Title: Kinematics and dynamics of barred spiral galaxies
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Permanent Link: http://ufdc.ufl.edu/UF00102770/00001
 Material Information
Title: Kinematics and dynamics of barred spiral galaxies
Physical Description: Book
Language: English
Creator: England, Martin Nicholas, 1954-
Copyright Date: 1986
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Bibliographic ID: UF00102770
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: ltuf - AEQ2504
oclc - 16656178

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"To the Reader Concerning the Hypothesis of this Work"

Andrew Osiander cl543 in

"De Revolutionibus Orbium Caelestium"

N. Copernicus

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.

Plato: Timaeus

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.






ABSTRACT .......


. . . . . . xiv



Selection Criteria... ....
Survey Galaxies.........
NGC 1073 . . . . . .
NGC 1300 . . . . .
NGC 3359 . . . . . .
NGC 3992 . . . . .


HI As A Kinematic Tracer .....
Aperture Synthesis Theory .....
Observing Strategy and Calibration
Map-Making and Image Processing ..

. . . 18

. . .


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 ...........

. 79

. 79






. . . . . . 185

The Beam Scheme ..........
Hydrodynamical Modeling of NGC 1300


. . .

. .

Triaxial Bar Models ....
Oval Distortion Models ...
Composite Models ......
Bulge Models ........



NGC 3359 . . . . .
Observational Results ..
Hydrodynamical Models ..
NGC 3992 . . . . .


. .

. .
. .

Observational Results ...
Hydrodynamical Models ...
NGC 1073 . . . . . .
Observational Results ...
Hydrodynamical Models ...


Observational Comparisons ...
Dynamical Properties .....

VIII. SUMMARY .. . . . . . .




Neutral Hydrogen Results for NGC 1300 .
Hydrodynamical Results ........
Dynamical Properties .........

. .283




BIBLIOGRAPHY . . . . . . . . . .








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



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



Martin Nicholas England

December 1986

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,

gas tracer.

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

al., 1986):

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

shock regions.

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.

Selection Criteria

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.

Survey Galaxies

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.

NGC 1073

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.




. : .- ..

E .

. ":C2


re Irs
I~." '?''.*r~5~3

"rIt I

Figure 1-1 cont.

(Part A).

Figure 1-1 cont.

(Part B3).


~ W

a le

Figure 1-1 cont. (Part C).



.~ *

,* *

Figure 1-1 cont.

(Part D).

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

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

NGC 3359

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


NGC 3992

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

into knots.

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

wavelength region.

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

lower level.

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

equilibrium. Thus,

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
-m (2-6)

TB(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.
In general,

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

results quoted.

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,

short spacings.

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
e T
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

summarized below.

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
o oad5
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

calibration procedure.

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

whole session.

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
secondary calibrators.


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)
aL,o a,a

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

be written

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

(Sramek, 1985).

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
NGC 1300.



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

spuriously extrapolated.

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,

and y<1.

3. Subtract the result of this convolution from the

dirty map.

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

brightness values.

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

various parameters:

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

krHz .

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

temperature using

-26 2
aSx10 C

v 2ks2

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
the beam~size.

NGC 1300
NGCl300 17L16.

28 L i

-1@ C O

32.o r/

30 .



36 .0 a o

o p

C 13 a 3
O .Q O,
70 -

RAF 3 17

Figure 2-3 cont.

(Pa~rt A).

NGC 1300
NGCl300 1725.

28 o

-1930 n O 'b




38 0

0 0

0/ o
o a--/C

92+ .

38 L O

n 10

RAF 3 17

Figure 2-3 cont.

(Part B).

NGC 1300
NGCl300 1705.

28 L O

I- 9 o a
310 - a
30 o


c> O

0. +

38 o,


9036 o O 0~'
4c *9


R A 3 17

Figure 2-3 cont.

(Part C).

I1 I




m I

d CI





+ 1 .


O s


o /O




r-) o

3 1)


3 17

Figure 2-3 cont.

(Part D).

NGC 1300



o O

13 3




32 L

39r L

36 C

O o

3 J



o >


42 L



3 17

Figure 2-3 cont.

NGC 1300


(Part E).


NGC 1300
NGlCl300 1693.

28 -,i Ii o
28 o

- 19.
3 0 _0

32 .

3 4 c o c 0 o

a 9, c

38 L o o

O o

R A 3 17

Figure 2-3 cont.

(Part F).


NGC 1300
NGlCl300 1622.

28 o
Oo o

-19.3 ;f o

c Ccr
o- a3 0
39 ? o

+ +
3G (3 o X

o a

alo o
o Co
92 O

L90 25 10 55
RAF 3 17

Figure 2-3 cont.

(Part G).

-I 1 I

_ _I


9 r" 3


-19"30 L


32 o"

a ,

38 L

0 ,1 3

U y


u0 L

L02 L


3 17

Figure 2-3 cont.

(Part H).

NGC 1300



NGC 1300
NGC1300 1581.

28 LO
G co

oa C
1930 -o

3 a

32 o

3Y C. 9
oG4` +3 c,
5 + o

X O.
36 "'

tlo o '

a 0 0 r;o
I, I 1 f

RAF 3 17

Figure 2-3 cont.

(Part I).




3, o
3 C




32 L


a 0




3 L~

o a ,


. 1

36 L

;7 ,
~j~ "

;O r'


38 C




. r




3 17

Figure 2-3 cont.

NGC 1300


(Part J) .

NGC 1300
NGC1300 15110.

28 o



O rO

+J 3
36 0I o

38 -- /

o 90

92 ~o

m1 a

L90 25 10 55
R A 3 17

Figure 2-3 cont.

(Part; K).

0 I i s


"e O


32 L

,oo /


36 L_

3a t~



92 L

O o -

3 17

Figure 2-3 cont.

(;Part L).

NGC 1300


I i

I -:I I





-19"30 1


o O




b P/


40 ti




3 17

Figure 2-3 cont.

(Part M).

NGC 1300


r I

?i c
3 O 3 \i
I - I ( .I




28 L

d 0



32 &

c~ L'
+ icl"r- ~i i

3 6 L

r C

c *



L92 L

B a

3 17


Figure 2-3 cont.

NGC 1300


(Part N).

NGC 1300
NGCl300 11958.

2 8 _0C;

P o
O o <


oo a a
32 so a

o a

oo 3
31 on

36 a
'O O "r

32 -, c

uo a
-0s 0

nL I

R A 3 17

Figure 2-3 cont.

(Part O).




D o


O r

0 0

30 L

o ai



38 0
'"~5 0


L j




? O

I "

I i

3 17

Figure 2-3 cont.

NGC 1300




(Part P).

I 1




-19 "30



+ O





36 L




n i:

42 W



3 17

Figure 2-3 cont.

(Part Q) .

NGC 1300


NGC 1300
NGCl300 1396.

28 L '
-19. o a

32 O
i; O O
"- D C

30 o

oX c

38 0

0~ 'o

C4 2 C

RAR 3 17

Figure 2-3 cont.

(Part R).

NGC 1300
NGCl300 1575.






36m 1

+ -

o O

3 17

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)





`D \I



o o



Q .

SO O 0

O c

0o o o

e0 a

. C.



0 P 0O


O o
3 a







o a



o o

o ,o

I ,

3 17

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

NGC 1300


) ~I 11Y L I

O *



a o





Oa o '

ac o

3 .*
D d
o B


- 1 9 .

25 L



o ~d
o O
os O

o0 .

oo oo

o O~

. e
o o a
4 (2

0 Go
o 6 b

o ~


L 5 1..

a ~"

0' a '


- 8






o o





o, 1


3 18

Figure 2-5. Channel Zero. Wide field map of channel
zero. Contour levels are at approximately twice rmrs noise

NGC 1300


a) Weighting:

b) Taper:

c) Convolving:

d) Cell size:

e) Image size:

2. Map Making (Wide

a) Weighting:

b) Taper:

c) Convolving:

d) Cell size:

e) Image size:

3. Clean

a) Flux cutoff:

b) Gain:


7k X(1484m) 30% level of Gaussian.

Spheroidal with m=6, n=0, a=1.



Field Maps)


7k X(1484m) 30% level of Gaussian.

Spheroidal with m=6, n=0, a=1.






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

given by

= .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

Hanning function

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

integrated maps.

3. The HI mass-integral.

4. The ratio of mass-integral to rms noise level in the

integrated maps.

The mass-integral is defined as (Mihalas and Binney,

1981 p490)

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

a maximum.

As a result of these tests the following procedures

were adopted:

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.

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