10km/sec; and Botinelli et al. (1970) find
1573+7km/sec. Channel 32 was chosen as the central channel
in order to avoid using the end channels in the 64 channel
spectrometer. Due to Gibbs phenomenon (oscillations in the
bandpass func ti on at the edge s of the bandpass) a few
channels at either end of the spectrometer are severely
degraded, and it was considered prudent to avoid these
channels. The mean velocity is a heliocentric ve locity
calculated using the definition
V = co p (2-15)
Thirty-one of the channels are narrow line channels
separated by 20.63km/sec, 97.656kI~z, with a full width at
half maximum (FWHM) of 25.2km/sec; the thirty-second is a
pseudo-continuum channel with a total width of 1000km/sec,
4.7MHz. This channel, designated channel zero, contains the
true continuum emission plus the line signal, utilizes ~ 75%
of the intermediate 6.25MHz broad band filter, and was used
primarily to calibrate the line channels. As this is a broad
band channel, the sensitivity to the calibration is much
greater (17x) than that for the line channels. Consequently,
the calibration procedure was carried out using channel zero
and then appli ed to the single line channels onc e a
satisfactory solution was found. This procedure is
The flux density for the primary calibrator, 3C48, is
forced to assume some "known" value at the frequency of the
observations (VLA calibration manual based on Baars et al.,
1977). Using the flux densities of the secondary calibrators
as free parameters, a solution for amplitude and phase for
each antenna in the array is computed as a function of time.
All the scans of the secondary calibrators are utilized for
this solution. Baselines with closure errors greater than
some specified limits in amplitude and phase ( -10% in
amplitude and 100 in phase) can then be identified and
rejected. If the assumption is made that the complex gain
for the antenna pair jk, C'jk 't can be represented by
amplitudes g p(t) and g p(t) and phases Oj (t) and 4kp~t
Gjkp~t 9jp(t)exp[i(4 +4kp GIkp~ t)+,jkp (2-16)
where c .P are the closure errors. Thus, the smaller these
closure errors the better the approximation becomes for the
actual complex gain. For the mode of observing employed for
these observations, few baselines had closure errors as
large as 10% and 10 ,and most were below the 7 n
range. After rejecting the baselines with unacceptable
closure errors, the antenna solution is repeated. This
iterative procedure is continued until acceptable solutions
have been found for the complex gains.
This procedure utilizes one antenna as a reference
antenna for the array. It is thus worthwhile repeating the
calibration using a different reference antenna to improve
the solution. The reference antenna should be particularly
stable compared with the rest of the array and should have
variations which are as slow as possible and not be
monotonic functions of either space or time. A good stable
antenna usually can be found by repeating the calibration
procedure for a few different antennae.
Once acceptable solutions for the complex gains have
been found using the primary calibration source, fluxes for
the secondary calibrators can be determined. These fluxes
are generally called bootstrappedd" fluxes and their errors
give a good indication of the stability of the atmosphere
during the observing run.
The bootstrapped fluxes can be applied to the entire
dataset, including the program object observations, by a
simple running mean, or "boxcar" interpolation of the
amplitude and phase gains of the individual antennae. At
every step of the process the database is inspected and
suspect signal data are flagged, hopefully leading to a
better solution from the next iteration and not seriously
degrading the overall quality of the dataset. The quality of
the dataset is usually not degraded very much as there is a
large duplication of baselines and rejecting a few data
points does not have a large overall effect on the database.
The final step in the calibration procedure is to calibrate
the bandpass by assuming a flat spectrum for the primary
calibrator over the total spectral-line bandwidth. The
purpose of the bandpass calibration is to correct for the
complex gain variations across the spectral channels. The
bandpass usually varies only slowly with time and usually
has to be measured only once during an observing run. The
data are now ready for Fourier inversion and image
Generally, the data for this project were unaffected by
any serious problems, and few baselines or scans had to be
flagged in the calibration procedures. However, the data
from Summer 1984, for the second half of the observing run,
exhibited some anomalous records at the beginning of each
sc an The source of the se anomalous records was not
discovered and the records were simply deleted from the
dataset. This improved the antenna solution noticeably and
allowed an acceptable solution to be calculated quickly.
Another problem with the more compact arrays when observing
a source with a low southerly declination, such as NGC 1300
(6=-19035') is "shadowing." This occurs when the projected
separation of two antennae is smaller than the physical
diameter of the antennae, 25m. This means that one antenna
is partially blocking the other's view of the source. A
correction for this effect can be applied, or the offending
antenna can simply be removed from the database for the
appropriate timerange. This "shadowing" also causes a more
subtle problem for the calibration procedure. When one
antenna is shadowing" ano the r, even slightly, the data
collected during that time range by the "shadowed" antenna
has a noticeable deterioration in quality. This "crosstalk"
arises when the shadowed antenna detects signals from the
electronics of its neighbour. As thi s effect can be
difficult to detect, the safest method to avoid "crosstalk"
is to flag all data from "shadowed" antennae. For NGC 1300
this amounted to approximately 2% of the data, the majority
being at the beginning and the end of the run,- at large hour
angles, or low elevation angles. The amount of data flagged
did not degrade seriously the overall qu al ity of the
database and allowed a good antenna solution to be
calculated. Apart from these two problems, which were easily
corrected, NGC 1300 showed no unpleasant surprises and a
good solution was arrived at in a few iterations of the
The galaxy NGC 1300 was observed using the D/C hybrid
configuration on the 9th and 12th July 1984. A total of 25
antennae, evenly distributed over the three arms, was used.
The north arm was in the C array configuration and the
southwest and southeast arms were in the more compact D
array configuration. This hybrid configuration allowed a
nearly circular beam to be synthesized and gave a maximum
unprojected separation of 2106.6m (9982X) and a minimum
unproj ected separation of 44.6m (213 X). Seven hours of
observing time were used on the 9th of July, 1984 and seven
hours on the 12th of July, 1984.
Calibration sources were observed at the beginning of
the session, every 40 minutes during the run, and again at
the end of the session. More frequent observations of the
calibrators were not deemed necessary as the timescale for
phase stability of the atmo sphe re at 21cm ( 1420MHz ) is
considered to be a good deal longer than the intervals
chosen here. The bandpass calibrator source, 3C48, was
observed three times during the session: at the beginning,
in the middle, and at the end. This also provided a check
on the overall stability of the system as it allowed a
comparison of the phase and amplitude response over the
The primary calibrator, 3C48, was used to calibrate the
receiver handpass and the flux densities of the secondary
calibrators. Two secondary calibrators, 0237-233 and
0420-014, were needed for NGC 1300 due to the relative
positions of available calibrators and the galaxy itself.
0237-233 was used for the first 4 hours of the observing run
and 0420-014 for the remaining 3 hours of the run. The
transition from one secondary calibrator to the other was
acc omp lished by using the primary calibrator as an
intermediate step between the two.
The 1985 observations employed 25 antennae in the C/B
hybrid configuration. The north arm was once again in the
higher resolution configuration and the antennae were evenly
distributed over the three arms. The configuration gave a
maximum unprojected separation of 6920m (32953 X) and a
minimum unprojected separation of 78m (372X). A total of 6
hours of observing were obtained using this hybrid array on
the 28th of June, 1985 and 7.5 hours on the 1st of July,
1985. The phase and amplitude calibration of the data were
done by using the same sources as for the D/C hybrid array;
the observing strategy was the same for both seasons. The
flux densities of these sources and the receiver bandpass
were once again calibrated using 3C48. Table 2-1 lists
calibrator positions and fluxes.
Map-Making and Image Processing
The fundamental result of the aperture synthe si s
description is the existence of a Fourier transform
relationship between the modified sky brightness and the
visibility observed with an interferometer,
I'(xy) = V'(u,v) exp~i2n(ux+vy)]du dv (2-17)
Calibrator P/S Frequency Epoch Array Flux Density
(1) (2) (3) (4) (5) (6)
3C48 P 1413.251 Jul 84 C/D 15.82
P 1413.240 Jun 85 B/C 15.82
0237-233 S 1413.251 Jul 84 C/D 6.25
S 1413.240 Jun 85 B/C 6.12
0420-014 S 1413.251 Jul 84 C/D 2.03
S 1413.240 Jun 85 B/C 2.22
3C48 01 34 49.8 (1950)
32 54 20.5
0237-233 02 37 52.7 (1950)
-23 22 06.4
0420-014 04 20 43.5 (1950)
-01 27 28.6
(1) Calibrator identification.
(2) Primary (P) or Secondary (S) calibrator.
(3) Frequency of observation (MHz).
(4) Epoch of observation.
(5) Array configuration employed for observations.
(6) Flux adopted for primary or determined for
Properties of Survey Calibrators
where I' is the product of the true bri ghtne ss
di stributi on, Io, and the single dish power pattern, A ,
equation (2-10) .
This result can be used to derive the source brightness
distribution from the observed interferometer visibilities.
These visibilities are observed at a number of discrete
(u,v) points. With a small number of points, model-fitting
of the points is feasible, but as a VLA spectral-line data-
base typically consists of ~ 500,000 points the most
practical way of constructing the brightness distribution is
to use Fourier inversion techniques.
There are two common ways of evaluating the Fourier
1. By direct evaluation of equation (2-17) at the
individual sample points, Dir~ect Fourier Transform,
2. By using a Fast Fourier algorithm, FFT.
The advantages of the DFT are that aliasing and
convolution introduced by the gridding procedure for the FFT
are avoided, but the disadvantage is that the number of
multiplications for an NxN grid of M data points cr2MN 2
which can be substantial for the large datasets usually
considered in spectral line observations. The use of the FFT
reduces the number of multiplications to N210gN2 which can
save a considerable amount of computing time. However, for
the FFT the data points must be on a rectangular grid,mxp,
where m and p are integer powers of two. The use .of FFT
algorithms can lead to the introduction of aliasing in the
maps. This aliasing results from the gridding process. The
gridded visibilities may be represented as
Vr(uIv) = III(u~v)* [C (uIv)RS(uIv)*V'I(uv)]) (2-18)
where III is a two- dimensional Shah function, S is a
sampling function, and C is a convolving function.
Due to the presence of the Shah function and the fact
that the Fourier Transform of C is not exactly zero beyond
the map limits, parts of the brightness distribution that
lie outside the primary map field will be aliased into the
primary field. The simplest way to tell if an image is
aliased is to remap the field with a different cell size.
The aliased source will appear to move while a primary
source will stay the same angular distance from the field
The most common grid for the FFT is a square grid (m=p)
with the (u,v) spacings comparable with the cell size; as
the observed data seldom lie on these grid points, some
interpolation method must be used to specify the
visibilities at the grid points. If a scheme which resembles
a convolution in the (u,v) plane is used, then the image
will have predictable distortions which can be corrected at
later stages of the reduction procedure. A convolution also
smoothes the data, providing a good estimate of the gridded
visibility from noisy input data.
The best way to avoid, or at least reduce, aliasing
problems is to use a convolving function, C, that results in
a fast drop-off beyond the edge of the image. This requires
that C be calculated over a large region in the (u,v) plane,
requiring a large amount of computing time Thu s, in
practice, a compromise between alias rejection and computing
time must be reached. The function C should ideally be flat
out to some distance and then drop off sharply without
having sidelobes beyond the edges of the map. The lack of
high sidelobes helps suppress the aliasing of sources lying
outside the map into the map. Aliasing of sources that lie
off the primary image back into the map is only part of the
problem. A primary image source will1 have sidelobes
extending beyond the edge of the image. These sidelobes will
be aliased back in, effectively raising the background and
resulting in a beam shape that is po si tion invariant
(Sramek, 1985 ). Thu s, the convolving function suppresses
aliasing due to replication of the image in the gridding
process. It suppresses aliasing but not sidelobe or ringlobe
responses from sources outside the area of the map. With
alias suppression of 102 or 10 3 at two or three map radii,
it is these sidelobe responses which may cause the dominant
spurious map features. As C is usually separable
C(u,v) = C' (u) C' (v) (2-19)
C'(x) = (1-n (x) /CB I(cen (x)),
ni(x) = x/max
(cn) = (1_2-a ) S (Cn.(2-20)
The function S (CI,) is a prolate spheroidal wave function
(Schwab, 1980) At the VLA the parameters used are
generally m=6, ar=1, n=0. Figure 2-1 shows the form of this
It is desirable not to have the product Nau so large
that the outer cells are all empty and the inner ones
heavily undersampled, nor so small that many po-ints at large
spacings are rejected. For the VLA spectral-line observing
mode an empirical relationship which produces good sampling
is that the synthesized beam be about three to four times
the cell size of the intensity images,Ae
Once the data have been convolved, the map must be
sampled to produce the gridded values. The sampling function
is a two dimensional Shah function (Bracewell, 1965),
III (u,v) = Au~v Cc 6[(u--j~u),(v--k~v)] (2-21)
where du,6v are the separations between grid points.
Unfo rtunate ly, the s ampli ng in baseline space by a
rotation synthesis array, such as the VLA, is non-uniform.
The projections (u,v) of sample points, with respect to a
reference direction, are therefore non-uniformly distributed
-015 \Y( o m= e
Figure 2-1. Spheroidal Convolving Function. Side-
lobe responses for the gridding function used in this study.
with varying density inside an irregular boundary, all of
which depend upon the source declination, see Figure 2-2 for
(u,v) coverage. Therefore some sort of weighting function,
W, is necessary to correct for this effect and to control
the synthesized beam shape. The sampling function can then
III(u,v) = Auav ym W6 [(u-j~u) ,(v-k~v)]i. (2-22)
The weighting function is usually expressed as the
product W=dt where d corrects for the varying number of
observed samples in each gridded cell, and t introduces a
taper to reduce the sidelobes. The beam usually consists of
a Gaussian core with broad sidelobes at a one to ten percent
levels. The shape of the sidelobes is simply the Fourier
transform of the unsampled spacings in the (u,v) plane out
to infinity. The taper, t, weights down the sparsely-sampled
outer region of the (u,v) plane and helps suppress the small
scale sidelobes at the expense of a broader beam. The
tapering function is usually a truncated Gaussian function
The other weighting function, d, is generally choose
from one of two extremes, natural or uniform weighting.
Natural weighting weights all observed samples equally: d=1.
Thus, the weight of each gridded visibility is proportional
to the number of observed visibilities contributing to that
samp le. Since the density of observed samples is always
Figure 2-2. (u,v) Coverage. Schematic representa-
tion of the (u,v) coverage obtained by the observations of
higher for the shorter baselines, this tends to produce a
beam with a broad low-level plateau (Sramek, 1982). However,
this type of weighting gives the best.signal-to-noise ratio
for detecting weak emission. Natural weighting is
undesirable for imaging sources with both large and small
scale structure, such as extended emission from galaxies.
Although the sensitivity is inc reased, the broad beam
degrades the resolution and the small scale structure will
become dependent on the beam shape. To remove the broad
plateau each gridded cell is weighted by the inverse of the
number of observed visibilities contributing to that cell:
d=1/N. This weighting is called uniform weighting and, since
not all visibilities are equally weighted, there will be a
degredation in si gnal -to -noi se ratio. Uni form wei ghti ng
gives the same weight to each cell in the gridded (u,v)
plane and the beam characteristics are controlled largely by
the tapering, t (Sramek, 1985).
In principle the procedure for producing the gridded
visibilities for the application of the FFT is
1. Convolve the observed vi sibili ty data points to
produce a continuous function.
2. Resample this continuous function at the grid points.
3. Apply the weighting and taper to the resampled data.
These gridded visibilities can now be Fourier inverted,
using equation 2-17, to produce an estimate of the source
brightness distribution. For NGC 1300 the Fourier inversion
was performed using a 6" cell size with a 7kh taper (1484m)
and uniform weighting producing 32 single channel "dirty
maps" and their associated "dirty beams." These dirty maps
are given by the true brightness distribution convolved with
the dirty beam.
Direct Fourier inversion of the observed visibilities,
with all unsampled visibilities set to zero, gives the
principal solution, or dirty image. Thus, the quality of the
image depends entirely upon the sampling in baseline space.
In general this sampling is non-uniform. It is obvious that
the true image cannot be as complex as this dirty image,
where the visibility vanishes at all positions -not sampled
by the observation. There must be image components invisible
to the instrument with non- ze ro vi sibili ti es at the
unsampled positions. The unsampled points in the (u,v) plane
give rise to the sidelobes of the dirty beam and reflect an
unavoidable confusion over the true brightness distribution.
S ome estimate of these uns amp led or i nvi sible image
components is necessary to augment the principal solution in
order to obtain an astronomically plausible image. The
scheme most widely used is the CLEAN algorithm introduced by
Hogbom (1974). CLEAN performs a func ti on re sembl1i ng
interpolation in the (u,v) plane.
The CLEAN algorithm uses the knowledge that radio
sources can be considered as the sum of a number of point
sources in an otherwise empty field of view. A simple
iterative procedure is employed to find the positions and
strengths of these point sources. The final image, or
"cle an" image, is the sum of these point components
convolved with a "clean" beam, usually a Gaussian, to de-
emphasize the higher spatial frequencies which are usually
The original Hogbom algorithm proceeds as follows:
1. Find the strength, M, and position of the point
brightest in absolute strength in the dirty image.
2. Convolve the dirty beam with a point source, at this
location, of amplitude yM, where v is the loop gain,
3. Subtract the result of this convolution from the
4. Repeat until the residual is below some predetermined
5. Convolve the point sources with an idealized clean
beam, usually an elliptical Gaussian fitted to the
core of the dirty beam.
6. Add the residuals of the dirty image to the clean
image Keeping the residuals avoids having an
amplitude cut-off in the structure corresponding to
the lowest subtracted component and also it provides
an indication of the level of uncertainty in the
Since the basis of this method is to interpolate
unobserved visibilities, the final image is the consequence
of preconceived astrophysical plausibility. Interpretation
of fine detail in clean maps should recognize this non-
uniqueness of the solution.
Clark (1980) developed a variant of the Hogbom
algorithm. The basic idea is to separate the operation of
peak locating from that of convolution-subtract and perform
the convolution-subtract step on a large number of point
sources simultaneously. The algorithm has a minor cycle in
approximate point source location using a truncated beam
patch, which includes the highest exterior sidelobe, and a
major cycle in proper subtraction of a set of point sources
(Clark, 1985; Cornwell, 1985).
It should be clear that CLEAN provides some sort of
estimate for unsampled (u,v) points. In most cases it does
this reasonably well. However, quite often it underestimates
the "zero-spacing" flux, the integral of the flux over the
clean image. This results in the source appearing to rest in
a "bowl of negative surface bri ghtne ss Provi di ng an
estimate of this flux (from single dish measurements for
example) can sometimes help (Cornwell, 1985).
In using CLEAN a decision has to be made concerning
1. Is the addition of the zero-spacing flux necessary?
2. Over what region of the image should the CLEAN be
3. How deep should the CLEANing go, i.e. at what level
should the cutoff be?
The solution to these que sti ons for NGC 1300 was
arrived at by considering the following:
1. The galaxy was observed using the D array. This array
contains short spacings and thu s, the un samp led
region in the (u,v) plane is small. Owing to this, no
zero- space ing flux was added. The decision was
justified as np evidence for a negative brightness
bowl was seen, meaning that CLEAN had provided a good
estimate for this flux. The total flux measured at
the VLA was 36.53Jy(km/sec) compared with
30.3Jy(km/sec) found by Reif et al. (1982).
2. All the line channels were examined and limits set on
the spatial extent of the signal in each channel.
This allowed only regions containing signal to be
used in CLEANing, thus avoiding the time-consuming
CLEANing of regions containing only random noise.
3. When CLEAN is applied to maps correctly the resultant
"blank" sky should show only random noise and no
sidelobe structure. The rms noise level should be
approximately the same from channel to channel and
should also be approximately equal to the expected
rms noise level for spectral-line maps with natural
weighting (Rots, 1982);
= a[N(N-1) T Av]-1/2 (2-23)
where N is the number of antennae used, a is a
constant, a=620 for 21cm, T. is the total on-source
integration time in hours, and av is the bandwidth in
For NGC 1300 using 25 antennae, N=25, Ti =20.58hr,
Av=97.656k~Hz gave an expected rms noise level of
0.6mJy/beam. The rms noise level for the dirty maps is
-0.8mJy/beam and for the clean maps ,0.7mJy/beam and thus
indicates that the CLEAN was acceptable. Table 2-2 shows the
rms noise level for the dirty maps and the rms noise level
for the clean maps. Once the expected rms value has been
reached the CLEANing is stopped; proceeding beyond this
point is tantamount to shuffling the noise around. For NGC
1300 this limit was generally reached after 1000 iterations,
as the loop gain was small, y =0.15.
Before the spectral-line channels are CLEANed some
method of subtracting the continuum emi ssion must be
utilized. Continuum emission usually consists of unresolved
point sources as well as some emission from the central
region of the nucleus. The method used for NGC 1300 was to
average a few "line-free" channels from both ends of the
line spectrum producing a continuum emission map. Thi s
continuum map was then subtracted from the dirty spectral-
line channel maps, producing a set of 18 dirty, continuum-
free, spectral-line maps. It is important to note that, due
Image Velocity rms Noise rms Noise rms Noise Peak Brightness
(km/sec) Dirty Map Clean Map Clean Map Clean Map
(mJ/beam) (mJ/beam) (K) (K)
6 1746.10 0.84 0.82 1.28 5.40
7 1725.49 0.75 0.73 1.14 7.44
8 1704.88 0.86 0.83 1.30 16.95
9 1684.27 0.81 0.78 1.22 14.93
10 1663.66 0.84 0.81 1.27 12.59
11 1643.05 0.79 0.75 1.17 13.60
12 1622.44 0.83 0.83 1.30 9.72
13 1601.83 0.82 0.79 1.24 11.21
14 1581.22 0.91 0.91 1.42 9.52
15 1560.61 0. 85 0.85 1.33 9.18
16 1540.00 0.84 0.83 1.30 8.87
17 1519.39 0.83 0.83 1.30 9.11
18 1498.78 0.78 0.77 1.20 11.08
19 1478.17 0.83 0.80 1.25 12.73
20 1457.56 0.83 0.80 1.25 15.82
21 1436.95 0.78 0.78 1.22 10.17
22 1416.34 0.80 0.78 1.22 5.51
23 1395.73 0.82 0.81 1.27 5.32
Continuum 0.30 0.27 0.43 12.63
Channel O 0.18 0.17 0.19-
Large Field 0.77 --
Image Signal and Noise Characteristics
to the non- lineariti es of the CLEAN algori thm, thi s
continuum subtraction must be performed before the maps are
CLEANed. Subtracting CLEAN maps can introduce noise at the
positions of continuum sources (van Gorkom, 1982). Table 2-2
has the noise levels for these continuum free line channels.
The dirty, continuum-free line channels and the continuum
map are now ready for the CLEANing procedure.
Figure 2-3 (a-s) shows the final CLEANed spectral line
channels and the continuum map. The effective resolution is
20.05"x19.53" (FWHM). The beam is indicated on the continuum
map, Figure 2-3 (s). As the astronomical significance of
these observations will be considered in a later section,
the CLEANed line channel maps are just presented here. The
emission from the galaxy appears to be clumpy and is not
very widespread. This hints at the structure found in the
integrated density map, well-defined arms with an extremely
low level disk component. The rms noise levels for the these
maps can be converted from mJy/beam to brightness
where hT_ is in Kelvin, aS is the rms noise level in
mJy/beam, C is the speed of light, is the frequency of
observation in MHz, k is the Boltzman constant, and R is the
beam solid angle in radians.
Figure 2-3. Spectral Line Channel Maps. CLEAN,
continuum-free spectral line channels for NGC 1300. The
+ mark the positions of fiducial stars and x the center
of the galaxy. The velocity is indicated in the top
right corner. All maps are plotted in intervals of twice
the rms noise level.
A-R. Spectral line channels 6 through 23.
S. CLEAN continuum map showing the center of the
galaxy (x), fiducial starts (+), H II regions (*) and
28 L i
-1@ C O
36 .0 a o
C 13 a 3
O .Q O,
RAF 3 17
Figure 2-3 cont.
-1930 n O 'b
38 L O
RAF 3 17
Figure 2-3 cont.
28 L O
I- 9 o a
310 - a
9036 o O 0~'
R A 3 17
Figure 2-3 cont.
+ 1 .
Figure 2-3 cont.
Figure 2-3 cont.
28 -,i Ii o
3 0 _0
3 4 c o c 0 o
a 9, c
38 L o o
R A 3 17
Figure 2-3 cont.
-19.3 ;f o
o- a3 0
39 ? o
3G (3 o X
L90 25 10 55
RAF 3 17
Figure 2-3 cont.
-I 1 I
9 r" 3
0 ,1 3
Figure 2-3 cont.
3Y C. 9
oG4` +3 c,
5 + o
tlo o '
a 0 0 r;o
I, I 1 f
RAF 3 17
Figure 2-3 cont.
o a ,
Figure 2-3 cont.
(Part J) .
36 0I o
38 -- /
L90 25 10 55
R A 3 17
Figure 2-3 cont.
0 I i s
O o -
Figure 2-3 cont.
I -:I I
Figure 2-3 cont.
3 O 3 \i
I - I ( .I
+ icl"r- ~i i
3 6 L
Figure 2-3 cont.
2 8 _0C;
O o <
oo a a
32 so a
'O O "r
32 -, c
R A 3 17
Figure 2-3 cont.
I I I
Figure 2-3 cont.
Figure 2-3 cont.
(Part Q) .
28 L '
-19. o a
i; O O
"- D C
C4 2 C
RAR 3 17
Figure 2-3 cont.
Figure 2-3 cont.
(Part S) .
Table 2-2 lists the rms noise levels for the CLEANed
maps in mJy/beam and in Kelvin. As can be seen these values
correspond very closely with the expected rms noise level
from equation 2-23.
In addition to the narrow band line channel maps in
Figure 2-3, several channels spaced over the whole velocity
range were mapped over a large field of view, 1.5ox1.50. The
effective resolution for the se large field maps is
24.25"x22.59" (FWHM). These maps were used to search for any
detections of satellite galaxies of NGC 1300 or any other
objects. An example of the inner portion of one of these
wide field maps is shown in Figure 2-4. This is a map of
channel 16 and is a typical result. The continuum emission
has not been subtracted from this map. No evidence was found
for any 21cm line emission from any source other than NGC
1300 in this or any other wide field map. Figure 2-5 shows a
wide field map of "channel zero" with the continuum emission
still present. Thi s, again, is only the inner portion of
the total field mapped. As no evidence was found for any
satellites or other objects, except unresolved continuum
point sources, only the inner portion is shown as an example
of the type of result obtained. The rms noise levels for
these maps are tabulated in Table 2-2.
In summary the map making and CLEANing parameters used
for NCC 1300 are
1. Map making (Channel and Continuum Maps)
SO O 0
0o o o
0 P 0O
Figure 2-4. Wide Field Nap. Inner portion of wide
field image of channel 16. The continuum emission is still
present. Contours are at intervals of twice the rms noise
) ~I 11Y L I
Oa o '
- 1 9 .
o o a
o 6 b
L 5 1..
0' a '
Figure 2-5. Channel Zero. Wide field map of channel
zero. Contour levels are at approximately twice rmrs noise
d) Cell size:
e) Image size:
2. Map Making (Wide
d) Cell size:
e) Image size:
a) Flux cutoff:
7k X(1484m) 30% level of Gaussian.
Spheroidal with m=6, n=0, a=1.
7k X(1484m) 30% level of Gaussian.
Spheroidal with m=6, n=0, a=1.
DETERMINATION OF THE NEUTRAL HYDROGEN PROPERTIES
Spectrum Integration Techniqiues
The final product of the acquisition, calibration and
processing of the 21cm VLA visibilities is a set of 18
continuum-free narrow spectral-line channels for NGC 1300.
The set consists of signal-free channels at either end of
the spectrum and a series of signal -ri ch spectral-line
channels. The channel separation is 20.63km/sec and each
channel has a width (FWHM) of 25.2km/sec. These continuum-
free channel maps can be used to infer the neutral hydrogen
distribution and its associated velocity field.
If we assume that the atomic hydrogen is optically
thin, then the column density Nh at some point (x,y) is
given by (Mihalas and Binney, 1981 p489)
18 Jm _xyd 31
N (x'Y) = 1.8226xl0
where the velocity, V, is in km/sec, and TB, the brightness
temperature, is in degrees Kelvin.
The mean temperature-weighted velocity at that point is
= .m (3-2)
These will be easily recognized as the zeroeth and first
moments of the bri ghtne ss temperature with re spec t to
In the absence of observational noise the evaluation of
these quantities would be straightforward, being a straight
summation over velocity at each (x,y) point. However, as
noise is always present, a method is required which is
capable of discriminating quickly between noise and line
signal, and rejecting the noise before integration. As the
channels cover only a limited spectral range the noise may
not average to zero and will give a definite contribution to
the summation, if the summation was naively carried out over
the full spectral range. The problem is then to define a
range, or window, in velocity space which contains only line
signal. Various methods have been proposed to define this
window (Bosma, 1978, 1981). Bosma considered four methods:
1. Study each spectrum visually and define limits in
2. Fit each spectrum with a preconceived shape.
3. Apply an acceptance level in intensity (the cut-off
4. Apply an acceptance level in velocity (the "window"
Bosma (1978) studied these various methods and concluded
that the optimum method is the "window" technique.
The method used here is a variation of the "window"
method. Bosma's procedure is followed with some additional
discriminating features. A narrow window in velocity is
initially defined ~and gradually expanded with the value
outside the window being compared with the value outside the
window calculated in the previous step. When these two
.values agree to within a specified tolerance level then all
the signal is considered to have been found and the
procedure is stopped for that pixel. Using this procedure
implies that the real signal is going to be present in a
single range of contiguous channels and any large spike at a
very discrepant velocity is considered to be noise. Also,
any large spike occurring in only one channel will1 be
rejected. The tolerance level depends upon the rms noise of
the single channel maps and on the number of points
remaining in the empirically defined continuum. Thi s
continuum is the mean level of points outside the window.
Two additional criteria are added to improve the signal
detection and noise rejection capabilities of the procedure.
The procedure requires at least n points in the spectrum to
be above a specified bri ghtne ss temperature If thi s
criterion is not satisfied then no further effort is spent
on that pixel and it is rejected from all further
consideration. However, if this criterion is satisfied then a
further test is applied to reject pixels which have the
required number of points above the specified brightness
temperature but do not actually contain line signal. The
total value at each pixel (equation 3-1) must be above a
specified cut-off l eveli, e.g. three times the rms noise
level. If the integrated value is below this cut-off level
then that pixel is considered not to contain line signal and
is rejected. Usually the procedure tests for at least two
points in the spectrum being above twice the rms noise level
in the single channel maps. The integrated value usually
must lie above three times the rms noise level of the single
channel maps or be discarded.
In an effort to ensure that all the low brightness gas
was used in the integration, various tests were carried out
utilizing different combinations of smoothed single channel
maps for signal discrimination and integration. Combinations
tested included testing for integrable signal on convolved
maps and integrating using Hanning smoothed maps, and
testing for signal on convolved maps and integrating the
same convolved image. Various-sized convolving functions
were tried, all two-dimensional Gaussian functions, in order
to determine the optimum convolving method. The tests also
were carried out using different cut-off values for the
integrated spectrum and different-size boxes surrounding the
region of HI emission.
The smoothing function used was a running, three-point
X= 0.25 X + 0.5 X + 0.25 X (3-3)
n n-1 n n+1'
This function was applied in velocity space to each
pixel position. The convolving function is designed to bring
out the low brightness features and to suppress the noise.
Various beam solid angles were tested and the convolution
was applied to each single channel map. Values ranging from
half the synthesized beamwidth (FWHM) to two and a half
times the synthesized beamwidth were tried. Below half the
synthesized beamwidth the convolution had essentially no
effect; whereas, above two and a half times the synthesized
beamwidth, the convolution became so broad as to render the
maps unu sable With this size c onvolIvi ng functi on, the
resolution was so severely degraded that essentially no fine
structure was visible; all that remained was a broad, beam-
smeared, disk-like feature.
The criteria used to determine which combinations of
smoothing and convolving functions, and cut-off values gave
the optimum results were
1. The number of spectra used in the integration.
2. The s signal -to-no ise ratio in the convolved and
3. The HI mass-integral.
4. The ratio of mass-integral to rms noise level in the
The mass-integral is defined as (Mihalas and Binney,
HIas = 2.35 x 105D2 /SdV (3-4)
where S is the flux in Jy, and D) is the distance in Mpc. The
integral is calculated for all pixel points deemed to belong
to the galaxy. The region in which thi s integral is
calculated is cho sen by inspection of the HI density
distribution map produced by the "window" procedure. The
ratio of mass-integral to rms noise level is calculated
using the value found by evaluating equation 3-4 inside the
box surrounding the galaxy, and the rms noise level
calculated outside the box. This ratio should be maximized
for the optimum combination of smoothing, convolving and
cutoff values. Table 3-1 shows some typical values from two
run s during thi s testing procedure Thi s shows quite
clearly that for these tests:
1. The rms noise level clearly goes through a minimum.
2. The mass-integral goes through a maximum.
3. The ratio of mass-integral to rms noise goes through
As a result of these tests the following procedures
1. Hanning smooth, in velocity space, the original
single channel maps with a running three-point
Signal Characteristics for Spectrum Integration
Npo~int Beamwidth rms Noise Mass-integral Ratio
(1) (2) (3) (4) (5)
1 0.5 24.29 1.1430 0.0471
1.0 30.78 1.1781 0.0383
1.5 22.07 1.1777 0.0534
2.0 20.38 1.1788 0.0578
2.5 20.41 1.1784 0.0577
2 0.5 17.68 1.1248 0.0636
1.0 24.75 1.1700 0.0473
1.5 14.92 1.1742 0.0787
2.0 14.67 1.1739 0.0800
2.5 15.00 1.1727 0.0782
(1) Minimum number of points in each spectrum required
to be above the cut-off value.
(2) Convolving beamsize, in units of synthesized
(3) Rms noise level outside a box surrounding the
galaxy in arbitrary units.
(4) Mass-integral for all points deemed to belong to
the galaxy in arbitrary units.
(5) Ratio of mass-integral to rms noise level.