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Mass determination of selected galaxies from small group statistics

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Title:
Mass determination of selected galaxies from small group statistics
Creator:
Erickson, Lance K., 1946-
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English
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v, 159 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Artificial satellites ( jstor )
Average linear density ( jstor )
Galaxies ( jstor )
Galaxy rotation curves ( jstor )
Mass ( jstor )
Milky Way Galaxy ( jstor )
Pixels ( jstor )
Simulations ( jstor )
Trucks ( jstor )
Velocity ( jstor )
Galaxies ( lcsh )
Radio astronomy ( lcsh )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Includes bibliographical references (leaves 156-158).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Lance Karl Erickson.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AFB3109 ( NOTIS )
18145292 ( OCLC )

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MASS DETERMINATION OF SELECTED GALAXIES FROM SMALL GROUP
STATISTICS






BY


LANCE KARL ERICKSON


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


1987

















TABLE OF CONTENTS


ABSTRACT . .

CHAPTER

I. INTRODUCTION . .

II. MASS ESTIMATE METHODS .

Binary Systems .
Small Groups .
Group Selection .

III. OBSERVATIONS . .

Single Dish and Interferometer Measurements
NGC 3893 . .
UGC 7089 . .
NGC 4258 . .
NGC 4303 . .

IV. DATA REDUCTION . .

Integrated Moments .
Rotation Curves .

V. SUPPLEMENTAL GROUPS .

NGC 224 (M31) .
NGC 1023 . .
NGC 1961 . .
NGC 3359 . .
NGC 3992 . .
NGC 4731 . .
NGC 5084 . .
Satellite Characteristics .


VI. MASS DISTRIBUTIONS .


Chi Distribution ..
Selection Bias .


PAGE

. iv



. 1

. 7

. 7
. 13
. 18

. 24

. 24
. 35
. 44
. 52
. 60

. 72

. 72
. 77

. 100

. 100
. 101
. 102
. 102
. 103
. 103
. 104
. 104

. 109


. 109
. 113










VII. DISCUSSION . 123

Halo Model . 123
Simulations . 125
Membership . 131

VIII. CONCLUSIONS . 137

APPENDIX

INTERFEROMETRY AND IMAGING 140

BIBLIOGRAPHY . 156

BIOGRAPHICAL SKETCH . 159













i.i





























iii
















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



MASS DETERMINATION OF SELECTED GALAXIES FROM SMALL GROUP
STATISTICS


By


Lance Karl Erickson


August 1987


Chairman: S. T. Gottesman
Cochairman: J. H. Hunter, Jr.
Major Department: Astronomy



The masses of carefully selected galaxies are measured

with both rotation curves and by orbiting dwarf satellites.

A comparison of the interior and the exterior masses of the

observed primary galaxies provides an estimate of the dark

matter, if any, in the surrounding regions.

A comparison is also made between the results of

similar studies which used binary galaxies and the small,

compact groups used in this investigation. A comparison of

the two distributions of exterior to interior galaxy mass

ratios indicates that the these distributions are not the

same.








Conclusions based on the n-body simulations of halo

mass distributions limit the dark matter to approximately

four times the group primary mass and 4 times the primary

disk radius. The measured mass ratio values which exceed the

model limit of 2 appear to be limited to the most massive or

peculiar spiral and barred spiral galaxies.
















CHAPTER I
INTRODUCTION


The rotation curve of a spiral galaxy provides a

reliable means of measuring the gravitational mass interior

to the region for which it was observed. Rotation curves

often show rigid body rotation near the central bulges of

spiral galaxies, describing the gravitational influence of

the central masses. The luminous material that can make up a

significant part of the central mass of the galaxy often

exhibits a rapid decrease near the outer edge of the

nucleus. If this rapid decrease in luminosity were to imply

a corresponding decrease in the mass density of the nucleus,

the rotational velocity of the disk should show a

corresponding decrease outside the luminous bulge. However,

these decreasing rotation curves are not common in spiral

and barred spiral galaxies. The most common shape of the

rotation curves for spiral galaxies decrease more slowly

towards the observable edge of the galaxy, or perhaps show

no decrease at all. It is argued that this implies a

gravitational influence from non-luminous, or dark matter,

beyond the measurable disk of the galaxy, in the form of a

massive halo (Faber and Gallagher, 1979).







2

Measurements to verify the existence of dark matter in

the surrounding region of a galaxy have been attempted by

extending the mass measurements of these galaxies beyond the

boundaries of the observed rotation curve. However, the

techniques devised to measure the exterior masses of

galaxies do not lead to consistent results. Such methods

range from binary galaxy analysis to galaxy cluster mass

estimates based on the Virial Theorem. The evidence for non-

luminous mass associated with galaxy clusters is limited

(Valtonen and Byrd, 1986) and is not, in general, applicable

to single galaxies. Binary mass estimates, on the other

hand, suffer from uncertainties in the bound state of the

orbits due, in part, to incomplete information on the true

separations and velocities of the members. The need to find

an improved technique to investigate the distribution of

suspected dark matter in galaxies is underscored by the

question of mass distribution on a much larger scale.

Associated with this is the need to find a cause for the

flat rotation curves of most spiral type galaxies. The

purpose of this dissertation is to develop a more reliable

technique for the study of mass distributions in selected

galaxies.

Galaxy mass estimates began with the use of optical

rotation curves for spiral galaxies and with the use of

velocity dispersion measurements for elliptical galaxies.

The flat rotation curves that were found as the measurements









became more precise led researchers to investigate the

region surrounding these galaxies for evidence of non-

luminous mass. Preliminary mass estimates of these exterior

regions came from earlier studies of binary galaxies that

were based on methods developed for determining binary star

orbits. Early work by Page (1951) provided the background

for a statistical assessment of selected galactic binary

systems. This statistical approach was necessary because

orbital parameters cannot be determined from a single

projected separation and velocity. More detailed analyses

were later carried out by Turner (1976), Peterson (1979),

and others. Gustaaf van Moorsel (1982), in his Ph.D.

dissertation, developed a very useful method for determining

the distribution of mass based on the projected separation

and velocity (squared) ratios. These projected mass

calculations provide a more reliable estimate than do Virial

Theorem calculations as Bahcall and Tremaine (1981) show.

This is due primarily to the weighting given small

separations between members by the virial method. The

expression used for the virial mass calculation is



M ( V2 (1-1)


while that used for the projected mass calculation is


M c V2R. (1-2)







4

The inverse separation expression for the virial calculation

of equation (1-1) can result in inconsistent or non-

convergent mass estimates, or a large variance in the

calculated masses.

The van Moorsel mass ratio expression is also useful

because it expresses the binary orbital projection

parameters in combined form. This projection function can be

modeled with random orbital elements and compared with a

selection of binary galaxies assumed to be random. A

statistical assessment of the selected pairs can then be

made directly from these two distributions. This method is

the primary tool used herein for estimating the mass

distributions in the selected galaxies.

The main body of this work involves the determination

of the mass distribution in selected spiral and barred

spiral galaxies. The binary galaxy techniques of van Moorsel

and others furnish the basis for the method used in

analyzing small, compact groups of galaxies. These small

groups are selected to provide satellite mass probes in the

volume surrounding the dominant galaxy. Each group has 1)

from one to four satellites, 2) satellite masses less than 5

% of the primary galaxy, 3) a separation limit of 30' from

the primary galaxy to the satellite. These criteria should

reduce the mass estimate uncertainties associated with the

previous binary studies. The galaxy group selection process

and the resulting list will be discussed in order to









establish procedures for analyzing selection biases and

their possible effects.

The details of the mass estimate method used in this

research are presented in the second chapter, while the

neutral hydrogen observations of the selected groups are

covered in Chapter III. These HI observations provide the

dynamical parameters leading to mass estimates for each of

the primary galaxies. The HI observations of the four galaxy

groups will be supplemented with observations of other

galactic groups obtained by other researchers. These

additional groups are discussed in Chapter V. Data reduction

procedures are also outlined in the Appendix which contains

a discussion of the techniques and instrumentation used for

the interferometric measurements.

The relatively small number of galaxy mass ratios used

in this study requires the use of statistical tests for

final estimates of the mass distribution. Statistical tests

are also needed for measuring the level of confidence in the

analysis. Chapter VI includes a discussion of the

statistical tests used and the arguments for using those

statistics, since the conclusions are based primarily on the

statistical interpretation of the data. Another statistical

assessment is the measurement of the likelihood of bound

membership. Valtonen and Byrd (1986) used an argument based

on the expected symmetry of redshifts in samples of binary

galaxies, groups of galaxies, and clusters of galaxies.







6

These are, of course, assumed to be gravitationally bound

systems. This same argument is used in Chapter VII to

assess the bound membership of the groups in this study, and

the membership in the van Moorsel sample.

A final discussion of the foregoing arguments and

results are also presented in Chapter VII. Comparisons are

made between the results of this work and the work of

previous researchers, even though the selected galaxies may

have dissimilar characteristics.















CHAPTER II
MASS ESTIMATE METHODS



Binary Systems

The technique used to measure the mass of a binary

galaxy is essentially the same as that used to measure the

mass of a stellar binary system. Elliptical binary orbits

describe the same physical relations of a bound system

whether for a planetary, stellar, or galactic system. For

binary systems in general, the plane of the orbit is

inclined to the plane of the sky, resulting in the orbital

elements being only partly determined by observation. In

addition, separation and relative radial velocity are the

only measurable quantities for binary galaxy orbits. Due to

the incomplete information for the orbital parameters, the

solution of any single orbit for a binary galaxy is not

possible. However, separation and velocity measurements of a

number of galaxy pairs can be used to infer a distribution

of mass for the entire set of binary galaxies. The indirect

nature required of this method was recognized, even in the

earliest studies of binary galaxy masses by Page (1951).

Improvements were made to this mass distribution technique

in order to reduce the uncertainties in binary membership

and to reduce the effects of selection bias. The development









of the separate mass distribution and orbit model

distribution by Gustav van Moorsel improved the binary mass

estimate method even more, with the ability to analyze the

biases in the distributions separately.

As shown by van Moorsel, the measurements of separation

and relative velocity for a random set of binary galaxies

can be compared to a model of random orbital values, because

both contain the same projection parameters. This comparison

is straightforward since the equation containing the

combined orbital parameters is separable from the measured

radial velocities and projected separations. Thus, to within

a constant, a random set of orbital projections should

produce the same distribution of values as a random set of

binary separations and velocities. The random orientations

of the samples for these galaxy pairs is necessary to avoid

biases in orientation, but cannot be guaranteed for the 23

measured samples used in this study. Therefore, these

observed samples must be checked for the effects of sampling

bias, as discussed in Chapter VI.

The separation and velocity measurements of the galaxy

pairs are used to produce the observed distribution of mass

values. To avoid the problems inherent in virial mass

estimates which express the member separations in an inverse

relation, Bahcall and Tremaine (1981), as well as van

Moorsel, use a mass formula based linearly on separation.

For a binary system this can be written as










M=a AV2 R/G. (2-1)


Here, a is a coefficient which reflects the mass dependence

upon the galaxy potential or the orbit projection, or both.

The difference in radial velocity between the two components

is AV and R is the projected separation of these components.

In order to modify this equation to include projection

effects due to the orientation of the orbit, we express the

measured separation and velocity in terms of the orbital

parameters and the true separation and velocity. If this

corrected orbital mass is then divided by the disk mass, a

ratio is produced for the exterior and interior masses

(multiplied by the coefficient of the combined orbital

parameters). The interior mass inferred from a rotation

curve is (Lequeux,1983)


m=V2(Rmax)Rmax/G (2-2)


where Rmax is the maximum true distance from the center of

the galaxy for which a reliable rotational velocity is

available, and V(Rmax) is the rotational velocity at Rmax'

The existence of dark matter can be determined by comparing

the number of observed mass ratios versus the value of the

mass ratio obtained (the observed distribution) with the

model distribution of the orbital elements; similar

distributions imply similar masses in the exterior and

interior regions of the galaxies under study.







10

A schematic representation of a binary galaxy system

and the corresponding orbital parameters is shown in Figure

2.1. The inclination angle, position angle of the line of

nodes, true anomaly, and semi-major axis have the same

convention as in stellar binary notation (see for example,

Aitken, 1935). The combined expression for the orbit

parameters will be referred to as X (chi), from the notation

of van Moorsel. His formulation of X follows from the

projected separation, r, and the velocity difference of the

components, Vr. The projected separation is


r=R(1-sin2i sin2 (v+))1/2 (2-3)


where R is the true separation, v is the true anomaly, and w

is the position angle of the line of nodes.

The observed (projected) velocity difference between

the two bodies is then


AV=- GM sini(cosO+ecoso)). (2-4)
a(1-e2)


Here, 0 is the sum of v and o and e is the orbital

eccentricity.

The orbital (exterior) mass which now includes the

projected variables can be written as


M=AV2r sini (cosv+ecos)2(_sin2i sin20)1/2. (2-5)
G(l+ecosv) s







































Figure 2.1 The orbital elements of an inclined binary
galaxy orbit.







12

Dividing the orbital mass by the interior (disk) mass

produces a distance independent expression of the exterior

to interior mass ratio multiplied by the X function. This

mass ratio is related to the observed mass ratio, Xobs, as

M
Xobs- (ml+m2)X (2-6)


where X represents the combined parameters in projection,


X sin2 (cosv+ecosO)2(1-sin2i sin2)1/2. (2-7)
(1+ecosv)



If the galaxy pairs selected for this investigation are

representative of a reasonably random set of projected

orientations, the observed number distribution, N(xobs),

should resemble the model number distribution, N(X), times

the exterior to interior mass ratio. Further, if these two

distributions are similar, the masses should be similar.

However, interpretations of any significant differences in

these two distributions are dependent on the assumptions

made about the physical system. Since these distributions

are intended to be used for dark mass estimates of the

galaxy systems studied, the assumption of bound orbits must

be made with reasonable confidence. A discussion of the

bound membership in these small groups is found in Chapter

VII.









The analytic comparison of the model and observed

distribututions is made using a "goodness of fit" test (see

Chapter VI), although a quick comparison can be done

visually. From Figure 2.2, one can see that the shape of the

model distribution is dependent on the orbital eccentricity,

or eccentricities, chosen. These model distributions,

however, have a bias towards small values of X owing to the

preponderence of orientations with large projection effects.

As seen from equation (2-7) and Figure 2.1, the values of X

range from 0 to 1+e. The cutoff in the distribution at e=0

for circular orbits is seen in this figure for X greater

than 1. The characteristic shape of the chi distribution at

a relatively high eccentricity (in this figure a maximum of

e=0.9 is shown) shows a gradual decrease with increasing

values of X. A random e squared distribution is included

because of the e squared dependence of Xobs on kinetic

energy, which includes the angular momentum squared. The

slightly elevated plateau in the e2 distribution is

displayed if Figure 2.2 near the circular orbit maximum of

X=l.



Small Groups

Mass estimates using carefully selected small groups of

galaxies have several advantages over the methods used for

binary galaxies as described above. The greatest advantage

for the small group analysis is the large mass ratio of the








































Figure 2.2


The random orientation distribution for three
eccentricities normalized to the same area.









primary to the secondary galaxies; this allows the dwarf

satellites to be treated as test particles. In addition,

each of the satellite-primary pairs can be considered as a

single binary system, and hence, the orbital analysis

developed for binary systems can be used for each of the

pairs within each group. An example of a group of five

satellites bound to a massive primary galaxy would produce

equivalent Xobs values of five individual binary pairs.

Although the absolute velocity difference (squared) between

the two objects may be the same for both the binary and the

primary-satellite cases, the radial velocity components are

not, in general, the same (although X has a maximum of l+e

for bound members). This is due to the motion of members of

the true pair about the barycenter. The advantages of using

a group dominated by one galaxy for the analysis include

simplified dynamics and the contribution of several Xobs

values for a single observation of a galaxy.

The expression for X is the same for the primary-

satellite pair as it is for the true binary because of the

negligible mass of the satellite galaxies. This is found in

the observed mass ratios expressions for the following three

cases.


M
Xobs =mlmX binary (2-8)
Xobs ml+m2


M
=mX singly dominated group (2-9)
m









M
=- MiN equivalent mass group. (2-10)



The XN distribution must be determined numerically since an

analytic expression is not available for more than two

bodies (see also section 6.2). A derivation for the XN

equation for the equivalent mass group (a group in which the

masses are approximately the same) is found in van Moorsel

(1982).

The interior masses, represented by the denominator of

the binary and dominant group expressions in equations (2-8)

and (2-10), are essentially the same, since the primary

galaxy mass, m, approximates the total mass of the group. It

is important to note that if the interior mass sum appearing

in the denominator of the binary expression of equation

(2-8) is actually the sum of masses of a larger number of

members as shown in equation (2-10) (a group of interacting

galaxies for example), the resulting value for Xobs can be

larger than is possible for the binary system. Such an

underestimate in the interior mass of a group that is

assumed to be a pair (due possibly to membership in a higher

order group) should therefore be avoided.

The most important difference in the binary versus

small group comparison concerns the gravitational potential

field associated with the masses. Binary galaxies with

relatively small total mass would exhibit greater dynamical







17

effect from the gravitational masses of neighboring galaxies

than would more massive binary systems. The larger

gravitational potential of a primary galaxy would also

provide more influence over an orbiting body than would two

bound galaxies with a mass comparable to the neighboring

galaxies. An analogous case of three bound members is shown

by the stability calculation for the restricted three body

problem. If we take the test mass to be appreciably less

than the other two orbiting masses (ml>>m3<
stability of the orbit for the test mass m3 is proven for

the inequality (Symon, 1975)


(ml+m2+m3) 2 > 27(mlm2+m m3+m2m3) (2-11)


or


ml > 24.96m2.


Although this example is valid for restrictions which cannot

be generalized for the small groups discussed herein, the

influence of the more massive primary galaxy is evident from

the inequality in equation (2-11). Isolated systems of

galaxies or systems with large mass potentials, such as the

compact, small groups, should improve the likelihood of bound

membership.

The requirement that group members be physically bound

is important for the mass estimate methods employed in this

investigation, just as it was in the binary studies. The







18

resulting observed and model distributions are dependent on

the validity of the assumed bound state of the system. The

selection criteria for the galaxy groups to be studied

reflect the efforts to help ensure accurate calculations of

the mass distribution and avoid interlopers or objects in

hyperbolic orbits.



Group Selection

The selection process is intended to find the small,

isolated galaxy groups which are described in the previous

section. These compact groups of galaxies include spiral

galaxies as primary members, in accordance with the binary

studies of other investigators. There are, however,

important characteristic differences between the galaxies

selected for this group study and the galaxies selected for

the binary studies, the most obvious is the dominant nature

of the primary galaxy. The desired characteristics of the

galaxy group are first specified in order to establish the

selection criteria. The defined selection parameters will

then allow an automated search through the Uppsala Catalog

(Nilson,1971) for the galaxy group members. The major

selection requirements for this group study are listed

below.


Dominance. In order to help ensure bound systems and

to select small groups with a single dominating mass, a

diameter and magnitude difference was chosen as follows: 1)







19

The primary galaxy must be 2' in diameter and larger than

any of the suspected satellites and 1' larger than any other

galaxy within 2 degrees. 2) The magnitude difference between

the primary and satellites must be 2m or greater and any

galaxy within 2 degrees must be greater than the primary

magnitude by Im or more. 3) The primary galaxy must be

equal to or brighter than magnitude 13, as listed in the

Uppsala Catalog. 4) A search of the Palomar Sky Survey

prints in a diameter of approximately 5 degrees surrounding

the primary galaxy should confirm the isolation of the group

and the dominance of the primary galaxy.

Groups which passed the selection process but may be

members of higher order groups or are listed as members of

clusters were avoided because of the complications of

interacting galaxies. Also, groups with more than ten

members within one degree of the primary galaxy were

normally rejected because of the uncertain dominance of any

single member, and because of the possibility that the

system may be composed of several interacting groups. A

lower limit of two members per group was chosen since the HI

observations may detect dwarf satellites in the region of

the primary (Gottesman et al., 1984). This single pair

criterion did not increase the final list of selected

groups.

These criteria will not guarantee the gravitational

dominance of the primary galaxy, but should provide









sufficient margin so that the dwarf satellites represent

only 5 10% of the total mass of the group. If we assume

(conservatively) constant MT /LB ratios for the primary and

satellite galaxies, the difference of two magnitudes would

provide a mass ratio greater than 6:1. Mass calculations

using the global spectrum of the HI (Casertano and Shostak,

1980) of the satellites, having spectra appearing in more

than one channel, indicated one satellite with approximately

21% of the mass of the primary. This was for the group UGC

7089, in which the dwarf satellite mass for UGC 7094 was

calculated from the global profile (Casertano and

Shostak,1980). The remaining measured dwarf masses were of

the order of 1.1% with a total average of 5.3%.


Separation. HI observations of the selected groups

were made at the NRAO VLA radio telescope in Socorro, New

Mexico. The limitations of the radio telescope which

affected the selection process were the sensitivity of the

system, the field of view, and the bandwidth of the

receiver-correlator system. At 21 cm, the field of view, or

primary beam diameter, is approximately 31'. Therefore, the

galactic groups were selected with an approximate maximum

separation of 30' for all members because of the generally

low HI mass and resulting low HI emission levels of dwarf

galaxies. Overlapping HI observations could be used to

expand the field of view, but would either reduce the

integration time on each observed field by the inverse of







21

the number of observed fields or require much more observing

time. The separation criteria for these members also include

a minimum separation value of one primary galaxy diameter.

This was established in order to avoid tidal interactions

and the obvious orbital complications that could result from

a closely interacting pair. Two such interacting systems,

M31-NGC 221 and NGC 3893-UGC 6781, are found in this study

but not included in the analysis. Seperate statistical tests

are also made for the M31-NGC 221 system because of the

possible tidal interaction.


Velocity. Velocity differences between the primary and

satellite galaxies were an important selection parameter

even though optical and HI velocities were only available

for approximately one third of the satellites. A large

velocity difference in the members of a particular group

could indicate the presence of an interloper or unbound

member. A velocity difference of at least 1000 km/sec for a

suspected satellite was indicative of an (unbound) optical

member, which should be omitted. The required mass for

binding members at such a large velocity difference is

greater even than that which is considered a very massive

galaxy. Turner (1976) found a useful maximum value of 500

km/sec for membership in binary pairs, while van Moorsel

found a velocity difference of 600 km/sec to be a maximum

value for physically associated pairs. Based on these

results and the Peterson estimate of 750 km/sec a maximum









difference of 600 km/sec was established to minimize the

possibility of including interlopers or unbound members.

The bandwidth restriction for the receiver system

limits the number of velocity channels available and the

velocity width of each channel. In order to cover a velocity

span of approximately 600 km/sec, 31 channels of 41

km/sec each were selected for these observations. Two

exceptions were made to this bandwidth choice, however.

Since velocity information was not available for one of the

suspected NGC 4303 satellites, and the maximum velocity

difference between two of the satellites was approximately

1000 km/sec, the total number of channels was increased to

64. The second exception, NGC 3893, has optical velocities

available for all of the likely satellites, with a maximum

velocity difference between primary and satellite of less

than 100 km/sec. Because of these relatively small velocity

differences, a channel bandwidth of 21 km/sec was chosen to

increase velocity resolution.


HI flux. The estimated HI flux of each primary galaxy

must be large enough to be detected easily within the time

allocated for observation. If detection of the primary

galaxy is desired within one hour, and the rms noise for 25

antennas at the 21 cm band is approximately 1.8 mJy/beam, a

3:1 signal to noise specification gives a limit of 5.4

mJy/beam for detection. Therefore, 10 mJy/beam was

established as the minimum detectable flux per beam suitable

for these HI observations.









To summarize the selection procedure, the primary

galaxy is first chosen according to specified morphological

types; the allowed types were SO, all spirals and barred

spirals, and blank entries in the Uppsala Catalog. No

primary galaxies which were listed as blank types satisfied

the remaining selection criteria, however. The primary and

possible satellites were then tested for group dominance

according to the following criteria:

1. The primary magnitude must be 13 or brighter, and 2

magnitudes brighter than the satellites, and 1

magnitude brighter than galaxies within 2 degrees.

2. The blue diameter must be 3.5' or larger, at least 2'

larger than the satellites, and at least 1' larger

than galaxies within 2 degrees.

3. The satellites must be arranged so that the group

members are situated within the 31' field of view and

no satellite can be less than one primary diameter

distant from the primary galaxy.

4. The velocity difference between the satellites and

the primary must be less than 600 km/sec.

5. The estimated flux for the primary galaxy must be 10

mJy or larger for the 43 km/sec channel width.

In addition, visual search of the group neighborhood (~5)

should not reveal nearby galaxies that are of the same order

of size as the primary galaxy.















CHAPTER III
OBSERVATIONS



Single Dish and Interferometer Measurements

HI is the atomic species of neutral hydrogen often used

to obtain spectral line maps of galaxies. It is a polar atom

with spin-spin interaction between the electron and the

proton which separates the ground state energy level. The

pervasive character of HI in the interstellar media of

spiral and barred spiral galaxies makes it ideal for

measuring the structural features of these galaxies. The

atomic form of hydrogen is of particular interest because

the dipole radiation produced by the neutral, atomic

hydrogen is more easily detected than the weaker quadrapole

radiation from the molecular species (Jackson,1975).

HI emission occurs as the spin state changes from

parallel to anti-parallel alignment of the electron and

proton (F=1-0 transition). Collisional excitation is the

predominant mechanism for exciting the hydrogen atom in most

of the interstellar medium. This excitation in turn produces

an equilibrium distribution of energy states. If the atom is

given sufficient energy to reverse the spin alignment of one

of the particles, the parallel spins will create a higher

energy state, equivalent to the magnetic interaction energy







25

between the particles. For neutral hydrogen this difference

is 5.8754xl0-5 ev, or 21.3893 cm (1.42041 GHz).

The spontaneous emission rate for 21 cm emission has an

Einstein transition probability coefficient, A21, of

2.85xl015 sec-1 corresponding to a lifetime of 3.51x1014 sec

or 1.lx107 years. The comparatively long lifetime of this

transition means that 3.5x1014 HI atoms are required to

produce one emission per second without collisional

excitation. If collisions populate the upper levels, the

transition lifetime is reduced to approximately 400 years

(Verschuur, 1974), producing a spin temperature equivalent

to the thermal or kinetic temperature of the surrounding

gas. An approximate HI mass necessary to produce an HI flux

level of one milliJansky (10-29 W m-2 Hz-1), which is

detected at a distance of 10 Mpc, using a 41 km/sec

bandwidth, is of the order of 1.Ox105 Mo (eq. 3-11).

Conversely, a one hour observation with 25 antennas of the

VLA interferometer at 21 cm, with a channel bandwidth of 41

km/sec, should detect an unresolved HI mass of approximately

5.4x105 Mo at 10 Mpc. This mass represents a 3:1 signal to

noise ratio above the sensitivity of the receiver, where the

sensitivity is given by (Appendix, equation (A-10))


AS= 45OmJy (3-1)
IN(N-l)dvdt







26

for a natural weighted map. The flux coefficient represents

the system response at 21 cm., N is the number of antennas

used, dv is the channel bandwidth in KHz, and dt is the

integration time in hours. A corresponding sensitivity in

brightness temperature units is



ATB=AS 2k (3-2)
s


For this conversion, k is Boltzmans constant, X is the

wavelength, and Os is the synthesized beam solid angle. The

brightness temperature expression assumes both a Rayleigh-

Jeans approximation for the wavelength dependence and an

unresolved emission source.

For these VLA spectral observations two independent

receiver systems of opposite polarization are combined and

then averaged with Hanning smoothing. This averaging

improves the rms sensitivity by approximately J2, as it

would if the integration time were doubled. Because the rms

noise level is inversely proportional to the square root of

the integration time, an observation of 10 hours should

allow the detection of a 1.7x105 Mo HI mass at 10 Mpc, with

a signal to noise ratio of 3:1, a bandwidth of 41 km/sec,

and with the source at the center of the primary beam.

The expression for brightness temperature which

describes the diffuse galactic HI emission detected in

spiral galaxies is the same as that used to describe the







27

emission temperature of of a diffuse gas cloud. In the more

general case, the cloud is considered to be illuminated by

another (continuum) emission source of temperature TS. The

brightness temperature, TB of a cloud (spin) temperature TC

and frequency v with an illuminating source of temperature

TS, is then (Kraus, 1966)


TB(v)=TSexp(-T,)+TC(l-exp(-T,)) (3-3)


where T, is the optical depth of the cloud at frequency v.

For diffuse emission, the optical depth is less than

one, and the exponential in equation (3-3) can be expanded

and then approximated with the first two terms of the

expansion. Since the optical depth is relatively constant

over the frequencies used for these observations, the

frequency dependence can be removed, hence


exp(-T)=l--, (T<

and the brightness temperature becomes


TB=TS(1-r) + rTC. (3-5)



The brightness temperature of the observed galactic HI

emission is more closely approximated without the background

source of illumination. Thus, the brightness temperature is

simply


T B=TC
*'B VC


(3-6)











The small HI optical depth in a typical low mass, dwarf

galaxy is due to the low density of neutral hydrogen. The

corresponding diffuse HI emission approximates the radiation

from the entire HI mass, since negligible re-absorption

occurs at these low hydrogen densities (Wright, 1974). The

galaxies which are large enough to be used as primary

galaxies in this study normally exhibit stronger emission

and may not be correctly approximated by a small optical

depth at all locations. Therefore, the surface emission is

only a lower limit on the HI mass estimate for these larger

galaxies (an estimate for the optical depth can be made from

equation (3-6) with Tc approximately 1000 K (Spitzer, 1978)

and r<< 1). Also, a larger column density, such as those

found in more edge-on galaxies, may not yield the correct

value when calculating the hydrogen mass.

The surface density of neutral hydrogen can be

calculated from the brightness temperature integrated over

the velocity bandwidth, according to (Wright, 1974)


nH=1.82x1018 J TB dV. (3-7)



The column density, NH, is measured in atoms per square

centimeter and dV is the channel bandwidth in km/sec. An

integrated hydrogen mass can also be found from either the

global profile (integrated flux) or from the surface density

map (integrated brightness temperature). For optically thin









emission, the hydrogen mass can be calculated from


MH=f nH (n)dG (3-8)



=1.82x1018 f T(0,V)dV dO. (3-9)
a--


Using the Rayleigh approximation we have


S(V)dV=-2k J T(U,V)da dV. (3-10)



Hence, the hydrogen mass can be expressed as


MH=2.356xlO5D2J SdV M (3-11)


where MH is in solar masses, D is the distance in Mpc, and S

is the integrated flux in Jy.

The galactic HI and continuum emissions can be mapped

spatially as well as in the velocity dimension either with

interferometer measurements or with single dish scans. The

theory and techniques used in both of these imaging methods

are lengthy, and will only be discussed briefly in the

Appendix. A useful reference for interferometric imaging is

found in the NRAO handbook on Synthesis Imaging (Perley,

1985). The emission maps used in this study are produced

from interferometer measurements made at the NRAO VLA.

Before the observations could be planned, however, estimates

of the expected HI flux levels for the primary galaxies were







30
necessary. These fluxes could then be used to determine if

the signal level from the primary galaxy source would be

sufficient to produce an emission map with the desired

signal to noise ratio, within an observation lasting one or

two hours. Although the actual integration time on the

individual groups was approximately 10 hours, the primary

galaxies were not, in general, in the center of the

observing field. The minimum flux requirement should ensure

ample signal from a primary galaxy whose flux strength may

be attenuated by the off-center beam response. This flux

limit should also ensure a large signal to noise ratio for

the peak emission features in the emission maps, as well as

a relatively low noise level in the temperature weighted

velocity field which was produced from these maps.

Of the five groups which satisfied the selection

process for this study, two primary galaxies had either no

flux listed in the literature or inconsistent flux values

listed. Observing requests for these two galaxies, NGC 4111

and NGC 5689, were submitted, and granted, for single dish

HI observations at the NRAO 92 m radio telescope in Green

Bank, West Virginia. These 92 m HI spectral observations

were completed in December of 1985.

Because of the transit configuration, the 92 m radio

telescope is limited in integration time to approximately 5

minutes*cosec(DEC) in a given 24 hour period. To increase

the spectral sensitivity, the scans from separate days are









averaged together. The positions corresponding primary

galaxies NGC 4111 and NGC 5689 were observed for five days,

each with approximately seven minute integration between

November, 1985 and December, 1985. All scans used the total

power mode, a procedure which compares off-source reference

data (blank sky) with on-source (galaxy) data in order to

subtract the system noise component and to improve the

spectral baselines (NRAO 300' Telescope Observers Guide,

1983). Cooled FET spectral amplifiers were used, providing a

system temperature of approximately 250 K for each of the

two receivers used. The detector for these spectral line

observations was a Model III auto-correlator spectrometer

with 384 channels. The spectral observations were split into

two identical receiver systems measuring oppositely

polarized (right circular and left circular) emissions.

Each of the 192 channels had a bandwidth of 50 Khz which

spanned the same frequency in both the right and the left

channels. The computed spectra for both the calibration

sources and the program sources were then averaged in each

overlapping channel. The baselines appearing in these

spectra were normally removed with a fourth order polynomial

fitting procedure available in the computer software at the

NRAO in Green Bank.

The observations of NGC 4111 indicate a peak HI flux

level of approximately 65 mJy as shown in Figure 3.1. This

HI emission would be detectable with the VLA interferometer









using a 42 km/sec bandwidth, assuming the source was

unresolved by the VLA array. However, the VLA observations

showed that the HI emission did not come from the primary

galaxy, but from a nearby galaxy identified as UGC 7089 (see

section 3.3). When one accounts for the off-center

attenuation of the 30' VLA primary beam and the attenuation

from the 92 m beam centered on NGC 4111 (12h 04m 30s, +430

20' 43"), the flux observed at Green Bank is consistent with

the global profile of UGC 7089 shown in Figure 3.1.

The velocity range used to observe UGC 7089 with the 92

m telescope was 0 to 2000 km/sec while the velocity range

spanned for NGC 5689 was 1200 to 3200 km/sec. However, the

primary galaxy, NGC 5689, and the associated group members

did not have sufficient HI flux to provide a reliable

spectral identification. The upper limit to the flux shown

for the five averaged integration in Figure 3.2 would be

less than 10 mJy. A theoretical calculation of the rms noise

can be made from (Appendix, eq. (A-9))


3.06T0 K
AT = sys (3-12)
rms V(2dvdt)


where T ysK is the system temperature in Kelvin (250 K for

the FET amplifiers), dv is the bandwidth in Mhz, and dt is

the integration time in minutes. Because the HI intensity

was too low to observe at the VLA within a reasonable

observing time, NGC 5689 was deleted from the list.

























S0.04 -
X
..J





-0.04 -



-0.08 I I I I I I
382.4 536.8 691.2 845.6 1000.0 IIS4.4

VELOCITY ( km/sec )





Figure 3.1 HI spectral plot of NGC 4111 showing the
emission of the neighboring galaxy UGC 7089.








34













0.04



0.02

-J

2 0.00


-0.02



-0.04 I I
1329 1628.6 1894.3 21s00 2425.7 2691.4 2957.1
VELOCITY ( km/sec )









Figure 3.2 HI spectral plot of NGC 5689 showing the
rms noise without baseline removal.









A total of four small groups were chosen using the

selection process described earlier, for spectral line

observations at the VLA. The HI observations were scheduled

from 1985 through 1987 because of the 15 month cycle time of

the antenna configuration at the VLA. The expected detection

levels for the dwarf satellites indicated that the

observations should be made with the two most sensitive

antenna configurations for extended sources, the C and D

arrays. The 10 mJy detection level is based on the previous

detections of dwarf satellites near spiral type galaxies

(Gottesman et al., 1984) and equation (3-1). Observations

were requested for both C and D array measurements of each

of the galaxy groups to improve both resolution and

sensitivity over a single configuration observation. The D

configuration observations were approved for all four groups

and an equivalent period of observation was approved for two

of the groups with the C array. NGC 3893 and NGC 4111 were

chosen for C array observations because of scheduling times

and the greater number of possible satellites within these

two groups.



NGC 3893

The galactic group associated with the Sc type galaxy,

NGC 3893, consists of the primary galaxy and the four dwarf

satellites which were detected with HI measurements. The

approximate 21 cm. primary beam coverage of the VLA







36
antennas at the 3dB level (30') is shown for this group in

Figure 3.3. A list of the physical characteristics of the

group appears in Table 3.1. Optical velocities for three of

the four detected satellites were available from the Uppsala

Catalog and resulted in a channel bandwidth selection of 21

km/sec. This provided a greater resolution in the galaxy

velocity field, which normally improves the rotation curve

accuracy.

The NGC 3893 group was observed with the VLA radio

telescope in the C array configuration during December,

1986, and with the D array in March of 1987. The visibility

data were combined in the map (image) plane rather than in

the visibility (uv) plane. This technique was used to

expedite the mapping process (van Gorkum, private

communication) and is explained briefly in the Appendix. A

total of 31 channels were used for a total velocity

bandwidth of 660 km/sec. Hanning smoothing and auto-

correlation normalization were used to stabilize the

receiver.

Figure 3.4 consists of contour plots the maps

(channels) exhibiting HI emission either from the primary or

satellite galaxies. These emission maps have had the

continuum emission subtracted and the emission features

cleaned (Hogbom, 1974; Clark, 1980) to remove the sidelobes

(see Appendix). The maps have a 6" pixel width and a

resolution (synthesized beam size) of 30.6" x 24.5", with































6 678 *
.
*,




.



**









Figure 3.3 Palomar Sky Survey field in the vicinity of
NGC 3893 (UGC 6778) showing the approximate pointing
center and primary field of view for the VLA 25 m
antennas (30').









TABLE 3.1

HI Summary For Observed Groups




NGC 3893 NGC 7089 NGC 4258 NGC 4303


Synthesized beam 30.1x24.9 30.1x24.9 54.3x51.8 66.5x57.8
FWHP (")

Beam position angle +88.0 +88.0 -76.0 -87.7
(degrees)

Channel Seperationa 20.75 41.44 41.46 41.3
(km/sec)

rms noise per channel 0.52 0.30 0.18 0.15
cleaned (o K)
(mJy) 0.65 0.39 0.84 0.93

Observed systemic 969 789 465 1561
velocity heliocentricc)
(km/sec) error 0.6 0.9 0.4 1.1

Inclination angle 43.0 68.0 71.8 18.8
(degrees) error 2.6 2.7 1.2 7.7

Position angle -1.6 215.0 29.0 -34.6
(degrees) error 0.5 0.9 0.8 1.0

Scale length b 0.26 1.80 0.14 0.47
(arc min.) error .02 .20 .09 .05

Global profile 253 218 505 249
(km/sec)

Hydrogen fluxd 1.62 0.19 7.64 1.56

a. Channel bandwidth = 1.22 channel separation.
b. Toomre n=0 model scale length parameter.
c. Full width at 0.2 of peak, not corrected for inclination.
d. Integrated hydrogen xl0 / D(Mpc)c.









the beam major axis rotated 88 degrees counter-clockwise

from the north. The three identification crosses appearing

in the contour maps indicate the positions of NGC 3893

(right), NGC 3906 (lower) and NGC 3928 (left) according to

the Dressel and Condon Catalog of Optical Positions of

Bright Galaxies (1976). Negative features are displayed as

dotted lines and are plotted at the same levels as those of

positive contours.

Several important features of the primary galaxy and

satellites are found in these narrow band spectral maps. The

most noticeable anomaly is the interaction of the satellite

UGC 6781 with the primary, NGC 3893, shown in Figure 3.4. As

a result of the strong interaction displayed by the HI

bridge in these maps, UGC 6781 is not used in the mass

analysis. Since the rotation curve of the primary galaxy

does not exhibit major anomalies, within a radius extending

as far as the opposite (North-West) HI boundary to the

interacting satellite, it was decided that the remaining

primary-satellite pairs were still useful for this study.

The emission maps in the first through fourth maps of Figure

3.4 show evidence of a fourth dwarf satellite which was not

listed in the Uppsala Catalog, nor seen in the Palomar Sky

Survey prints.

The position of the primary beam center during

observation was II11h 47m 35s RA, +480 55' 00" DEC. This

position is off-set from that of the primary galaxy so that































Figure 3.4 The spectral line emission maps of NGC 3893
plotted at 3 sigma (0.16 mJy) (upper left to bottom right).
Velocities are heliocentric. Identification crosses appear
at the optical positions of NGC 3893 (right), NGC 3906
(lower), and NGC 3928 (left).










































































































40 4 47 40


I11? km/se



10 .



-
so "-l


t11


4* 4 4? 40


-I -I I I-












--K







+
1104 kis/es
































*' ** '.




063 km/see
*.































0 +
**
-I I 1 ,

-1 ,I "1 I I "t
~1104 kllll~e .
;





eL
+




,










g
*










a
- ._ I I
I 110 I/ C
-' "a k i '


























**
-4 J "% 1 "1 I
**
~ ~ *







*.




n ta/ *


I I 1 I I

1126 km/e *

.

10 *





00*




so-


10 -
















46





1s -
11 M


I I Iu


+





I I I -


I


A


.


7


















I I


1042 km/ses


so -





*a --


NO I I 1 .. "

1001 kU/ue .



to
*
10 -




**








*o -' *
.. .












40
8 *m* .
.0- "







*. .










~ ** ** '



.o .* ..
10 *


.


.. .
B."






















IJ *.- .-
, 1* III-*
** 3 --- -1 ; *' I* ____ I I


48 47 46


I i


i -iI I I I

"I I 1 .
WO tam/3M

.. *




.* .








; ~ ~ -.- *" '- *:






'4 -






1~ S
*


S- -

















*
46*














*
i*- r* .. i "o -- I --
















tM f/M *
m O .
















**II*** l -I
: 47




;- ." **. .




.'" "* '


Figure 3.4-continued


40 10


1 I I I I
1022 bkm /on


+ ..


*


11 GO




















.r .~'i. I


( I o*.
867 la/set



* *



"o
















+ *"



0

o .* .


*9.-


- 9.

-a.


41 48 47 46 11 8


156 ki6/-












*' F
"* I I I I

~, *


'
o .
/ -




**
.* ,.



"







'*







'. I 1 *
o .*








*
-*







*







.*


.' .. .
o "


- I. .. .
., m'mo













**

*~ 's

S
.
*-
...

*
-I '* ,' ,I 1-- --- A -l --


48 48 47 44


076 km/See













S.. *'~*







9



9. *





*1 I I


u 4L-


48


o0


400


1. ,* k /M
'-. I* i i -


63 a *
-L 0 u*
i' .t,
4 p




*
I +





..
? .



*
..* a


**' *, *




*' .

*. '
o *


1 .1 I


Figure 3.4-continued


--


rr


I









the primary and the satellites could be observed

simultaneously. The emission features in these maps have not

been corrected for the off-center attenuation of the primary

beam. Contour increments for these maps are 3a with a=

0.52K (o=/2)

Figure 3.5 shows a contour plot of the continuum

emission which has been cleaned to a level of 3a. Continuum

emission for the dwarf satellites is not evident at 3o

significance level. Strong continuum sources (S > 100 mJy)

were not apparent in a field much larger than the original

512x512 pixel maps. The continuum map is an average of

channels 3 to 5 and 27 to 29. Channels 1, 2, 30, and 31 were

discarded because of the degraded response in these end

channels which were caused by the passband.



UGC 7089

NGC 4111 is an SO type spiral galaxy, with three

suspected satellites appearing with the group on the Palomar

Sky Survey print shown in figure 3.6. The single dish HI

spectral plot in Figure 3.1 was made from observations

obtained with the NRAO 92 m radio telescope in November,

1985. The peak emission appears at approximately 800 km/sec.

However, the first HI maps, which were made from VLA C array

data, showed an unexpected lack of spectral emission from

NGC 4111 in all channel maps. The HI emission originally

thought to be from the primary galaxy, NGC 4111, is actually

















49 20






10




D
E
C 00
L
N
A
T
I
0
N 48 50





40


NGC3893 IPOL 1416.819 MHZ N38CONCV.CONCV.1


30[- I 'I I I
11 50 49 48 47 46
RIGHT ASCENSION
PEAK FLUX 3.9005E-03 JY/BEAM
LEVS 1.5000E-04 ( -6.00, 6.000, 12.00,
18.00, 24.00, 30.00, 36.00)


Figure 3.5


Cleaned continuum map of NGC 3893 plotted
at the 3 sigma level. Negative emission
features appear as dotted lines.









from the nearby galaxy UGC 7089. The lack of detectable

emission from the primary galaxy was compensated by the

identification of a second primary-satellite pair, UGC 7089

(primary) and UGC 7094. The details of this and all other

groups used in this study are presented in Table 3.2.

The channel maps shown in Figure 3.7 are contour maps

for the UGC 7089 group plotted at 3 a intervals, with a=

0.300 K. The data sets obtained with the C and D array

observations were combined in the same fashion as for NGC

3893, that is, in the image plane. The identification

crosses appear at the Dressel and Condon optical positions

for UGC 7089 (right center) and UGC 7112 (left center). The

field of view consists of 6" square pixels with a resolution

30.6" by 24.9". The pointing center of the maps is at 12h

03m 49s RA and +430 24' 00" DEC. The dwarf satellite UGC

7094 appears in the seventh and eighth maps of Figure 3.7.

A later spectral observation of NGC 4111 in HI was made

with the 92 m NRAO transit telescope by Otto Richter in

April, 1987 (Richter, private communication). The single

five minute scan from 0 and 4000 km/sec indicates a maximum

flux level of approximately 20 mJy which is too low to be

considered for observation with the VLA. A slightly higher

emission level may exist at approximately 2400 km/sec. If

this were the recessional velocity of NGC 4111, however, it

would be far too distant in velocity space to be considered

a member of the group.








47














*Ad&
.



SII*



























the galaxy NGC 4111.
*, ---. *. *










Figure 3.6 The UGC 7089 group from the Palomar Sky
Survey pring showing the 30' VLA field of view and
the galaxy NGC 4111.









TABLE 3.2

Primary Galaxy Characteristics


System Typea V sys Dist incb Rmax V(Rmax) M(V ~
km/sec Mpc deg Kpc km/sec x10~0 Mo


NGC 224 SAS3 -301 0.7 78 28.0 230 34.4
SbI-II

NGC 1023 SBO- 610 7.5 80 9.5 251 13.9
SBO1(5)

NGC 1961 SXT5 3935 41.3 50 24.0 400 75.0
Sb(rs)IIpec

NGC 3359 SBT5 1009 11.0 51 20.8 140 9.5
SBc(s)I.8pec

NGC 3893 SXT5 969 10.4 34 7.8 180 8.1
Sc(s)I.2 2.0

UGC 7089d 789 8.4 68 5.5 82 0.9
Sc 2.7

NGC 3992 SBT4 1046 14.2 53 19.0 240 25.4
SBb(rs)I


NGC 4258 SXS4 465 5.2 72 19.0 208 19.5
Sb(s)II 1.2

NGC 4303 SXT4 1561 12.9 19 12.2 221 14.0
Sc(s)I.2 7.7

NGC 4731 SBS6 1490 10.5 54 13.0 150 6.8
SBc(s)III

NGC 5084 L(-2) 1721 15.0 >86 34.0 328 85.0
S01(8)


a. From de Vaucouleurs et al. (1976) and Sandage and
Tammann (1981).
b. Inclination angle and error, if given,
c. Disk mass calculated from 2.325xl0 V' Rmax/G.
d. Type from Nilsen (1973).
































Figure 3.7 Contour maps of the emission features of
UGC 7089 and the associated satellite UGC 7094.
NGC 4111 is not apparent at the 3 sigma minimum of
these plots (sigma=0.39 mJy). Identification crosses
appear at the optical positions of UGC 7112 (left)
and NGC 4111.

















I I .
1041 klam e
s o _m -,





- -t-






6I .










. --

*































I I I I


43 41 *.


40



as -









to -





0 -



aI-
sa,"s


os 04 03o a it as


I I 1


.1* r __ I --- I --- I -

I I I I
W4 hkj/a.







*











84 /**
I + I I I





















+












*
" I I "




















*


*




- II I


o 04 oa 0o


1007 km/ise


-3 kml/Me


. I


i &I












S* *I /



- I' .I I
800 km/seC















aS



.+
0
S*















I I I I. a
S













| I I '1 .
'+ .








- + *"


18 O A 04 08 0 18 0s


Figure 3.7-continued


TS3 kiI/e









- '4.
1 1 I | I






~ "I I l '


_' l* '







52
Figure 3.8 shows the continuum map which was averaged

from channels 3, 4, 5, 27, 28, and 29, and cleaned to a

level of 3a. The weak continuum emission seen at the

position of NGC 4111 is typical of many SO galaxies. The

weak continuum emission at the position of UGC 7089 is also

suggestive of a lower mass object. Several other weak, un-

resolved continuum sources are also shown in the field of

view, unassociated with the satellite detected in HI. Strong

sources were not observed in a field much larger than the

original 512x512 pixel maps.



NGC 4258

A number of investigators have obtained single dish and

interferometric HI measurements of NGC 4258. The most

comprehensive study of this large Sc galaxy was the Ph.D.

dissertation of G. D. van Albada (van Albada, 1978). None of

the observations found in the literature, however, included

observations of the nearby satellites which are necessary in

this study for the evaluation of the orbital masses.

Therefore, VLA observations were requested in order to

measure the orbital parameters of the satellites and to

obtain the rotation curve of the primary galaxy. A section

of the Palomar Sky Survey print in the region of the NGC

4258 group is shown in Figure 3.9.

VLA observations were made of this group with the D

array in December, 1985. Since velocity data were not

















NGC4111 IPOL 1418.555 MHZ


N41CLCONCV.CLCV.1


0 d1


* C..


10


0-.-.


0


* ,0


0 ~
0 '~


4' -


a 0


06 05 04 03
RIGHT ASCENSION
PEAK FLUX 1.2407E-03 JY/BEAM
LEVS 1.0000E-04 ( -6.00, 6.000, 12.00,
18.00, 24.00)


Figure 3-8


Cleaned continuum map of the UGC 7089 group
at 3 sigma. Negative emissions appear as
dotted lines.


43 45



40


35 [-


* a


251 -


4
4
S *


* t.


0O


I.-
0


10


a.
.7


05 -


.f,


0
"1I


12


ra


. I















































Figure 3.9 The NGC 4258 group from the Palomar Sky
Survey pring showing the 30' VLA field of view.









available for one of the three suspected satellites, UGC

7304, a velocity bandwidth of 41 km/sec with 31 channels was

selected. The channel bandwidth of 41 km/sec is larger than

that used by other researchers, including van Albada. With

this bandwidth, however, the velocity range of

630km/sec is spanned with 31 channels, and the rms

sensitivity is increased owing to the larger bandwidth.

Hanning averaging and autocorrelation normalization were not

used during the observations. The stronger emissions from

NGC 4258 were of the same order as the continuum emission,

approximately 700 mJy.

The contour maps shown in Figure 3.11 are plotted at

intervals of 3a, with a=0.18K. The maps are made with 6"

pixels and 54.3" by 51.8" resolution (-760). The pointing

center of these maps is 12h 15m 30s RA and +470 37' 00" DEC.

The two crosses appearing in these maps are located at the

optical centers (Dressel and Condon, 1976) of NGC 4258 and

the satellite NGC 4248.

The relatively high -continuum emission features within

this field are shown in the cleaned continuum map, Figure

3.10. The strongest emissions are from the central region of

the primary galaxy, and from two unresolved sources of

approximately 200 mJy/beam each. Because the emission from

NGC 4258 was extended and was comparable in strength to the

continuum emission in a number of these maps, it was decided

that the stronger continuum emission should be removed































Figure 3.10 Spectral maps of the NGC 4258 group plotted
at 3 sigma (sigma=0.84 mJy). The optical positions
of NGC 4258 (left) and NGC 4248 (right) are shown
with identification crosses.















-,,- i ]i I I" I -
922 km/*"




*
S-

+ 0 .






-




0 0
0
0


0




739 kmi/se







o 4 o ,




.0





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m/













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o a







oa




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11 IS













- I I I
o 574 km/see
,






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/
I I <:f I _,I .
o 0~ 5 o st 91ia/


0 *0






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00 io 30 o0 6 o3 00o 4 io


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I 1,3 0. 1 0 0


Figure 3.10-continued


-t I


. 0 4 O


*0
0


C- i-- --3 -
L co I .I,.
410 uII./Me 0


Q) 0 0


0 0 .




0^ 7
0000







*0

0 0
no O 0 o

CI *


I k


11 o0 17 s0


00 10 so 00 10 30 00 14 30












































































00 1 0 00 1i o


00 14 0 18 00 17 30 00 1 0o 00 1 0


Figure 3.]0-continued


47 50 -




40












2 -







*0
0-







47 sioL


40










21


t0






I 15 00


o 244 km/seu
0 o


L3 ,


0o

.0

S0






0 *

- 0

. I 1 I I I I


I I I I I
So ."






0o
0 +
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i 0




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o6 .
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a 0
0

o o

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1? 30


00 14 30







60

before the final mapping was accomplished. This was done to

minimize the effects of calibration inaccuracies and also to

remove the sidelobe effects from the beam response. Only

one satellite, NGC 4248, is apparent at the 3o level in

these channel maps. The second satellite listed in Table 3.2

is visible in the Palomar Sky Survey prints as a small

object which may be a dwarf satellite. An HI spectrum of

this small object, using the NRAO 92 m telescope, was made

by Thuan and Seitzer (1979). However, as the authors

suggest, this emission is spatially confused with the

emission from NGC 4258. A calculation of the emission flux

expected from NGC 4258, with the beam response of the 92 m

telescope centered on this satellite, is comparable to the

spectral plot of the small satellite in the Thuan and

Seitzer survey. Emission from this small satellite does not

appear at other velocities in these maps at the 3a level,

and so, it can be assumed that the observed emission is from

NGC 4258.

The continuum map of the strong continuum sources,

which were removed in the visibility plane, is shown in

Figure 3.11. The map is cleaned to and plotted in increments

of 3a. Note the assymetrical continuum emission from the

primary galaxy center.


NGC 4303





















47 45


NGC4258 IPOL 1417.398 MHZ


12 16 30 00 15 30 00 14 30
RIGHT ASCENSION
PEAK FLUX 6.5457E-02 JY/BEAM
LEVS 7.5000E-04 ( -3.00, 3.000, 8.000,
12.00, 24.00, 48.00, 98.00)





Figure 3.11 Cleaned continuum map of the NGC 4258 group
showing the strongest features of the continuum
radiation which was removed from the visibility data.
Contour levels are in 3 sigma intervals.


N4258CC.CLN2.1









NGC 4303

Several HI observations have been made for NGC 4303,

but again, the emission features of the satellite galaxies

are not available at sufficiently sensitive detection

levels. This group is a member of the Virgo cluster and

would have normally been excluded from consideration.

However, as the group is isolated, and the selection

criteria were satisfied, it was retained for observation.

Nonetheless, the possible effects of the cluster environment

should not be forgotten. Figure 3.12 shows the NGC 4303

group environment with the 30' VLA field of view.

The relatively high continuum flux in the field of view

suggested that the continuum be subtracted from the

visibility data before the final map-making procedure for

the same reasons as those given for NGC 4258. Both the

original set of maps and the set of maps produced with the

continuum subtracted were made with 6" square pixels with a

512x512 pixel field and a resolution of 66.5" x 57.8". The

continuum channels which were used for continuum subtraction

were 8 to 15 and 48 to 55, from a total of 63 channels.

Channels 1 to 7 and 56 to 63 were discarded because of the

non-linear passband response in these end channels. The

contour plots of the emission at the 3 a level are shown in

Figure 3.13 with a= 0.150 K. One satellite (UGC 7439),

which was detected at the 3o level, is also shown. The

centers of NGC 4303 and UGC 7404 are plotted with the







63










"m I < "''v *" X

/ *" '" .,; MU.1 "
.* v ,&- V .
." 1 -/ ,






*04
.. .S ,

















approximate pointing center and field of view for the
VLA observations.
VLA observations.









optical positions from the Dressel and Condon catalog. The

pointing center for these maps is 12h 19m 18s, +04o 5' 00".

The channel bandwidth of the maps in these figures is

41 km/sec, with a total bandpass of 2600 km/sec for 63

channels. Since 31 channels were either discarded or used

for continuum, the actual velocity bandwidth is 1334 km/sec

for the remaining 32 channels. Each of the continuum

channels was searched for line emission before the continuum

averaging and again after subtracting the continuum from all

but the discarded channels. The continuum map which was

produced to provide the original visibilities for

subtraction, cleaned to the 3o level, is shown in Figure

3.14. Weak continuum emission is observed for the primary

and detected satellite galaxies, while the stronger, un-

resolved continuum emission appears without an associated

visible object in the Palomar Sky Survey prints.

































Figure 3.13 Spectral line emission maps of the
NGC 4303 group showing the primary and satellite
emissions above 3 sigma (sigma=0.93 mJy). The
identification crosses appear at the positions of
NGC 4303 (left) and UGC 7404 (undetected, right).
















































I I,
0 0 '*-






S-










0






*0*
o I
o 0S
fi Q l l


142 kM/soc 0.








0 +

0





OA'









I I l


01 of


00


04 II



so



46



40




as









00






so



45


40


as


M
I1 aS 00


00 s1 30 0o o 3


72I i m/e
I725 lu/s.C .


be Is a


00 is s0


0


*8f


p



a


1"83 km/Sec









0 +










*o











* 0









0
aI-











0
6 o












0

o




o





*















0 _
*
,f


a


'1.4
S
"A


0


I I


00 t s0


1 1 I 1 I
.' ISMS ki --


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0

0 0
a o 0 0



S-






o o
-~ 0






0
- o -




0 I



0 0


-


-



-



-



-






-















I I
14






00 -
04 5



S* 0









so -
0 0





a




o I 4,
I----- -- >- -

Is
04 as -

o
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o
04u O
o








40 -



U -


.
so 1 I I



Is aO -O o





0r













0





It 11 50 as 0 00 1 3O0



Figure 3.13-continued


oo 00


SI I I I
liaT kmise/


0

o -
o 0



S* -

o +

0- a 0


S0


o




I n I I I I I


I I
75 km/SOC










0*






0







I I-


1 I t o I
1431 km/mc

L
0






0 0+
o 0
o





0






o -
o


1 I,
)0ot km/SOC


6'
0



-



*


0

0 0


-


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o as so o 11 U




















0


0


00 -






so -
S.



46



40


3.


0 I I I
1 *1 00 20 30 00 0s 0




Figure 3.13-continued


oo 0 10o


Skm/e
E6 kl /n


o







0
*



*



a






*4


o40












NGC4303 IPOL 1407.702 MHZ N43CONCLN.CLN.1


05 20


D
E
C
L
I
N
A
T
I
0
N 04


30so I I lif I I
12 21 00 20 30 00 19 30 00 18 30 00
RIGHT ASCENSION
PEAK FLUX 1.3707E-01 JY/BEAM
LEVS 3.0000E-03 ( -3.00, 3.000, 6.000,
12.00. 18.00, 24.00, 30.00, 38.00, 48.00,
60.00)


Figure 3.14 Cleaned continuum of the NGC 4303 region
plotted at 3 sigma showing the positions of the
continuum removed from the visibility data base.









TABLE 3.3

Galaxy Group Characteristics


Group vela del Vb sep. M(d V)c chi mag
Satellite km/sec km/sec Kpc x10 Mo (M/m)


NGC 224d
NGC 147
NGC 185
NGC 205
NGC 221

NGC 1023
North
South

NGC 1961

A
B
Bl
UGC 3342
UGC 3349

NGC 3359
dwarf

NGC 3893
UGC 6797
UGC 6834
dwarf

UGC 7089
UGC 7094

NGC 3992
UGC 6923
UGC 6940
UGC 6969

NGC 4258
UGC 7335

NGC 4303
UGC 7439

NGC 4731
RNGC 4731a


-301
-168
-208
-240
-216

610
905
695

3935

4108
3895
3800
3927
4282

1009
962

969
963
987
1102

789
769

1046
1062
1112
1115

448
479

1561
1275

1490
1505


+142
+100
+62
+84


+295
+85


-173
+40
+135
+8
+347


-47


-6
+18
+133


-18


+16
+66
+69


-31


-286


90.2
85.6
7.5
4.9


34.2
17.5


91
106
122
157
240


48.5


47.5
30.3
37.2


24.1


60.6
35.4
45.3


19.1


39.1


+15 32.1


42.3
20.0
0.7
0.8


1.23
0.58
0.02
0.02


85.9 6.18
3.7 0.26


62.6
3.9
52.4
0.2
671.9


0.84
0.05
0.70
0.00
8.96


2.5 0.26


.04
.23
15.5


0.01
0.04
2.68


4.3
12.0
11.0
9.4
9.2

10.5



12.2




15.4
14.4

11.0


10.6
14.1
13.1


14.8
.18 0.25 15.6


0.4
3.6
5.0


0.02
0.14
0.20


10.7
14.1
16.0
15.5


9.6
0.4 0.02 13.9

10.9
74.4 4.75 14.9

6.0
0.2 0.03 -














Table 3.3-continued


Group vela del Vb sep. M(d V)c chi mag
Satellite km/sec km/sec Kpc x10 Mo (M/m)


NGC 5084 1721
dwarf 2089 +368 65.6 206.5 2.43

a. Heliocentric velocity reference.
b Vs ttl1ite- Vss
c. orbita mass according to 2.325x105 (del V)2 R/G.
d. A.S.R. velocity reference from Einasto and Lynden-Bell
(1982).















CHAPTER IV
DATA REDUCTION



Integrated Moments

The spectral line maps made from the VLA HI observation

are the primary source of data for the analysis of the mass

distribution of these galactic groups. The HI emission

features found in the maps are used to identify the

dynamical characteristics of the primary and secondary

galaxies. In addition, the spectral emission features in

these maps are summed over the velocity channels

(integrated) to provide an estimate of the hydrogen mass for

each of the galaxies and a velocity field for the primary

galaxies. The exterior mass estimates are made from the

satellite velocity measurements and the position

measurements on the emission maps. The disk masses of the

primary galaxies are calculated from the rotation curve

solution to the velocity field the first moment of the

integrated flux.

The HI emission maps produced from the spectral

interferometer observations contain artifacts from the

synthesized beam response to the received signals. In

contrast, the sidelobes of the primary beam (the sidelobe

spacing of the primary beam is determined by the diameter of








73
the antennas, while the sidelobe spacing for the synthesized

beam is determined by the maximum projected spacings of the

antenna array and the u-v coverage) do not contribute

significantly to errors in the image plane unless a strong

source is located within several degrees of the field of

view. This is primarily due to the spatial incoherence in

signals received from beyond the HPBW of primary beam,

especially in observations which require wide bandwidths

(Bridle, 1985). The map images can be improved, or cleaned,

by removing much of the synthesized beam sidelobe features,

which include the irregular patterns produced with

incomplete u-v coverage. The cleaning process can also

interpolate between the measured u-v values. This appears as

a smoothing of the emission (or absorbtion) features in the

image plane.

The velocity of a satellite used in the dynamical mass

calculations can be estimated to better than the width of a

channel simply by inspecting the emission maps. However, the

systemic velocity for the satellite is best estimated using

the integrated flux and first moment (either the flux/beam

or temperature weighted velocity) calculations. To find

these integrated moments of flux and flux weighted velocity,

the images are summed in each pixel over all channels

(maps). The surface density of the HI emission in each pixel

is the summation over all maps exhibiting line emission, of

the brightness temperature at that pixel position, above a









specified minimum level. This level of significance is

normally chosen as three times the rms noise in the

continuum-free, cleaned maps. A lower limit to the hydrogen

mass can be calculated from the surface emission calculation

shown in equation 3.11. This calculation is made from the

integration, over each pixel position in the map, of the

surface density. Thus the hydrogen mass estimate is

expressed as


MH=1.82x108D2f f TBi dV d& (4-1)
nV


TBi is the brightness temperature, dV is the velocity

bandwidth of the images, D is the distance in Mpc. The

first (velocity) moment is calculated using the same

summation procedure as that used to calculate the total

flux. The temperature weighted (or flux/beam weighted if the

units are in flux/beam) mean velocity is expressed as the

summed product of the temperature at pixel i, times the

velocity of the map at the corresponding position, over the

sum of the temperatures. Thus, for a position i (which can

also be expressed as an x,y coordinate), the temperature

weighted velocity can be expressed as the first moment by


J ViTB dV
= -O (4-2)
S TB dV
~00









for brightness temperature units. In order to avoid

contributions due to noise, the rms noise value can serve as

a cut-off level. However, this cut-off technique includes a

number of unwanted biases. These biases and improved methods

of integrating the temperature weighted moments are

discussed by Bosma in his Ph.D. dissertation (1978).

The Bosma window method employs a calculated velocity

range for each pixel as well as a signifance level for the

emission feature. The velocity window is established by

iterating outwards from the channel in which the spectral

emission is the strongest and summing the emission spectra

which contribute an amount greater than a convergence

limit. The emission must also be above a specified minimum

to be considered significant, normally 3 a. The iterative

sum of emission for each individual pixel is stopped when

the addition of emission from channels farther from the peak

channel no longer contributes significantly to the sum.

Thus, the velocity window is calculated to be the range of

velocity channels that contains significant spectral

emission at the individual pixel position. The emission

which remains outside this window is considered continuum

emission and is not included in the spectral emission sum.

This continuum emission can be summed separately to provide

a continuum map. Tests made by Bosma for summing spectral

emission using a simple cut-off method, individual profile

fitting of spectra, and the window method, indicate that the







76

window method to be the most accurate summing procedure that

did not require excessive computing time. The selective

window method reduces the contribution of emission from

noise and excludes spikes which may appear in single

channels. A further reduction in noise and an increase in

the smoothing of emission features of the summed emision map

is made by smoothing in the velocity plane, as Hanning

smoothing. This procedure reduces the rms noise in each

emission map by approximately 12 by averaging adjacent

channels in the ratio of .5:1:.5, in effect, doubling the

bandwidth. Additional smoothing in the spatial plane, by

convolving each emission map with a specified Gaussian beam

function, produces smoother integrated emission maps, and

improves sensitivity in detection observations (England,

1986). Integrated moments were produced from these data

with software available from the NRAO VLA that did not

employ the Bosma window method. A second algorithm used on

the observed galaxy group data, based on the Bosma window

method, was developed by Gottesman (England, 1986), and was

found to be optimal for detecting weak emission features.

The integration scheme includes the Hanning velocity

smoothing, spatial smoothing with a Gaussian beam of twice

the clean beam dimensions, a significance level of 2 a for

cut-off, and a minimum number of channels in which channels

above a minimum rms occurs sequentially.









The integrated HI (surface density) and velocity

moments for each group, calculated with the NRAO integration

algorithm, are overlayed in alternating figures, from Figure

4.1 through 4.7. The velocity moments are plotted for each

primary galaxy after each overlayed moment, alternating in

Figures 4.2 to 4.8. The velocity contours for the overlayed

plots in increments of 20 km/sec for NGC 3893 and 40 km/sec

for NGC 4111, NGC 4258, and NGC 4303, which represents the

approximate velocity operation of the channels. The total HI

mass, calculated from the surface emission for each primary

galaxy, is given in Table 3.1.



Rotation Curves

The interior masses for the primary galaxies are

calculated from the rotational velocities at the last

(reliable) observed velocity point of each galaxy. This

interior, or disk, mass is a measure of the total mass

interior to the radius of the last velocity point, after the

velocities are corrected for the projection of the galaxy

onto the sky. The individual velocities and radial

separations are taken from the velocity field map and must

be corrected for the inclination projection in order to

produce the rotation curve. The velocities must also be

corrected for the recessional (systemic) velocity of the

primary galaxy by subtracting the systemic velocity. The

rotation curve can then be computed by averaging the











GREY: NGC3893
CONT: NGC3893
1 I 7


49 04




02


D
E
C 00
L

N
A
T 48 58
I
0
N

56




54




52


IPOL N380NLY.MOMO.1
IPOL N380NLY.MOM1.1


11 46 45 30 15 00 45 45
RIGHT ASCENSION
GREY SCALE FLUX RANGE- O.OOOOE+00 1.2993E+03 JY/B'M/S
PEAK CONTOUR FLUX 1.1667E+06 MIS
LEVS 1.0000E+05 ( 8.000, 8.200, 8.400,
8.600, 8.800, 9.000, 9.200, 9.400, 9.8600,
9.800, 10.00, 10.20, 10.40, 10.60, 10.80,
11.00, 11.20, 11.40, 11.60, 11.80)



Figure 4.1 Integrated zeroth and first moments
for NGC 3893 with increments to velocity contours
and grey scale in MKS. The cutoff level for
integration was 3 times rms noise.




















NGC3893 0.0
Ia ., I .......... M ... 1 Q t1 1 .......... .. .


L N



U,

z


C3


13.800 MINUTES OF ARC.


MAXIMUM CONTOUR I556

MINIMUM CONTOUR IS 0O

CINT = 20.600

XINT = 0.100








Figure 4.2 Velocity field for NGC 3893 displaying
the tidal assymmetry from UGC 6781. The maximum
velocity shown is the maximum contour level
times the velocity increment.













GREY: NGC4111
CONT: NGC4111


IPOL N41110NLY.MOMO.1
IPOL N41110NLY.MOM1.1


43 32


30 -


28-


24 --


221-


12 03 45 30 s15
RIGHT ASCENSION
GREY SCALE FLUX RANGE- O.O000E+00 9.2855E+
PEAK CONTOUR FLUX 8.5969E+05 M/S
LEVS 1.0000E+05 ( 6.000, 6.400, 6.800,
7.200, 7.600, 8.000, 8.400, 8.800, 9.200,
9.600)


02 JY/B*M/S


Figure 4.3 The overlayed HI and velocity moments
for the primary galaxy UGC 7089 with 3 sigma cutoff.












NGC41 11 0.0


. .







Kl


0^


r


9.600 MINUTES OF ARC.

MAXIMUM CONTOUR IS20
MINIMUM CONTOUR ISO
CINT = 41.400
XINT = 0.100




Figure 4.4 Velocity field for the primary galaxy
UGC 7089 integrated at 3 sigma.


I I I "I II I














GREY: NGC4258
CONT: NGC4258


IPOL N4258GRP.MOMO.4
IPOL N4258GRP.MOM1 .4


I I I .1- :-[


IN ~ Ar







-.7 ~


35 -


30- -


25 -


17 30 00 168 30 00 156 30 00
RIGHT ASCENSION
GREY SCALE FLUX RANGE- -9.2443E+01 6.3637E+03 JY/B*M/S
PEAK CONTOUR FLUX 7.4006E+05 M/S
LEVS 1.0000E+04 ( 5.000, 9.000, 13.00,
17.00, 21.00, 25.00, 29.00, 33.00, 37.00,
41.00, 48.00, 49.00, 83.00, 87.00, 61.00,
65.00, 69.00, 73.00, 77.00, 81.00, 85.00)


Figure 4.5 Integrated HI plot and velocity
field for NGC 4258 and the satellite NGC 4248.


47 50


451-


40












NGC4258 0.0
usumiginu inuntimil linrtimilinilumimillunf~ilisstnliigit~ntriil~rllutl1tI'M 0 1111110pu 111-1 ti "Mitni 1al 1,1n1al Ii M IMEn 1illfli1it...l.. uilitshin


25.800 MINUTES OF ARC.


MAXIMUM

MINIMUM

CINT =
XINT x


CONTOUR 15 15

CONTOUR IS 0

41.500

0.100


Figure 4.6 Velocity field for NGC 4258
integrated at the 3 sigma level.












GREY: NGC4303
CONT: NGC4303


04 55


IPOL N43O3ONLY.MOMO.2
I POL N43O3ONLY.MOM1 .2


12 20 00 19 45 30 15 00
RIGHT ASCENSION
GREY SCALE FLUX RANGE- -1.3330E+01 3.8490E+03 JY/B*M/S
PEAK CONTOUR FLUX 1.6443E+06 M/S
LEVS 1.0000E+04 ( 132.0, 136.0, 140.0,
144.0, 148.0, 152.0, 156.0, 180.0. 184.0.
188.0. 172.0, 178.0, 180.0, 184.0)





Figure 4.7 Integrated HI and velocity
field for NGC 4303 and the satellite
NGC 4303a.



















NGC4303 0.0
............. ............. Ini ...................... R isnki ni u m m ne umu ..


21.300 MINUTES OF ARC.


MAXIMUM
MINIMUM

CINT =

XINT =


CONTOUR IS 39

CONTOUR ISO
41.300
0.100


Figure 4.8 Velocity field for NGC 4303
integrated at the 3 sigma level.







86

corrected velocities over increments in radial distance from

the galaxy center. The disk mass can, in turn be found from

the rotation curve. The disk mass is expressed as

(Lequeux,1983)


MdiskV(Rmax)2Rmax/G. (4-4)


with Rmax the maximum radius of separation. Lequex points

out that this disk mass is actually a measure of the

combined disk structure mass, the mass of the nucleus, and

any halo mass which may be interior to Rmax This makes the

"disk" mass an ideal measure of the interior mass of the

galaxy for this mass distribution study, because all of the

mass components are measured.

The procedure for calculating the rotation curve from

the galaxy velocity field requires a simultaneous fit of all

the velocity and radial separation values to assumed

projection parameters. To accomplish this, a fit of both

the observed and the model velocities (for a given rotation

model) is accomplished using an iterative least squares

procedure. The linearized expression for the rotation law,

shown by equation (4-7), is employed in the least squares

minimization. The projection parameterers, maximum

rotational velocity, and model rotation parameters are

adjusted by the least squares solution in order for the

observed velocity data to fit the model rotation curve. The

parameter fit is halted when the corrections are below a

convergence limit.







87
The rotation curve of the primary galaxy requires the

de-projected velocities of the observed velocity field.

However, the projected velocities near the minor axis of the

galaxy have a small observed radial velocity, and may be

dominated by non-circular or random velocities. The region

near the minor axis should therefore be omitted from the

velocity averaging, if these irregular velocities are to be

avoided. This can be accomplished using cosine weighting

(weighting velocities with the cosine of the angle from the

major axis) or by simply using the velocities within a

specified angular separation from the major axis.

The accurate calculation of a rotation curve also

requires an accurate position for the center of rotation of

the galaxy. A list of accurate optical positions for the

centers of larger galaxies is available from several authors

(Gallouet et al., 1973; Dressel and Condon, 1976). The

optical positions of Dressel and Condon were used to find

the centers for the four primary galaxies observed in this

study. Efforts to establish more accurately the center of

the velocity field (more accurate than the 4" rms error

quoted for the Dressel and Condon positions), using

dynamical center calculations which are based on rotational

symmetry, had limited success owing to the assymetrical

structure and the high inclination of three of the four

primary galaxes, and were abandoned.







88

Calculating the de-projected rotation curve solution

for the primary galaxy first requires known or estimated

projection parameters, which describe the orientation of the

galaxy in the plane of the sky. The parameters describing an

arbitrary position within a galaxy, in both the plane of the

sky and the plane of the galaxy, are shown in figure 4.9.

The coordinate references for angles measured in the plane

of the galaxy are 1) from the positive Y axis, counter-

clockwise for position and 2) from the positive Y axis

counter-clockwise to the positive velocity reference of the

position angle of the galaxy. If the angle p represents the

angle from the major axis (M in Figure 4.9) of the orbit in

the plane of the galaxy, to the position reference point at

a scalar distance R, the corresponding position angle to the

reference point in the plane of the sky will be v. The

scalar distance form the center of the galaxy to the same

reference point in the plane of the sky, r, is


r=Rco ip/cosy. (4-5)


The relationship between the position angles in the two

planes is then


tan(p=tanycosi (4-6)


where inclination angle i is measured between +0 and +90

degrees.















































Figure 4.9 Position parameters within an inclined
galaxy. The major axis is parallel to the line from
the focal point to position M. The projected major
axis passes through the focal point and position
M' .







90
The circular rotational velocity is described in the

plane of the sky as


V=Vsys+ V(r) cosp sini. (4-7)


V(r) is the rotational velocity at distance r from the

galaxy center and Vsys is the systemic velocity of the

galaxy.

The solution for the yet undetermined projection

parameters of the rotation law is produced from a least

squares fit to these variable parameters, i, P (the position

angle of the line of nodes), Vsys, V(r), and in this

analysis, b, the scale length of the assumed model rotation

curve, which is the Toomre n=0 model. This model rotation

velocity expression is (Hunter et al., 1984)


V(r)=Vmax 1- r2 (4-8)
Nr -+b


Vmax is the maximum rotational velocity, more often referred

to as C for the Toomre models.

The non-linear rotational velocity equation (4-7)

requires a linear expansion for input into the first order

least squares minimization routine. A first order expansion

in the Taylor derivatives of this velocity equation can be

used with a few caveats. First, the requirements for

convergence of a least squares solution to the parameter

variables may not be satisfied by the observed velocity







91
data, as cautioned by Jefferies (Jefferies, 1980), Eichhorn

(Eichhorn and Clary, 1973), and others. More specifically, a

large variance (noise) in the velocity field data may

produce residuals in the least squares fitting that are of

the same order as the adjusted parameters of the model, in

this case the five rotation parameters, i, P, Vmax, Vsys,

and b. A second order expansion may be necessary for a more

accurate solution, or in some cases, for a convergent

solution at all. Second, correlated variables should be

treated with a covariance expression which separates the

related residuals if a more rigorous solution is expected.

The equation of observed rotational velocity, (4-7), shows

an explicit example of correlated variables, the

interdependence of the circular velocity component (V -

Vsys) on both the inclination angle and the maximum

rotational velocity, Vmax The effects of the inclination

variations in a model rotation field are inseparable from

the maximum rotational velocity variations of the same

model, except for any elliptical projection of the circular

disk structure in a spiral galaxy. This may be the primary

reason that larger non-circular velocities produce higher

inclination angle solutions in least squares tests of

simulated galactic rotation fields, especially for shallow

inclinations. The third cautionary note concerns the

preliminary estimates for the input model parameters. These

should be reasonably close to the actual parameter values if






92

convergence is expected without more a detailed algorithm

which corrects for large adjustments to the residuals.

For each position in the velocity field, the observed

velocities are subtracted from the linearized model of the

rotation law at the same distance from the galaxy center and

then minimized for an optimum fit of the model to the data.

For a first order expression of the model rotation law, the

total derivative of the rotational velocity model, equation

(4-7), is used. This first order derivative is




dV= V/V SysdVys + V/aV dVma + 8V/8pVo sinp sini di


+ 3V/(p Vo cosi cosp dP + WV/Ob cosp sini db


=dV + V/V dV -V sinp cosp tanidi
sys max max (cosy cosi)


2
+ V sing cosP dP V P db. (4-9)
(cosy cosi) V(r)(p2+b2)3/2


The adjustments to the parameters used for de-projection and

for describing the rotation curve, i.e. di, dP, dVsys,

dVmax, and db, are made from the changes in the least

squares fit of the rotation model with the observed

velocities (as projected by the adjusted parameters).

Mimimizing the differences of the observed velocity and

the first order model velocity shown above, is accomplished

by the least squares routine, which returns the adjustments







93

to the input variables. The actual fitting algorithm used

was developed by Howard Cohen of the University of Florida

(private communication), which is based on the method of

Banachwiewicz (1942).

The resulting projection parameters are used to

calculate the de-projected velocities in the galaxy field

and the averages of the observed velocities for the given

velocity field over a radial interval. This interval must be

chosen large enough to avoid reducing the resolution of the

velocity field unless such a reduction is desired. These

averaged intervals are the values that define the velocity

curve, V(r). The errors calculated by the least squares

procedure are standard deviations in the input variables,

which, in turn, can be used to estimate the uncertainty in

the calculated rotation curve. Uncertainty in the rotation

curve solution can also be calculated with the variance of

the velocity field in each of the radial increments that are

averaged. This is a more direct procedure for calculating

the rotation curve uncertainty. However, the deviations in

the individual adjusted parameters give a better figure of

uncertainty of the overall rotation curve and also for the

uncertainty in the linear and angular variables.

Tests with the first order least squares fitting

routine using synthesized velocity fields show a strong

dependence of the minimum inclination angle on either non-

circular or random noise velocities in the velocity field,







94
although the product of the maximum rotational velocity and

the sine of the inclination angle remains constant. A number

of tests were performed with random velocities or non-

symmetrical motion to the velocities or as an offset in the

actual position in the velocity field center. Sample tests

show solutions for shallow inclination angles in a velocity

field of approximately 50 pixel diameter whose center is

offset by 2 pixels, will increase the inclination angle

solution by 10 to 15 degrees. Position angle and systemic

velocity are not appreciably affected by an offset in the

center, although the solution to the scale length b of the

Toomre n=0 model shows increasing variations with increasing

random noise or with center position off-set. Similar tests

on synthesized velocity fields with an added random

(uniformly distributed, not Gaussian) velocity component

increase the inclination angle solution for a shallow

velocity field by the ratio of the non-circular velocity to

the maximum rotational velocity to a maximum inclination of

approxiamtely 50. The inclination values listed in Table

3.2 are assumed to be upper limits because of the noise

dependence of the inclination angle solution.

The rotation curves and the results from the least

squares fitting procedure to the four primary galaxies are

shown in Figures 4.10 to 4.13. The calculated disk mass for

each of the primary galaxies is taken from each of these de-

projected rotation curves at the most distant point with a







95
reasonable signal to noise ratio for the rotational

velocity. The linear scale of each rotation plot is in

angular units of arc minutes. The adopted distance of each

of the galaxy groups is shown in Table 3.2.




Full Text
MASS DETERMINATION OF SELECTED GALAXIES FROM SMALL GROUP
STATISTICS
BY
LANCE KARL ERICKSON
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
1987

TABLE OF CONTENTS
PAGE
ABSTRACT iv
CHAPTER
I. INTRODUCTION 1
II. MASS ESTIMATE METHODS 7
Binary Systems 7
Small Groups 13
Group Selection 18
III. OBSERVATIONS 24
Single Dish and Interferometer Measurements . . 24
NGC 3893 35
UGC 7089 44
NGC 4258 52
NGC 4303 60
IV. DATA REDUCTION 72
Integrated Moments 72
Rotation Curves 77
V. SUPPLEMENTAL GROUPS 100
NGC 224 (M31) 100
NGC 1023 101
NGC 1961 102
NGC 3359 102
NGC 3992 103
NGC 4731 103
NGC 5084 104
Satellite Characteristics 104
VI. MASS DISTRIBUTIONS 109
Chi Distribution 109
Selection Bias 113
ii

VII. DISCUSSION
123
Halo Model 123
Simulations 125
Membership 131
VIII. CONCLUSIONS 137
APPENDIX
INTERFEROMETRY AND IMAGING 140
BIBLIOGRAPHY 156
BIOGRAPHICAL SKETCH 159
*6
*
tf-
k
U
ñ
i*
li
iii

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
MASS DETERMINATION OF SELECTED GALAXIES FROM SMALL GROUP
STATISTICS
By
Lance Karl Erickson
August 1987
Chairman: S. T. Gottesman
Cochairman: J. H. Hunter, Jr.
Major Department: Astronomy
The masses of carefully selected galaxies are measured
with both rotation curves and by orbiting dwarf satellites.
A comparison of the interior and the exterior masses of the
observed primary galaxies provides an estimate of the dark
matter, if any, in the surrounding regions.
A comparison is also made between the results of
similar studies which used binary galaxies and the small,
compact groups used in this investigation. A comparison of
the two distributuions of exterior to interior galaxy mass
ratios indicates that the these distributions are not the
same.
IV

Conclusions based on the n-body simulations of halo
mass distributions limit the dark matter to approximately
four times the group primary mass and 4 times the primary
disk radius. The measured mass ratio values which exceed the
model limit of 2 appear to be limited to the most massive or
peculiar spiral and barred spiral galaxies.
v

CHAPTER I
INTRODUCTION
The rotation curve of a spiral galaxy provides a
reliable means of measuring the gravitational mass interior
to the region for which it was observed. Rotation curves
often show rigid body rotation near the central bulges of
spiral galaxies, describing the gravitational influence of
the central masses. The luminous material that can make up a
significant part of the central mass of the galaxy often
exhibits a rapid decrease near the outer edge of the
nucleus. If this rapid decrease in luminosity were to imply
a corresponding decrease in the mass density of the nucleus,
the rotational velocity of the disk should show a
corresponding decrease outside the luminous bulge. However,
these decreasing rotation curves are not common in spiral
and barred spiral galaxies. The most common shape of the
rotation curves for spiral galaxies decrease more slowly
towards the observable edge of the galaxy, or perhaps show
no decrease at all. It is argued that this implies a
gravitational influence from non-luminous, or dark matter,
beyond the measurable disk of the galaxy, in the form of a
massive halo (Faber and Gallagher, 1979).
1

2
Measurements to verify the existence of dark matter in
the surrounding region of a galaxy have been attempted by
extending the mass measurements of these galaxies beyond the
boundaries of the observed rotation curve. However, the
techniques devised to measure the exterior masses of
galaxies do not lead to consistent results. Such methods
range from binary galaxy analysis to galaxy cluster mass
estimates based on the Virial Theorem. The evidence for non-
luminous mass associated with galaxy clusters is limited
(Valtonen and Byrd, 1986) and is not, in general, applicable
to single galaxies. Binary mass estimates, on the other
hand, suffer from uncertainties in the bound state of the
orbits due, in part, to incomplete information on the true
separations and velocities of the members. The need to find
an improved technique to investigate the distribution of
suspected dark matter in galaxies is underscored by the
question of mass distribution on a much larger scale.
Associated with this is the need to find a cause for the
flat rotation curves of most spiral type galaxies. The
purpose of this dissertation is to develop a more reliable
technique for the study of mass distributions in selected
galaxies.
Galaxy mass estimates began with the use of optical
rotation curves for spiral galaxies and with the use of
velocity dispersion measurements for elliptical galaxies.
The flat rotation curves that were found as the measurements

3
became more precise led researchers to investigate the
region surrounding these galaxies for evidence of non-
luminous mass. Preliminary mass estimates of these exterior
regions came from earlier studies of binary galaxies that
were based on methods developed for determining binary star
orbits. Early work by Page (1951) provided the background
for a statistical assessment of selected galactic binary
systems. This statistical approach was necessary because
orbital parameters cannot be determined from a single
projected separation and velocity. More detailed analyses
were later carried out by Turner (1976), Peterson (1979),
and others. Gustaaf van Moorsel (1982), in his Ph.D.
dissertation, developed a very useful method for determining
the distribution of mass based on the projected separation
and velocity (squared) ratios. These projected mass
calculations provide a more reliable estimate than do Virial
Theorem calculations as Bahcall and Tremaine (1981) show.
This is due primarily to the weighting given small
separations between members by the virial method. The
expression used for the virial mass calculation is
M oc
V
2
<1/R>
(1-1)
while that used for the projected mass calculation is
M oc V2R.
(1-2)

4
The inverse separation expresión for the virial calculation
of equation (1-1) can result in inconsistent or non-
convergent mass estimates, or a large variance in the
calculated masses.
The van Moorsel mass ratio expression is also useful
because it expresses the binary orbital projection
parameters in combined form. This projection function can be
modeled with random orbital elements and compared with a
selection of binary galaxies assumed to be random. A
statistical assessment of the selected pairs can then be
made directly from these two distributions. This method is
the primary tool used herein for estimating the mass
distributions in the selected galaxies.
The main body of this work involves the determination
of the mass distribution in selected spiral and barred
spiral galaxies. The binary galaxy techniques of van Moorsel
and others furnish the basis for the method used in
analyzing small, compact groups of galaxies. These small
groups are selected to provide satellite mass probes in the
volume surrounding the dominant galaxy. Each group has 1)
from one to four satellites, 2) satellite masses less than 5
% of the primary galaxy, 3) a separation limit of 30' from
the primary galaxy to the satellite. These criteria should
reduce the mass estimate uncertainties associated with the
previous binary studies. The galaxy group selection process
and the resulting list will be discussed in order to

5
establish procedures for analyzing selection biases and
their possible effects.
The details of the mass estimate method used in this
research are presented in the second chapter, while the
neutral hydrogen observations of the selected groups are
covered in Chapter III. These HI observations provide the
dynamical parameters leading to mass estimates for each of
the primary galaxies. The HI observations of the four galaxy
groups will be supplemented with observations of other
galactic groups obtained by other researchers. These
additional groups are discussed in Chapter V. Data reduction
procedures are also outlined in the Appendix which contains
a discussion of the techniques and instrumentation used for
the interferometric measurements.
The relatively small number of galaxy mass ratios used
in this study requires the use of statistical tests for
final estimates of the mass distribution. Statistical tests
are also needed for measuring the level of confidence in the
analysis. Chapter VI includes a discussion of the
statistical tests used and the arguments for using those
statistics, since the conclusions are based primarily on the
statistical interpretation of the data. Another statistical
assessment is the measurement of the likelihood of bound
membership. Valtonen and Byrd (1986) used an argument based
on the expected symmetry of redshifts in samples of binary
galaxies, groups of galaxies, and clusters of galaxies.

6
These are, of course, assumed to be gravitationally bound
systems. This same argument is used in Chapter VII to
assess the bound membership of the groups in this study, and
the membership in the van Moorsel sample.
A final discussion of the foregoing arguments and
results are also presented in Chapter VII. Comparisons are
made between the results of this work and the work of
previous researchers, even though the selected galaxies may
have dissimilar characteristics.

CHAPTER II
MASS ESTIMATE METHODS
Binary Systems
The technique used to measure the mass of a binary
galaxy is essentially the same as that used to measure the
mass of a stellar binary system. Elliptical binary orbits
describe the same physical relations of a bound system
whether for a planetary, stellar, or galactic system. For
binary systems in general, the plane of the orbit is
inclined to the plane of the sky, resulting in the orbital
elements being only partly determined by observation. In
addition, separation and relative radial velocity are the
only measurable quantities for binary galaxy orbits. Due to
the incomplete information for the orbital parameters, the
solution of any single orbit for a binary galaxy is not
possible. However, separation and velocity measurements of a
number of galaxy pairs can be used to infer a distribution
of mass for the entire set of binary galaxies. The indirect
nature required of this method was recognized, even in the
earliest studies of binary galaxy masses by Page (1951).
Improvements were made to this mass distribution technique
in order to reduce the uncertainties in binary membership
and to reduce the effects of selection bias. The development
7

8
of the separate mass distribution and orbit model
distribution by Gustav van Moorsel improved the binary mass
estimate method even more, with the ability to analyze the
biases in the distributions separately.
As shown by van Moorsel, the measurements of separation
and relative velocity for a random set of binary galaxies
can be compared to a model of random orbital values, because
both contain the same projection parameters. This comparison
is straightforward since the equation containing the
combined orbital parameters is separable from the measured
radial velocities and projected separations. Thus, to within
a constant, a random set of orbital projections should
produce the same distribution of values as a random set of
binary separations and velocities. The random orientations
of the samples for these galaxy pairs is necessary to avoid
biases in orientation, but cannot be guaranteed for the 23
measured samples used in this study. Therefore, these
observed samples must be checked for the effects of sampling
bias, as discussed in Chapter VI.
The separation and velocity measurements of the galaxy
pairs are used to produce the observed distribution of mass
values. To avoid the problems inherent in virial mass
estimates which express the member separations in an inverse
relation, Bahcall and Tremaine (1981), as well as van
Moorsel, use a mass formula based linearly on separation.
For a binary system this can be written as

9
M = a AV2 R/G.
(2-1)
Here, a is a coefficient which reflects the mass dependence
upon the galaxy potential or the orbit projection, or both.
The difference in radial velocity betwen the two components
is AV and R is the projected separation of these components.
In order to modify this equation to include projection
effects due to the orientation of the orbit, we express the
measured separation and velocity in terms of the orbital
parameters and the true separation and velocity. If this
corrected orbital mass is then divided by the disk mass, a
ratio is produced for the exterior and interior masses
(multiplied by the coefficient of the combined orbital
parameters). The interior mass inferred from a rotation
curve is (Lequeux,1983)
(2-2)
max
where is the maximum true distance from the center of
the galaxy for which a reliable rotational velocity is
available, and V(Rmax) is the rotational velocity at .
The existence of dark matter can be determined by comparing
the number of observed mass ratios versus the value of the
mass ratio obtained (the observed distribution) with the
model distribution of the orbital elements; similar
distributions imply similar masses in the exterior and
interior regions of the galaxies under study.

10
A schematic representation of a binary galaxy system
and the corresponding orbital parameters is shown in Figure
2.1. The inclination angle, position angle of the line of
nodes, true anomaly, and semi-major axis have the same
convention as in stellar binary notation (see for example,
Aitken, 1935). The combined expression for the orbit
parameters will be referred to as x (chi), from the notation
of van Moorsel. His formulation of x follows from the
projected separation, r, and the velocity difference of the
components, Vr. The projected separation is
(2-3)
where R is the true separation, v is the true anomaly, and o
is the position angle of the line of nodes.
The observed (projected) velocity difference between
the two bodies is then
(2-4)
Here, 0 is the sum of v and co and e is the orbital
eccentricity.
The orbital (exterior) mass which now includes the
projected variables can be written as
...2 . 2.
AV r sin i
G(1+ecosv)
(2-5)

11
Figure 2.1 The orbital elements of an inclined binary
galaxy orbit.

12
Dividing the orbital mass by the interior (disk) mass
produces a distance independent expression of the exterior
to interior mass ratio multiplied by the x function. This
mass ratio is related to the observed mass ratio, X0£S* as
Xobs__(^T+m2TX (2~6)
where x represents the combined parameters in projection,
2 .
—sin 1—(cosv + ecosO)2 (l-sin2i sin26)1/2. (2-7)
K (1+ecosv)
If the galaxy pairs selected for this investigation are
representative of a reasonably random set of projected
orientations, the observed number distribution, ^tx^g)/
should resemble the model number distribution, N(x), times
the exterior to interior mass ratio. Further, if these two
distributions are similar, the masses should be similar.
However, interpretations of any significant differences in
these two distributions are dependent on the assumptions
made about the physical system. Since these distributions
are intended to be used for dark mass estimates of the
galaxy systems studied, the assumption of bound orbits must
be made with reasonable confidence. A discussion of the
bound membership in these small groups is found in Chapter
VII.

13
The analytic comparison of the model and observed
distribututions is made using a "goodness of fit" test (see
Chapter VI), although a quick comparison can be done
visually. From Figure 2.2, one can see that the shape of the
model distribution is dependent on the orbital eccentricity,
or eccentricities, chosen. These model distributions,
however, have a bias towards small values of x owing to the
preponderence of orientations with large projection effects.
As seen from equation (2-7) and Figure 2.1, the values of %
range from 0 to 1+e. The cutoff in the distribution at e=0
for circular orbits is seen in this figure for x greater
than 1. The characteristic shape of the chi distribution at
a relatively high eccentricity (in this figure a maximum of
e=0.9 is shown) shows a gradual decrease with inceasing
values of x- A random e squared distribution is included
because of the e squared dependence of x0jjS on kinetic
energy, which includes the angular momentum squared. The
o
slightly elevated plateau in the e distribution is
displayed if Figure 2.2 near the circular orbit maximum of
X=1 •
Small Groups
Mass estimates using carefully selected small groups of
galaxies have several advantages over the methods used for
binary galaxies as described above. The greatest advantage
for the small group analysis is the large mass ratio of the

14
X
Figure 2.2 The random orientation distribution for three
eccentricities normalized to the same area.

15
primary to the secondary galaxies; this allows the dwarf
satellites to be treated as test particles. In addition,
each of the satellite-primary pairs can be considered as a
single binary system, and hence, the orbital analysis
developed for binary systems can be used for each of the
pairs within each group. An example of a group of five
satellites bound to a massive primary galaxy would produce
equivalent x0bs values of five individual binary pairs.
Although the absolute velocity difference (squared) between
the two objects may be the same for both the binary and the
primary-satellite cases, the radial velocity components are
not, in general, the same (although x has a maximum of 1+e
for bound members). This is due to the motion of members of
the true pair about the barycenter. The advantages of using
a group dominated by one galaxy for the analysis include
simplified dynamics and the contribution of several
values for a single observation of a galaxy.
The expression for x is the same for the primary-
satellite pair as it is for the true binary because of the
negligible mass of the satellite galaxies. This is found in
the observed mass ratios expressions for the following three
cases.
M
7-obs ml+m2X
binary
(2-8)
singly dominated group (2-9)

16
= — Xm equivalent mass group. (2-10)
Im^ w
The x^ distribution must be determined numerically since an
analytic expression is not available for more than two
bodies (see also section 6.2). A derivation for the x^j
equation for the equivalent mass group (a group in which the
masses are approximately the same) is found in van Moorsel
(1982).
The interior masses, represented by the denominator of
the binary and dominant group expressions in equations (2-8)
and (2-10), are essentially the same, since the primary
galaxy mass, m, approximates the total mass of the group. It
is important to note that if the interior mass sum appearing
in the denominator of the binary expression of equation
(2-8) is actually the sum of masses of a larger number of
members as shown in equation (2-10) (a group of interacting
galaxies for example), the resulting value for X^g can be
larger than is possible for the binary system. Such an
underestimate in the interior mass of a group that is
assumed to be a pair (due possibly to membership in a higher
order group) should therefore be avoided.
The most important difference in the binary versus
small group comparison concerns the gravitational potential
field associated with the masses. Binary galaxies with
relatively small total mass would exhibit greater dynamical

17
effect from the gravitational masses of neighboring galaxies
than would more massive binary systems. The larger
gravitational potential of a primary galaxy would also
provide more influence over an orbiting body than would two
bound galaxies with a mass comparable to the neighboring
galaxies. An analogous case of three bound members is shown
by the stability calculation for the restricted three body
problem. If we take the test mass to be appreciably less
than the other two orbiting masses (m^>>m3< stability of the orbit for the test mass m3 is proven for
the inequality (Symon, 1975)
2
(mi + n^ + m^) > 27(m1m2 + m1m3 + m2m3) (2-11)
or
m^ > 24.96m2.
Although this example is valid for restrictions which cannot
be generalized for the small groups discussed herein, the
influence of the more massive primary galaxy is evident from
the inequality in equation (2-11). Isolated systems of
galaxies or systems with large mass potentials, such as the
compact, small groups, should improve the liklihood of bound
membership.
The requirement that group members be physically bound
is important for the mass estimate methods employed in this
investigation, just as it was in the binary studies. The

18
resulting observed and model distributions are dependent on
the validity of the assumed bound state of the system. The
selection criteria for the galaxy groups to be studied
reflect the efforts to help ensure accurate calculations of
the mass distribution and avoid interlopers or objects in
hyperbolic orbits.
Group Selection
The selection process is intended to find the small,
isolated galaxy groups which are described in the previous
section. These compact groups of galaxies include spiral
galaxies as primary members, in accordance with the binary
studies of other investigators. There are, however,
important characteristic differences between the galaxies
selected for this group study and the galaxies selected for
the binary studies, the most obvious is the dominant nature
of the primary galaxy. The desired characteristics of the
galaxy group are first specified in order to establish the
selection criteria. The defined selection parameters will
then allow an automated search through the Uppsala Catalog
(NiIson,1971) for the galaxy group members. The major
selection requirements for this group study are listed
below.
Dominance. In order to help ensure bound systems and
to select small groups with a single dominating mass, a
diameter and magnitude difference was chosen as follows: 1)

19
The primary galaxy must be 2' in diameter and larger than
any of the suspected satellites and 1' larger than any other
galaxy within 2 degrees. 2) The magnitude difference between
the primary and satellites must be 2m or greater and any
galaxy within 2 degrees must be greater than the primary
magnitude by lm or more. 3) The primary galaxy must be
equal to or brighter than magnitude 13, as listed in the
Uppsala Catalog. 4) A search of the Palomar Sky Survey
prints in a diameter of approximately 5 degrees surrounding
the primary galaxy should confirm the isolation of the group
and the dominance of the primary galaxy.
Groups which passed the selection process but may be
members of higher order groups or are listed as members of
clusters were avoided because of the complications of
interacting galaxies. Also, groups with more than ten
members within one degree of the primary galaxy were
normally rejected because of the uncertain dominance of any
single member, and because of the possibility that the
system may be composed of several interacting groups. A
lower limit of two members per group was chosen since the HI
observations may detect dwarf satellites in the region of
the primary (Gottesman et al., 1984). This single pair
criterion did not increase the final list of selected
groups.
These criteria will not guarantee the gravitational
dominance of the primary galaxy, but should provide

20
sufficient margin so that the dwarf satellites represent
only 5 - 10% of the total mass of the group. If we assume
(conservatively) constant MT /Lg ratios for the primary and
satellite galaxies, the difference of two magnitudes would
provide a mass ratio greater than 6:1. Mass calculations
using the global spectrum of the HI (Casertano and Shostak,
1980) of the satellites, having spectra appearing in more
than one channel, indicated one satellite with approximately
21% of the mass of the primary. This was for the group UGC
7089, in which the dwarf satellite mass for UGC 7094 was
calculated from the global profile (Casertano and
Shostak,1980). The remaining measured dwarf masses were of
the order of 1.1% with a total average of 5.3%.
Separation. HI observations of the selected groups
were made at the NRAO VLA radio telescope in Socorro, New
Mexico. The limitations of the radio telescope which
affected the selection process were the sensitivity of the
system, the field of view, and the bandwidth of the
receiver-correlator system. At 21 cm, the field of view, or
primary beam diameter, is approximately 31'. Therefore, the
galactic groups were selected with an approximate maximum
separation of 30' for all members because of the generally
low HI mass and resulting low HI emission levels of dwarf
galaxies. Overlapping HI observations could be used to
expand the field of view, but would either reduce the
integration time on each observed field by the inverse of

21
the number of observed fields or require much more observing
time. The separation criteria for these members also include
a minimum separation value of one primary galaxy diameter.
This was established in order to avoid tidal interactions
and the obvious orbital complications that could result from
a closely interacting pair. Two such interacting systems,
M31-NGC 221 and NGC 3893-UGC 6781, are found in this study
but not included in the analysis. Seperate statistical tests
are also made for the M31-NGC 221 system because of the
possible tidal interaction.
Velocity. Velocity differences between the primary and
satellite galaxies were an important selection parameter
even though optical and HI velocities were only available
for approximately one third of the satellites. A large
velocity difference in the members of a particular group
could indicate the presence of an interloper or unbound
member. A velocity difference of at least 1000 km/sec for a
suspected satellite was indicative of an (unbound) optical
member, which should be omitted. The required mass for
binding members at such a large velocity difference is
greater even than that which is considered a very massive
galaxy. Turner (1976) found a useful maximum value of 500
km/sec for membereship in binary pairs, while van Moorsel
found a velocity difference of 600 km/sec to be a maximum
value for physically associated pairs. Based on these
results and the Peterson estimate of 750 km/sec a maximum

22
difference of 600 km/sec was established to minimize the
possibility of including interlopers or unbound members.
The bandwidth restriction for the receiver system
limits the number of velocity channels available and the
velocity width of each channel. In order to cover a velocity
span of approximately ± 600 km/sec, 31 channels of 41
km/sec each were selected for these observations. Two
exceptions were made to this bandwidth choice, however.
Since velocity information was not available for one of the
suspected NGC 4303 satellites, and the maximum velocity
difference between two of the satellites was approximately
1000 km/sec, the total number of channels was increased to
64. The second exception, NGC 3893, has optical velocities
available for all of the likely satellites, with a maximum
velocity difference between primary and satellite of less
than 100 km/sec. Because of these relatively small velocity
differences, a channel bandwidth of 21 km/sec was chosen to
increase velocity resolution.
HI flux. The estimated HI flux of each primary galaxy
must be large enough to be detected easily within the time
allocated for observation. If detection of the primary
galaxy is desired within one hour, and the rms noise for 25
antennas at the 21 cm band is approximately 1.8 mJy/beam, a
3:1 signal to noise specification gives a limit of 5.4
mJy/beam for detection. Therefore, 10 mJy/beam was
established as the minimum detectable flux per beam suitable
for these HI observations.

23
To summarize the selection procedure, the primary
galaxy is first chosen according to specified morphological
types; the allowed types were SO, all spirals and barred
spirals, and blank entries in the Uppsala Catalog. No
primary galaxies which were listed as blank types satisfied
the remaining selection criteria, however. The primary and
possible satellites were then tested for group dominance
according to the following criteria:
1. The primary magnitude must be 13 or brighter, and 2
magnitudes brighter than the satellites, and 1
magnitude brighter than galaxies within 2 degrees.
2. The blue diameter must be 3.5' or larger, at least 2'
larger than the satellites, and at least 1' larger
than galaxies within 2 degrees.
3. The satellites must be arranged so that the group
members are situated within the 31' field of view and
no satellite can be less than one primary diameter
distant from the primary galaxy.
4. The velocity difference between the satellites and
the primary must be less than 600 km/sec.
5. The estimated flux for the primary galaxy must be 10
mJy or larger for the 43 km/sec channel width.
In addition, visual search of the group neighborhood (~5°)
should not reveal nearby galaxies that are of the same order
of size as the primary galaxy.

CHAPTER III
OBSERVATIONS
Single Dish and Interferometer Measurements
HI is the atomic species of neutral hydrogen often used
to obtain spectral line maps of galaxies. It is a polar atom
with spin-spin interaction between the electron and the
proton which separates the ground state energy level. The
pervasive character of HI in the interstellar media of
spiral and barred spiral galaxies makes it ideal for
measuring the structural features of these galaxies. The
atomic form of hydrogen is of particular interest because
the dipole radiation produced by the neutral, atomic
hydrogen is more easily detected than the weaker quadrapole
radiation from the molecular species (Jackson,1975).
HI emission occurs as the spin state changes from
parallel to anti-parallel alignment of the electron and
proton (F=l-*0 transition). Collisional excitation is the
predominant mechanism for exciting the hydrogen atom in most
of the interstellar medium. This excitation in turn produces
an equilibrium distribution of energy states. If the atom is
given sufficient energy to reverse the spin alignment of one
of the particles, the parallel spins will create a higher
energy state, equivalent to the magnetic interaction energy
24

25
between the particles. For neutral hydrogen this difference
is 5.8754x10”^ ev, or 21.3893 cm (1.42041 GHz).
The spontaneous emission rate for 21 cm emission has an
Einstein transition probability coefficient, A21, of
2.85x10^^ sec-^ corresponding to a lifetime of 3.51x10^ sec
7
or 1.1x10' years. The comparatively long lifetime of this
14
transititon means that 3.5x10 HI atoms are required to
produce one emission per second without collisional
excitation. If collisions populate the upper levels, the
transition lifetime is reduced to approximately 400 years
(Verschuur, 1974), producing a spin temperature equivalent
to the thermal or kinetic temperature of the surrounding
gas. An approximate HI mass necessary to produce an HI flux
-29 -2 -1
level of one milliJansky (10 W m Hz ), which is
detected at a distance of 10 Mpc, using a 41 km/sec
bandwidth, is of the order of 1.0x10^ MQ (eq. 3-11).
Conversely, a one hour observation with 25 antennas of the
VLA interferometer at 21 cm, with a channel bandwidth of 41
km/sec, should detect an unresolved HI mass of approximately
5.4x10^ MQ at 10 Mpc. This mass represents a 3:1 signal to
noise ratio above the sensitivity of the receiver, where the
sensitivity is given by (Appendix, equation (A-10))
AS =
450mJy
VN(N^1)dvdt
(3-1)

26
for a natural weighted map. The flux coefficient represents
the system response at 21 cm. , N is the number of antennas
used, dv is the channel bandwidth in KHz, and dt is the
integration time in hours. A corresponding sensitivity in
brightness temperature units is
atb=as
(3-2)
For this conversion, k is Boltzmans constant, X is the
wavelength, and is the synthesized beam solid angle. The
brightness temperature expression assumes both a Rayleigh-
Jeans approximation for the wavelength dependence and an
unresolved emission source.
For these VLA spectral observations two independent
receiver systems of opposite polarization are combined and
then averaged with Hanning smoothing. This averaging
improves the rms sensitivity by approximately -J2, as it
would if the integration time were doubled. Because the rms
noise level is inversely proportional to the square root of
the integration time, an observation of 10 hours should
allow the detection of a 1.7x10“* MQ HI mass at 10 Mpc, with
a signal to noise ratio of 3:1, a bandwidth of 41 km/sec,
and with the source at the center of the primary beam.
The expression for brightness temperature which
describes the diffuse galactic HI emission detected in
spiral galaxies is the same as that used to describe the

27
emission temperature of of a diffuse gas cloud. In the more
general case, the cloud is considered to be illuminated by
another (continuum) emission source of temperature Tg. The
brightness temperature, TB of a cloud (spin) temperature Tc
and frequency v with an illuminating source of temperature
Tg, is then (Kraus, 1966)
TB (v) = Tgexp ( -tv) +Tc ( l-exp( -tv ) ) (3-3)
where iv is the optical depth of the cloud at frequency v.
For diffuse emission, the optical depth is less than
one, and the exponential in equation (3-3) can be expanded
and then approximated with the first two terms of the
expansion. Since the optical depth is relatively constant
over the frequencies used for these observations, the
frequency dependence can be removed, hence
exp(-t) = 1-T, (t<<1 ) (3-4)
and the brightness temperature becomes
Tb = Ts(1-t) + tTc. (3-5)
The brightness temperature of the observed galactic HI
emission is more closely approximated without the background
source of illumination. Thus, the brightness temperature is
simply
TB tTC'
(3-6)

28
The small HI optical depth in a typical low mass, dwarf
galaxy is due to the low density of neutral hydrogen. The
corresponding diffuse HI emission approximates the radiation
from the entire HI mass, since negligible re-absorption
occurs at these low hydrogen densities (Wright, 1974). The
galaxies which are large enough to be used as primary
galaxies in this study normally exhibit stronger emission
and may not be correctly approximated by a small optical
depth at all locations. Therefore, the surface emission is
only a lower limit on the HI mass estimate for these larger
galaxies (an estimate for the optical depth can be made from
equation (3-6) with Tc approximately 100° K (Spitzer, 1978)
and x<< 1). Also, a larger column density, such as those
found in more edge-on galaxies, may not yield the correct
value when calculating the hydrogen mass.
The surface density of neutral hydrogen can be
calculated from the brightness temperature integrated over
the velocity bandwidth, according to (Wright, 1974)
1 O oo
nH=1.82xl0'LO J Tb dV. (3-7)
— 00
The column density, Njj is measured in atoms per square
centimeter and dV is the channel bandwidth in km/sec. An
integrated hydrogen mass can also be found from either the
global profile (integrated flux) or from the surface density
map (integrated brightness temperature). For optically thin

29
emission, the hydrogen mass can be calculated from
Mjj = J rijj (Q)dfl
(3-8)
n
IQ oo
= 1.82xl(jaJ J T(ÍI,V)dV dfl.
(3-9)
Using the Rayleigh approximation we have
S(V)dV=^J- Í T(Q,V)dil dV.
X2 «
(3-10)
Hence, the hydrogen mass can be expressed as
M-j= 2.356x105D2 ? SdV M
rl JO
(3-11)
— oo
where MH is in solar masses, D is the distance in Mpc, and S
is the integrated flux in Jy.
The galactic HI and continuum emissions can be mapped
spatially as well as in the velocity dimension either with
interferometer measurements or with single dish scans. The
theory and techniques used in both of these imaging methods
are lengthy, and will only be discussed briefly in the
Appendix. A useful reference for interferometric imaging is
found in the NRAO handbook on Synthesis Imaging (Perley,
1985). The emission maps used in this study are produced
from interferometer measurements made at the NRAO VLA.
Before the observations could be planned, however, estimates
of the expected HI flux levels for the primary galaxies were

30
necessary. These fluxes could then be used to determine if
the signal level from the primary galaxy source would be
sufficient to produce an emission map with the desired
signal to noise ratio, within an observation lasting one or
two hours. Although the actual integration time on the
individual groups was approximately 10 hours, the primary
galaxies were not, in general, in the center of the
observing field. The minimum flux requirement should ensure
ample signal from a primary galaxy whose flux strength may
be attenuated by the off-center beam response. This flux
limit should also ensure a large signal to noise ratio for
the peak emission features in the emission maps, as well as
a relatively low noise level in the temperature weighted
velocity field which was produced from these maps.
Of the five groups which satisfied the selection
process for this study, two primary galaxies had either no
flux listed in the literature or inconsistent flux values
listed. Observing requests for these two galaxies, NGC 4111
and NGC 5689, were submitted, and granted, for single dish
HI observations at the NRAO 92 m radio telescope in Green
Bank, West Virginia. These 92 m HI spectral observations
were completed in December of 1985.
Because of the transit configuration, the 92 m radio
telescope is limited in integration time to approximately 5
minutes*cosec(DEC) in a given 24 hour period. To increase
the spectral sensitivity, the scans from separate days are

31
averaged together. The positions corresponding primary
galaxies NGC 4111 and NGC 5689 were observed for five days,
each with approximately seven minute integrations between
November, 1985 and December, 1985. All scans used the total
power mode, a procedure which compares off-source reference
data (blank sky) with on-source (galaxy) data in order to
subtract the system noise component and to improve the
spectral baselines (NRAO 300' Telescope Observers Guide,
1983). Cooled FET spectral amplifiers were used, providing a
system temperature of approximately 25° K for each of the
two receivers used. The detector for these spectral line
observations was a Model III auto-correlator spectrometer
with 384 channels. The spectral observations were split into
two identical receiver systems measuring oppositely
polarized (right circular and left circular) emissions.
Each of the 192 channels had a bandwidth of 50 Khz which
spanned the same frequency in both the right and the left
channels. The computed spectra for both the calibration
sources and the program sources were then averaged in each
overlapping channel. The baselines appearing in these
spectra were normally removed with a fourth order polynomial
fitting procedure available in the computer software at the
NRAO in Green Bank.
The observations of NGC 4111 indicate a peak HI flux
level of approximately 65 mJy as shown in Figure 3.1. This
HI emission would be detectable with the VLA interferometer

32
using a 42 km/sec bandwidth, assuming the source was
unresolved by the VLA array. However, the VLA observations
showed that the HI emission did not come from the primary
galaxy, but from a nearby galaxy identified as UGC 7089 (see
section 3.3). When one accounts for the off-center
attenuation of the 30' VLA primary beam and the attenuation
from the 92 m beam centered on NGC 4111 (12*1 04m 30s, +43°
20' 43"), the flux observed at Green Bank is consistent with
the global profile of UGC 7089 shown in Figure 3.1.
The velocity range used to observe UGC 7089 with the 92
m telescope was 0 to 2000 km/sec while the velocity range
spanned for NGC 5689 was 1200 to 3200 km/sec. However, the
primary galaxy, NGC 5689, and the associated group members
did not have sufficient HI flux to provide a reliable
spectral identification. The upper limit to the flux shown
for the five averaged integrations in Figure 3.2 would be
less than 10 mJy. A theoretical calculation of the rms noise
can be made from (Appendix, eq. (A-9))
AT
rms
3.06T° K
sys
V (2dvdt)
(3-12)
where T°ysK is the system temperature in Kelvin (25° K for
the FET amplifiers), dv is the bandwidth in Mhz, and dt is
the integration time in minutes. Because the HI intensity
was too low to observe at the VLA within a reasonable
observing time, NGC 5689 was deleted from the list.

BEAM FLUX ( Jy )
33
Figure 3.1 HI spectral plot of NGC 4111 showing the
emission of the neighboring galaxy UGC 7089.

BEAM FLUX ( Jy )
34
0.04 U
0.02 h
0.00
-0.02 I-
-0.04
I36Í9
1628.6
1894.3 2160.0 2423.7
VELOCITY ( km/*tc )
2691.4
2957.1
Figure 3.2 HI spectral plot of NGC 5689 showing the
rms noise without baseline removal.

35
A total of four small groups were chosen using the
selection process described earlier, for spectral line
observations at the VLA. The HI observations were scheduled
from 1985 through 1987 because of the 15 month cycle time of
the antenna configuration at the VLA. The expected detection
levels for the dwarf satellites indicated that the
observations should be made with the two most sensitive
antenna configurations for extended sources, the C and D
arrays. The 10 mJy detection level is based on the previous
detections of dwarf satellites near spiral type galaxies
(Gottesman et al., 1984) and equation (3-1). Observations
were requested for both C and D array measurements of each
of the galaxy groups to improve both resolution and
sensitivity over a single configuration observation. The D
configuration observations were approved for all four groups
and an equivalent period of observation was approved for two
of the groups with the C array. NGC 3893 and NGC 4111 were
chosen for C array observations because of scheduling times
and the greater number of possible satellites within these
two groups.
NGC 3893
The galactic group associated with the Sc type galaxy,
NGC 3893, consists of the primary galaxy and the four dwarf
satellites which were detected with HI measurements. The
approximate 21 cm. primary beam coverage of the VLA

36
antennas at the 3dB level (30') is shown for this group in
Figure 3.3. A list of the physical characteristics of the
group appears in Table 3.1. Optical velocities for three of
the four detected satellites were available from the Uppsala
Catalog and resulted in a channel bandwidth selection of 21
km/sec. This provided a greater resolution in the galaxy
velocity field, which normally improves the rotation curve
accuracy.
The NGC 3893 group was observed with the VLA radio
telescope in the C array configuration during December,
1986, and with the D array in March of 1987. The visibility
data were combined in the map (image) plane rather than in
the visibility (uv) plane. This technique was used to
expedite the mapping process (van Gorkum, private
communication) and is explained briefly in the Appendix. A
total of 31 channels were used for a total velocity
bandwidth of ±660 km/sec. Hanning smoothing and auto¬
correlation normalization were used to stabilize the
receiver.
Figure 3.4 consists of contour plots the maps
(channels) exhibiting HI emission either from the primary or
satellite galaxies. These emission maps have had the
continuum emission subtracted and the emission features
cleaned (Hogbom, 1974; Clark, 1980) to remove the sidelobes
(see Appendix). The maps have a 6" pixel width and a
resolution (synthesized beam size) of 30.6" x 24.5",
with

37
Figure 3.3 Palomar Sky Survey field in the vicinity of
NGC 3893 (UGC 6778) showing the approximate pointing
center and primary field of view for the VLA 25 m
antennas (30').

38
TABLE 3.1
HI Summary For Observed Groups
NGC 3893
NGC 7089
NGC 4258
NGC 4303
Synthesized beam 30.
1x24.9
30.1x24.9
54.3x51.8
66.5x57.8
FWHP (")
Beam position angle
+ 88.0
+ 88.0
-76.0
-87.7
(degrees)
Channel Seperationa
20.75
41.44
41.46
41.3
(km/sec)
rms noise per channel
0.52
0.30
0.18
0.15
cleaned (° K)
(mJy)
0.65
0.39
0.84
0.93
Observed systemic
969
789
465
1561
velocity (heliocentric)
(km/sec) error 0.6
0.9
0.4
1.1
Inclination angle
43.0
68.0
71.8
18.8
(degrees) error
2.6
2.7
1.2
7.7
Position angle
-1.6
215.0
29.0
-34.6
(degrees) error
0.5
0.9
0.8
1.0
Scale length*3 b
0.26
1.80
0.14
0.47
(arc min.) error
.02
.20
.09
.05
Global profile0
253
218
505
249
(km/sec)
Hydrogen flux0*
1.62
0.19
7.64
1.56
a. Channel bandwidth = 1.22 channel separation.
b. Toomre n=0 model scale length parameter.
c. Full width at 0.2 of peak, not corrected for inclination.
d. Integrated hydrogen xlO'/ D(Mpc) .

39
the beam major axis rotated 88 degrees counter-clockwise
from the north. The three identification crosses appearing
in the contour maps indicate the positions of NGC 3893
(right), NGC 3906 (lower) and NGC 3928 (left) according to
the Dressel and Condon Catalog of Optical Positions of
Bright Galaxies (1976). Negative features are displayed as
dotted lines and are plotted at the same levels as those of
positive contours.
Several important features of the primary galaxy and
satellites are found in these narrow band spectral maps. The
most noticeable anomaly is the interaction of the satellite
UGC 6781 with the primary, NGC 3893, shown in Figure 3.4. As
a result of the strong interaction displayed by the HI
bridge in these maps, UGC 6781 is not used in the mass
analysis. Since the rotation curve of the primary galaxy
does not exhibit major anomalies, within a radius extending
as far as the opposite (North-West) HI boundary to the
interacting satellite, it was decided that the remaining
primary-satellite pairs were still useful for this study.
The emission maps in the first through fourth maps of Figure
3.4 show evidence of a fourth dwarf satellite which was not
listed in the Uppsala Catalog, nor seen in the Palomar Sky
Survey prints.
The position of the primary beam center during
observation was ll*1 47m 35s RA, +48° 55* 00" DEC. This
position is off-set from that of the primary galaxy so that

Figure 3.4 The spectral line emission maps of NGC 3893
plotted at 3 sigma (0.16 mJy) (upper left to bottom right).
Velocities are heliocentric. Identification crosses appear
at the optical positions of NGC 3893 (right), NGC 3906
(lower), and NGC 3928 (left).

41

42
Figure 3.4-continued

43
Figure 3.4-continued

44
the primary and the satellites could be observed
simultaneously. The emission features in these maps have not
been corrected for the off-center attenuation of the primary
beam. Contour increments for these maps are 3a , with a =
0.52°K (a=1/2).
Figure 3.5 shows a contour plot of the continuum
emission which has been cleaned to a level of 3a. Continuum
emission for the dwarf satellites is not evident at 3a
significance level. Strong continuum sources (S > 100 mJy)
were not apparent in a field much larger than the original
512x512 pixel maps. The continuum map is an average of
channels 3 to 5 and 27 to 29. Channels 1, 2, 30, and 31 were
discarded because of the degraded response in these end
channels which were caused by the passband.
UGC 7089
NGC 4111 is an SO type spiral galaxy, with three
suspected satellites appearing with the group on the Palomar
Sky Survey print shown in figure 3.6. The single dish HI
spectral plot in Figure 3.1 was made from observations
obtained with the NRAO 92 m radio telescope in November,
1985. The peak emission appears at approximately 800 km/sec.
However, the first HI maps, which were made from VLA C array
data, showed an unexpected lack of spectral emission from
NGC 4111 in all channel maps. The HI emission originally
thought to be from the primary galaxy, NGC 4111, is actually

45
NGC3893 I POL 1416.819 MHZ N38CONCV.CONCV.1
11 50 49 48 47 46
RIGHT ASCENSION
PEAK FLUX - 3.9005E-03 JY/BEAM
LEVS - 1.5000E-04 • ( -6.00, 6.000, 12.00,
18.00. 24.00, 30.00, 36.00)
Figure 3.5 Cleaned continuum map of NGC 3893 plotted
at the 3 sigma level. Negative emission
features appear as dotted lines.

46
from the nearby galaxy UGC 7089. The lack of detectable
emission from the primary galaxy was compensated by the
identification of a second primary-satellite pair, UGC 7089
(primary) and UGC 7094. The details of this and all other
groups used in this study are presented in Table 3.2.
The channel maps shown in Figure 3.7 are contour maps
for the UGC 7089 group plotted at 3 a intervals, with a =
0.30° K. The data sets obtained with the C and D array
observations were combined in the same fashion as for NGC
3893, that is, in the image plane. The identification
crosses appear at the Dressel and Condon optical positions
for UGC 7089 (right center) and UGC 7112 (left center). The
field of view consists of 6” square pixels with a resolution
30.6" by 24.9". The pointing center of the maps is at 12h
03m 49s RA and +43° 24' 00" DEC. The dwarf satellite UGC
7094 appears in the seventh and eighth maps of Figure 3.7.
A later spectral observation of NGC 4111 in HI was made
with the 92 m NRAO transit telescope by Otto Richter in
April, 1987 (Richter, private communication). The single
five minute scan from 0 and 4000 km/sec indicates a maximum
flux level of approximately 20 mJy which is too low to be
considered for observation with the VLA. A slightly higher
emission level may exist at approximately 2400 km/sec. If
this were the recessional velocity of NGC 4111, however, it
would be far too distant in velocity space to be considered
a member of the group.

47
Figure 3.6 The UGC 7089 group from the Palomar Sky
Survey pring showing the 30' VLA field of view and
the galaxy NGC 4111.

48
TABLE 3.2
Primary Galaxy Characteristics
System
Type3 \
r sys
Dist
incb
Rmax V(Rmax)
M(vio
xl0lu Mq
km/sec
Mpc
deg
Kpc
km/sec
NGC
224
SAS3
Sbl-II
-301
0.7
78
28.0
230
34.4
NGC
1023
SB0-
SB01(5)
610
7.5
80
9.5
251
13.9
NGC
1961
SXT5
3935
41.3
50
24.0
400
75.0
Sb(rs)IIpec
NGC
3359
SBT5
1009
11.0
51
20.8
140
9.5
SBc(s)I
. 8pec
NGC
3893
SXT5
969
10.4
34
7.8
180
8.1
Sc(s)I.
2
2.0
UGC
7089d
—
789
8.4
68
5.5
82
0.9
Sc
2.7
NGC
3992
SBT4
1046
14.2
53
19.0
240
25.4
SBb(rs)I
NGC
4258
SXS4
465
5.2
72
19.0
208
19.5
Sb(s)II
1.2
NGC
4303
SXT4
1561
12.9
19
12.2
221
14.0
Sc(s)I.
2
7.7
NGC
4731
SBS6
1490
10.5
54
13.0
150
6.8
SBc(s)III
NGC
5084
L(-2)
SOI(8)
1721
15.0
>86
34.0
328
85.0
a. From de Vaucouleurs et al. (1976) and Sandage and
Tammann (1981).
b. Inclination angle and error, if given.
c. Disk mass calculated from 2.325x10^ Rmax/G.
d. Type from Nilsen (1973).

Figure 3.7 Contour maps of the emission features of
UGC 7089 and the associated satellite UGC 7094.
NGC 4111 is not apparent at the 3 sigma minimum of
these plots (sigma=0.39 mJy). Identification crosses
appear at the optical positions of UGC 7112 (left)
and NGC 4111.

50

IS
10
os
00
4S
40
36
30
26
20
1S
10
OS
00
12
r
r-—:—r
1
1
nr-’ r
1
r r
800 km/ssc
_
•
759 km/ssc .
,
•*
•
-
, â– 
.
-
• a
-
• v
_
•
*
•
•
•
*
~
-
-
+ .
•
-
-
-
+ •
-
+
-
' â–  .
-
•
-
•
-
•
. •
-
-
-
1
1
1. *
1
Zl
. *i
1
1 • • • 1
-
• 1
1 1
1
i
|
—: r-
r
1 "T-
-
717 km/ttc
â–  -
â– ?
• 674 km/toc
-
•
•
-i
. -
•
-
*
%
. / +
* +
*
• +
-
+
-
* * '
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â– 
i_
,i
. 1 “
|
1
1 ' 1
OS
04
03
02
12
08
OS
04
03 02
3.7-continued

52
Figure 3.8 shows the continuum map which was averaged
from channels 3, 4, 5, 27, 28, and 29, and cleaned to a
level of 3cy. The weak continuum emission seen at the
position of NGC 4111 is typical of many SO galaxies. The
weak continuum emission at the position of UGC 7089 is also
suggestive of a lower mass object. Several other weak, un¬
resolved continuum sources are also shown in the field of
view, unassociated with the satellite detected in HI. Strong
sources were not observed in a field much larger than the
original 512x512 pixel maps.
NGC 4258
A number of investigators have obtained single dish and
interferometric HI masurements of NGC 4258. The most
comprehensive study of this large Sc galaxy was the Ph.D.
dissertation of G. D. van Albada (van Albada, 1978). None of
the observations found in the literature, however, included
observations of the nearby satellites which are necessary in
this study for the evaluation of the orbital masses.
Therefore, VLA observations were requested in order to
measure the orbital parameters of the satellites and to
obtain the rotation curve of the primary galaxy. A section
of the Palomar Sky Survey print in the region of the NGC
4258 group is shown in Figure 3.9.
VLA observations were made of this group with the D
array in December, 1985. Since velocity data were not

53
NGC4111 I POL 1418.555 MHZ N41CLC0NCV.CLCV.1
RIGHT ASCENSION
PEAK FLUX - 1.2407E-03 JY/BEAM
LEVS - 1.0000E-04 • ( -6.00. 6.000, 12.00,
18.00, 24.00)
Figure 3-8 Cleaned continuum map of the UGC 7089 group
at 3 sigma. Negative emissions appear as
dotted lines.

Figure 3.9 The NGC 4258 group from the Palomar Sky
Survey pring showing the 30' VLA field of view.

55
available for one of the three suspected satellites, UGC
7304, a velocity bandwidth of 41 km/sec with 31 channels was
selected. The channel bandwidth of 41 km/sec is larger than
that used by other researchers, including van Albada. With
this bandwidth, however, the velocity range of
±630km/sec is spanned with 31 channels, and the rms
sensitivity is increased owing to the larger bandwidth.
Hanning averaging and autocorrelation normalization were not
used during the observations. The stronger emissions from
NGC 4258 were of the same order as the continuum emission,
approximately 700 mJy.
The contour maps shown in Figure 3.11 are plotted at
intervals of 3a, with a = 0.18°K. The maps are made with 6"
pixels and 54.3” by 51.8" resolution (-76°). The pointing
center of these maps is 12*1 15m 30s RA and +47° 37' 00" DEC.
The two crosses appearing in these maps are located at the
optical centers (Dressel and Condon, 1976) of NGC 4258 and
the satellite NGC 4248.
The relatively high continuum emission features within
this field are shown in the cleaned continuum map, Figure
3.10. The strongest emissions are from the central region of
the primary galaxy, and from two unresolved sources of
approximately 200 mJy/beam each. Because the emission from
NGC 4258 was extended and was comparable in strength to the
continuum emission in a number of these maps, it was decided
that the stronger continuum emission should be removed

Figure 3.10 Spectral maps of the NGC 4258 group plotted
at 3 sigma (sigma=0.84 mJy). The optical positions
of NGC 4258 (left) and NGC 4248 (right) are shown
with identification crosses.

57

47 50
45
40
35
30
25
20
IS
47 SO
46
40
35
30
25
20
16
47 50
46
40
36
30
26
20
16
! 16 00
58
0 ‘¿f QfboePo ¿-0 5 J
L_I [*T- Ó ° l^O? 1
—7 3 H
409 km/sM * 0
C
o
O
&J GD I CL
Oi U-
■;¿k °.
O o *0*
• * o „ •
> ^ O'
a 0 • . •
„ ° o • »
° . o ® . cs
K ° I ° I'.^.3^ íLl° a I /> I
t r
1 77
T^1 HT
^ 490 1UÜ/64C
° r7?
;-iKgk
’Bp
*0 e
i 0°
» 0
2J
o.
L£L
J L
o'
I â– 
17 30 00 16 30 00 16 30 00 14 30 It 00 17 30 00 1t 30 00 16 30 00 14 30
3.10-continued

so
4S
40
35
30
2S
20
IS
SO
46
40
3S
30
25
20
IS
59
wrr
" a
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V o«o •
• <3 i <0 6
^ .
K
O 8
• o
>(r
'-o -I _L
_L
J L
_L
C 1 r-71
o
t 1 1 1 :
202 km/MC
c
O
o . ° °.o
■ 4**, •
° wV'
o • "
O
Ó
0 + 0
O ,
°o
0
•
O
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• 0
0
•
• •
0
j i «j
1 1 L 1
17 30 00 It 30 00 16 30 00 14 30 16 00 17 30 00 It JO 00 16 SO 00 14 30
3. ]O-continued

60
before the final mapping was accomplished. This was done to
minimize the effects of calibration inaccuracies and also to
remove the sidelobe effects from the beam response. Only
one satellite, NGC 4248, is apparent at the 3o level in
these channel maps. The second satellite listed in Table 3.2
is visible in the Palomar Sky Survey prints as a small
object which may be a dwarf satellite. An HI spectrum of
this small object, using the NRAO 92 m telescope, was made
by Thuan and Seitzer (1979). However, as the authors
suggest, this emission is spatially confused with the
emission from NGC 4258. A calculation of the emission flux
expected from NGC 4258, with the beam response of the 92 m
telescope centered on this satellite, is comparable to the
spectral plot -of the small satellite in the Thuan and
Seitzer survey. Emission from this small satellite does not
appear at other velocities in these maps at the 3a level,
and so, it can be assumed that the observed emission is from
NGC 4258.
The continuum map of the strong continuum sources,
which were removed in the visibility plane, is shown in
Figure 3.11. The map is cleaned to and plotted in increments
of 3a. Note the assymetrical continuum emission from the
primary galaxy center.
NGC 4303

61
NGC4258 I POL 1417.398 MHZ N4258CC.CLN2.1
Figure 3.11 Cleaned continuum map of the NGC 4258 group
showing the strongest features of the continuum
radiation which was removed from the visibility data.
Contour levels are in 3 sigma intervals.

62
NGC 4303
Several HI observations have been made for NGC 4303,
but again, the emission features of the satellite galaxies
are not available at sufficiently sensitive detection
levels. This group is a member of the Virgo cluster and
would have normally been excluded from consideration.
However, as the group is isolated, and the selection
criteria were satisfied, it was retained for observation.
Nonetheless, the possible effects of the cluster environment
should not be forgotten. Figure 3.12 shows the NGC 4303
group environment with the 30' VLA field of view.
The relatively high continuum flux in the field of view
suggested that the continuum be subtracted from the
visibility data before the final map-making procedure for
the same reasons as those given for NGC 4258. Both the
original set of maps and the set of maps produced with the
continuum subtracted were made with 6" square pixels with a
512x512 pixel field and a resolution of 66.5" x 57.8". The
continuum channels which were used for continuum subtraction
were 8 to 15 and 48 to 55, from a total of 63 channels.
Channels 1 to 7 and 56 to 63 were discarded because of the
non-linear passband response in these end channels. The
contour plots of the emission at the 3 o level are shown in
Figure 3.13 with a= 0.15° K. One satellite (UGC 7439),
which was detected at the 3a level, is also shown. The
centers of NGC 4303 and UGC 7404 are plotted with the

63
Figure 3.12 NGC 4303 Palomar Sky Survey showing the
approximate pointing center and field of view for the
VLA observations.

64
optical positions from the Dressel and Condon catalog. The
pointing center for these maps is 12*1 19m 18s, +04° 5' 00".
The channel bandwidth of the maps in these figures is
41 km/sec, with a total bandpass of 2600 km/sec for 63
channels. Since 31 channels were either discarded or used
for continuum, the actual velocity bandwidth is 1334 km/sec
for the remaining 32 channels. Each of the continuum
channels was searched for line emission before the continuum
averaging and again after subtracting the continuum from all
but the discarded channels. The continuum map which was
produced to provide the original visibilities for
subtraction, cleaned to the 3a level, is shown in Figure
3.14. Weak continuum emission is observed for the primary
and detected satellite galaxies, while the stronger, un¬
resolved continuum emission appears without an associated
visible object in the Palomar Sky Survey prints.

Figure 3.13 Spectral line emission maps of the
NGC 4303 group showing the primary and satellite
emissions above 3 sigma (sigma=0.93 mJy). The
identification crosses appear at the positions of
NGC 4303 (left) and UGC 7404 (undetected, right).

66
oo ao ao oo io so oo is so

67
O

—a r 1
130* km/ooc q
o
06 06
/N •
_
0
&
0
00
-
0
04 66
-
m +
0
•
o ^
46
o
-
O
t
o
O
40
7)
• o *
-
o
•
36
m
-
•
* o
30
b
2 .
1 1 . 1
L—z
12 21 00 20 30 00 10 30 00 13 30
1 1
—i—r~
1
~1—
•
I2«7
ko/tM
e
o
o
-
o
-
3
•
• o 0
-
. o
0 •
-
•
o
+
0
o ‘
0
•
o'°
-
0
o •
0
0
o
-
-
0 0
o
i
-
-
_L_q n - 1-
_l
i
1 -
00 20 30 00 It 30 00 It 30
Figure 3.13-continued

68
Figure 3.13-continued

69
NGC4303 I POL 1407.702 MHZ N43CONCLN.CLN.1
RIGHT ASCENSION
PEAK FLUX - 1.3707E-01 JY/BEAM
LEVS - 3.0000E-03 • ( -3.00, 3.000, 8.000.
12.00, 18.00, 24.00, 30.00, 36.00, 48.00,
60.00)
Figure 3.14 Cleaned continuum of the NGC 4303 region
plotted at 3 sigma showing the positions of the
continuum removed from the visibility data base.

70
TABLE 3.3
Galaxy Group Characteristics
Group
Satellite
vela
km/sec
del Vb
km/sec
sep.
Kpc
M(delV)c
xl0lu Mo
chi
(M/m)
mag
NGC 224d
-301
4.3
NGC 147
-168
+ 142
90.2
42.3
1.23
12.0
NGC 185
-208
+ 100
85.6
20.0
0.58
11.0
NGC 205
-240
+ 62
7.5
0.7
0.02
9.4
NGC 221
-216
+ 84
4.9
0.8
0.02
9.2
NGC 1023
610
10.5
North
905
+ 295
34.2
85.9
6.18
-
South
695
+ 85
17.5
3.7
0.26
-
NGC 1961
3935
12.2
A
4108
-173
91
62.6
0.84
_
B
3895
+ 40
106
3.9
0.05
-
B1
3800
+ 135
122
52.4
0.70
-
UGC 3342
3927
+ 8
157
0.2
0.00
15.4
UGC 3349
4282
+ 347
240
671.9
8.96
14.4
NGC 3359
1009
11.0
dwarf
962
-47
48.5
2.5
0.26
-
NGC 3893
969
10.6
UGC 6797
963
-6
47.5
.04
0.01
14.1
UGC 6834
987
+ 18
30.3
.23
0.04
13.1
dwarf
1102
+ 133
37.2
15.5
2.68
-
UGC 7089
789
14.8
UGC 7094
769
-18
24.1
.18
0.25
15.6
NGC 3992
1046
10.7
UGC 6923
1062
+ 16
60.6
0.4
0.02
14.1
UGC 6940
1112
+ 66
35.4
3.6
0.14
16.0
UGC 6969
1115
+ 69
45.3
5.0
0.20
15.5
NGC 4258
448
9.6
UGC 7335
479
-31
19.1
0.4
0.02
13.9
NGC 4303
1561
10.9
UGC 7439
1275
-286
39.1
74.4
4.75
14.9
NGC 4731
1490
6.0
RNGC 4731a
1505
+ 15
32.1
0.2
0.03
_

71
Table 3.3-continued
Group
vela
del Vb
sep.
M(de1V)c
chi mag
Satellite
km/sec
km/sec
Kpc
xl0lu Mo
(M/m)
NGC 5084
1721
dwarf
2089
+ 368
65.6
206.5
2.43
a.
b.
c.
d.
Heliocentric velocity reference.
^satellite “ ^sys
orbital mass according to
A.S.R. velocity reference
(1982).
2.325xl05 (del V)2 R/G.
from Einasto and Lynden-Bell

CHAPTER IV
DATA REDUCTION
Integrated Moments
The spectral line maps made from the VLA HI observation
are the primary source of data for the analysis of the mass
distribution of these galactic groups. The HI emission
features found in the maps are used to identify the
dynamical characteristics of the primary and secondary
galaxies. In addition, the spectral emission features in
these maps are summed over the velocity channels
(integrated) to provide an estimate of the hydrogen mass for
each of the galaxies and a velocity field for the primary
galaxies. The exterior mass estimates are made from the
satellite velocity measurements and the position
measurements on the emission maps. The disk masses of the
primary galaxies are calculated from the rotation curve
solution to the velocity field the first moment of the
integrated flux.
The HI emission maps produced from the spectral
interferometer observations contain artifacts from the
synthesized beam response to the received signals. In
contrast, the sidelobes of the primary beam (the sidelobe
spacing of the primary beam is determined by the diameter of
72

73
the antennas, while the sidelobe spacing for the synthesized
beam is determined by the maximum projected spacings of the
antenna array and the u-v coverage) do not contribute
significantly to errors in the image plane unless a strong
source is located within several degrees of the field of
view. This is primarily due to the spatial incoherence in
signals received from beyond the HPBW of primary beam,
especially in observations which require wide bandwidths
(Bridle, 1985). The map images can be improved, or cleaned,
by removing much of the synthesized beam sidelobe features,
which include the irregular patterns produced with
incomplete u-v coverage. The cleaning process can also
interpolate between the measured u-v values. This appears as
a smoothing of the emission (or absorbtion) features in the
image plane.
The velocity of a satellite used in the dynamical mass
calculations can be estimated to better than the width of a
channel simply by inspecting the emission maps. However, the
systemic velocity for the satellite is best estimated using
the integrated flux and first moment (either the flux/beam
or temperature weighted velocity) calculations. To find
these integrated moments of flux and flux weighted velocity,
the images are summed in each pixel over all channels
(maps). The surface density of the HI emission in each pixel
is the summation over all maps exhibiting line emission, of
the brightness temperature at that pixel position, above a

74
specified minimum level. This level of significance is
normally chosen as three times the rms noise in the
continuum-free, cleaned maps. A lower limit to the hydrogen
mass can be calculated from the surface emission calculation
shown in equation 3.11. This calculation is made from the
integration, over each pixel position in the map, of the
surface density. Thus the hydrogen mass estimate is
expressed as
Mr= 1.82x1018D2J J TBi dV dQ
n v 1
(4-1)
Tg^ is the brightness temperature, dV is the velocity
bandwidth of the images, D is the distance in Mpc. The
first (velocity) moment is calculated using the same
summation procedure as that used to calculate the total
flux. The temperature weighted (or flux/beam weighted if the
units are in flux/beam) mean velocity is expressed as the
summed product of the temperature at pixel i, times the
velocity of the map at the corresponding position, over the
sum of the temperatures. Thus, for a position i (which can
also be expressed as an x,y coordinate), the temperature
weighted velocity can be expressed as the first moment by
oo
Í ViTB dV
- 00
(4-2)
00
Í tb dv

75
for brightness temperature units. In order to avoid
contributions due to noise, the rms noise value can serve as
a cut-off level. However, this cut-off technique includes a
number of unwanted biases. These biases and improved methods
of integrating the temperature weighted moments are
discussed by Bosma in his Ph.D. dissertation (1978).
The Bosma window method employs a calculated velocity
range for each pixel as well as a signifance level for the
emission feature. The velocity window is established by
iterating outwards from the channel in which the spectral
emission is the strongest and summing the emission spectra
which contribute an ammount greater than a convergence
limit. The emission must also be above a specified minimum
to be considered significant, normally 3 a. The iterative
sum of emission for each individual pixel is stopped when
the addition of emission from channels farther from the peak
channel no longer contributes significantly to the sum.
Thus, the velocity window is calculated to be the range of
velocity channels that contains significant spectral
emission at the individual pixel position. The emission
which remains outside this window is considered continuum
emission and is not included in the spectral emission sum.
This continuum emission can be summed seperately to provide
a continuum map. Tests made by Bosma for summing spectral
emission using a simple cut-off method, individual profile
fitting of spectra, and the window method, indicate that the

76
window method to be the most accurate summing procedure that
did not require excessive computing time. The selective
window method reduces the contribution of emission from
noise and excludes spikes which may appear in single
channels. A further reduction in noise and an increase in
the smoothing of emission features of the summed emisión map
is made by smoothing in the velocity plane, as Hanning
smoothing. This procedure reduces the rms noise in each
emission map by approximately -J2 by averaging adjacent
channels in the ratio of .5:1:.5, in effect, doubling the
bandwidth. Additional smoothing in the spatial plane, by
convolving each emission map with a specified Gaussian beam
function, produces smoother integrated emission maps, and
improves sensitivity in detection observations (England,
1986). Integrated moments were produced from these data
with software available from the NRAO VLA that did not
employ the Bosma window method. A second algorithm used on
the observed galaxy group data, based on the Bosma window
method, was developed by Gottesman (England, 1986), and was
found to be optimal for detecting weak emission features.
The intergration scheme includes the Hanning velocity
smoothing, spatial smoothing with a Gaussian beam of twice
the clean beam dimensions, a significance level of 2 a for
cut-off, and a minimum number of channels in which channels
above a minimum rms occurs sequentially.

77
The integrated HI (surface density) and velocity
moments for each group, calculated with the NRAO integration
algorithm, are overlayed in alternating figures, from Figure
4.1 through 4.7. The velocity moments are plotted for each
primary galaxy after each overlayed moment, alternating in
Figures 4.2 to 4.8. The velocity contours for the overlayed
plots in increments of 20 km/sec for NGC 3893 and 40 km/sec
for NGC 4111, NGC 4258, and NGC 4303, which represents the
approximate velocity speration of the channels. The total HI
mass, calculated from the surface emission for each primary
galaxy, is given in Table 3.1.
Rotation Curves
The interior masses for the primary galaxies are
calculated from the rotational velocities at the last
(reliable) observed velocity point of each galaxy. This
interior, or disk, mass is a measure of the total mass
interior to the radius of the last velocity point, after the
velocities are corrected for the projection of the galaxy
onto the sky. The individual velocities and radial
separations are taken from the velocity field map and must
be corrected for the inclination projection in order to
produce the rotation curve. The velocities must also be
corrected for the recessional (systemic) velocity of the
primary galaxy by subtracting the systemic velocity. The
rotation curve can then be computed by averaging the

78
GREY: NGC3893 I POL N380NLY.MOMO.1
CONT: NGC3893 I POL N380NLY.MOM1.1
49 04
11 46 45 30 15 00 45 45
RIGHT ASCENSION
GREY SCALE FLUX RANGE- 0.0000E+00 1.2993E+03 JY/B*M/S
PEAK CONTOUR FLUX - 1.1667E+06 M/S
LEVS - 1.0000E+05 • ( 8.000, 8.200, 8.400.
8.
600,
8.
800,
9.
000,
9.
200,
9.
400,
9.600,
9.
800,
10
.00,
10
.20,
10
.40,
10
.60,
10.80.
11
.00,
11
.20,
11
.40,
11
.60,
11
.80)
Figure 4.1 Integrated zeroth and first moments
for NGC 3893 with increments to velocity contours
and grey scale in MKS. The cutoff level for
integration was 3 times rms noise.

79
NGC3893 0.0
MAXIMUM CONTOUR IS 56
MINIMUM CONTOUR IS 0
CINT = 20.600
X I NT * 0.100
Figure 4.2 Velocity field for NGC 3893 displaying
the tidal assymmetry from UGC 6781. The maximum
velocity shown is the maximum contour level
times the velocity increment.

80
GREY: NGC4111
CONT: NGC4111
IPOL N41110NLY.MOMO.1
IPOL N41110NLY.MOM1.1
43 32
30
28
D
12 03 45 30 15
RIGHT ASCENSION
GREY SCALE FLUX RANGE- 0.0000E+00 9.2855E+02 JY/B»M/S
PEAK CONTOUR FLUX - 8.5969E+05 M/S
LEVS - 1.0000E+05 • { 6.000. 6.400, 6.800,
7.200, 7.600. 8.000, 8.400. 8.800, 9.200,
9.600)
Figure 4.3 The overlayed HI and velocity moments
for the primary galaxy UGC 7089 with 3 sigma cutoff.

13.000 MINUTES OF ARC
NGCUlll 0.0
MAXIMUM CONTOUR IS 20
MINIMUM CONTOUR IS 0
CINT = Ul.UOO
X I NT = 0.100
Figure 4.4
Velocity field for the primary galaxy
UGC 7089 integrated at 3 sigma.

82
GREY: NGC4258
CONT: NGC4258
I POL N4258GRP.MOMO.4
I POL N4258GRP.MOM1.4
12 17 30 00 16 30 00 15 30 00
RIGHT ASCENSION
GREY SCALE FLUX RANGE- -9.2443E+01 6.3537E+03 JY/B»M/S
PEAK CONTOUR FLUX - 7.4005E+05 M/S
LEVS - 1.0000E+04 • ( 5.000. 9.000, 13.00.
17.00, 21.00, 25.00, 29.00, 33.00, 37.00,
41.00, 45.00. 49.00, 53.00, 57.00, 61.00,
65.00, 69.00, 73.00, 77.00, 81.00. 85.00)
Figure 4.5 Integrated HI plot and velocity
field for NGC 4258 and the satellite NGC 4248

25.800 MINUTES OF RRC.
NGC4258
0.0
25.800 MINUTES OF RRC.
MAXIMUM CONTOUR ISIS
MINIMUM CONTOUR IS O
CINT = NI.500
XINT * 0.100
Figure 4.6 Velocity field for NGC
integrated at the 3 sigma level.
4258

84
GREY: NGC4303
CONT: NGC4303
I POL N43030NLY.MOMO.2
I POL N43030NLY.MOM1.2
04 55
RIGHT ASCENSION
GREY SCALE FLUX RANGE- -1.3330E+01 3.8490E+03 JY/B*M/S
PEAK CONTOUR FLUX - 1.6443E+06 M/S
LEVS - 1.0000E+04 • ( 132.0, 136.0, 140.0,
144.0, 148.0, 152.0, 156.0. 160.0, 164.0.
168.0. 172.0. 176.0, 180.0. 184.0)
Figure 4.7 Integrated HI and velocity
field for NGC 4303 and the satellite
NGC 4303a.

20.300 MINUTES OF ARC.
85
NGC4303 0.0
MAXIMUM CONTOUR IS 39
MINIMUM CONTOUR IS 0
CINT = 41.300
X I NT = 0.100
Figure 4.8 Velocity field for NGC 4303
integrated at the 3 sigma level.

86
corrected velocities over increments in radial distance from
the galaxy center. The disk mass can, in turn be found from
the rotation curve. The disk mass is expressed as
(Lequeux,1983)
M,. . =V(R )2R /G. (4-4)
disk v max' max' v '
with the maximum radius of separation. Lequex points
out that this disk mass is actually a measure of the
combined disk structure mass, the mass of the nucleus, and
any halo mass which may be interior to . This makes the
"disk" mass an ideal measure of the interior mass of the
galaxy for this mass distribution study, because all of the
mass components are measured.
The procedure for calculating the rotation curve from
the galaxy velocity field requires a simultaneous fit of all
the velocity and radial separation values to assumed
projection parameters. To accomplish this, a fit of both
the observed and the model velocities (for a given rotation
model) is accomplished using an iterative least squares
procedure. The linearized expression for the rotation law,
shown by equation (4-7), is employed in the least squares
minimization. The projection parameterers, maximum
rotational velocity, and model rotation parameters are
adjusted by the least squares solution in order for the
observed velocity data to fit the model rotation curve. The
parameter fit is halted when the corrections are below a
convergence limit.

87
The rotation curve of the primary galaxy requires the
de-projected velocities of the observed velocity field.
However, the projected velocities near the minor axis of the
galaxy have a small observed radial velocity, and may be
dominated by non-circular or random velocities. The region
near the minor axis should therefore be omitted from the
velocity averaging, if these irregular velocities are to be
avoided. This can be accomplished using cosine weighting
(weighting velocities with the cosine of the angle from the
major axis) or by simply using the velocities within a
specified angular separation from the major axis.
The accurate calculation of a rotation curve also
requires an accurate position for the center of rotation of
the galaxy. A list of accurate optical positions for the
centers of larger galaxies is available from several authors
(Gallouet et al., 1973; Dressel and Condon, 1976). The
optical positions of Dressel and Condon were used to find
the centers for the four primary galaxies observed in this
study. Efforts to establish more accurately the center of
the velocity field (more accurate than the 4” rms error
quoted for the Dressel and Condon positions), using
dynamical center calculations which are based on rotational
symmetry, had limited success owing to the assymetrical
structure and the high inclination of three of the four
primary galaxes, and were abandoned.

88
Calculating the de-projected rotation curve solution
for the primary galaxy first requires known or estimated
projection parameters, which describe the orentation of the
galaxy in the plane of the sky. The parameters describing an
arbitrary position within a galaxy, in both the plane of the
sky and the plane of the galaxy, are shown in figure 4.9.
The coordinate references for angles measured in the plane
of the galaxy are 1) from the posotive Y axis, counter¬
clockwise for position and 2) from the positive Y axis
counter-clockv.'i se to the positive velocity reference of the
position angle- of the galaxy. If the angle

angle from the major axis (M in Figure 4.9) of the orbit in
the plane of the galaxy, to the position reference point at
a scalar distance R, the corresponding position angle to the
reference point in the plane of the sky will be y. The
scalar distance form the center of the galaxy to the same
reference point in the plane of the sky, r, is
r = Rco 5

The relationship between the position angles in the two
planes is then
tanp = tanycosi (4-6)
where inclination angle i is measured between +0 and +90
degrees.

89
Figure 4.9 Position parameters within an inclined
galaxy. The major axis is parallel to the line from
the focal point to position M. The projected major
axis passes through the focal point and position
M' .

90
The circular rotational velocity is described in the
plane of the sky as
V = VSyS+ V(r) cosip sini.
(4-7)
V(r) is the rotational velocity at distance r from the
galaxy center and VgyS is the systemic velocity of the
galaxy.
The solution for the yet undetermined projection
parameters of the rotation law is produced from a least
squares fit to these variable parameters, i, P (the position
angle of the line of nodes), VSyS, V(r), and in this
analysis, b, the scale length of the assumed model rotation
curve, which is the Toomre n=0 model. This model rotation
velocity expression is (Hunter et al., 1984)
V (r) = V
max
yf
r2 + b2
(4-8)
Vmax is maximum rotational velocity, more often refered
to as C for the Toomre models.
The non-linear rotational velocity equation (4-7)
requires a linear expansion for input into the first order
least squares minimization routine. A first order expansion
in the Taylor derivatives of this velocity equation can be
used with a few caveats. First, the requirements for
convergence of a least squares solution to the parameter
variables may not be satisfied by the observed velocity

91
data, as cautioned by Jefferies (Jefferies, 1980), Eichhorn
(Eichhorn and Clary, 1973), and others. More specifically, a
large variance (noise) in the velocity field data may
produce residuals in the least squares fitting that are of
the same order as the adjusted parameters of the model, in
this case the five rotation parameters, i, P,
in d a. ! sys r
and b. A second order expansion may be necessary for a more
accurate solution, or in some cases, for a convergent
solution at all. Second, correlated variables should be
treated with a covariance expression which separates the
related residuals if a more rigorous solution is expected.
The equation of observed rotational velocity, (4-7), shows
an explicit example of correlated variables, the
interdependence of the circular velocity component (V -
VSyS) on both the inclination angle and the maximum
rotational velocity, Vmax . The effects of the inclination
variations in a model rotation field are inseparable from
the maximum rotational velocity variations of the same
model, except for any elliptical projection of the circular
disk structure in a spiral galaxy. This may be the primary
reason that larger non-circular velocities produce higher
inclination angle solutions in least squares tests of
simulated galactic rotation fields, especially for shallow
inclinations. The third cautionary note concerns the
preliminary estimates for the input model parameters. These
should be reasonably close to the actual parameter values if

92
convergence is expected without more a detailed algorithm
which corrects for large adjustments to the residuals.
For each position in the velocity field, the observed
velocities are subtracted from the linearized model of the
rotation law at the same distance from the galaxy center and
then minimized for an optimum fit of the model to the data.
For a first order expression of the model rotation law, the
total derivative of the rotational velocity model, equation
(4-7), is used. This first order derivative is
dV = ¿V/¿VsysdVsys + <3V/«3VmaxdVmax + cV/<3(pVo sincp sini di
+ <5V/<3

dV + V/V dV -V sin

sys ' max max (cosy cosi)
+ V
sincp cosy
(cosy cosi)
•dP - V
V(r) (p2 + b2)3/2
db. (4-9)
The adjustments to the parameters used for de-projection and
for describing the rotation curve, i.e. di, dP, dVc,,„
t>y o,
dVmax and d*3' are macie from changes in the least
squares fit of the rotation model with the observed
velocities (as projected by the adjusted parameters).
Mimimizing the differences of the observed velocity and
the first order model velocity shown above, is accomplished
by the least squares routine, which returns the adjustments

93
to the input variables. The actual fitting algorithm used
was developed by Howard Cohen of the University of Florida
(private communication), which is based on the method of
Banachwiewicz (1942).
The resulting projection parameters are used to
calculate the de-projected velocities in the galaxy field
and the averages of the observed velocities for the given
velocity field over a radial interval. This interval must be
chosen large enough to avoid reducing the resolution of the
velocity field unless such a reduction is desired. These
averaged intervals are the values that define the velocity
curve, V(r). The errors calculated by the least squares
procedure are standard deviations in the input variables,
which, in turn, can be used to estimate the uncertainty in
the calculated rotation curve. Uncertainty in the rotation
curve solution can also be calculated with the variance of
the velocity field in each of the radial increments that are
averaged. This is a more direct procedure for calculating
the rotation curve uncertainty. However, the deviations in
the individual adjusted parameters give a better figure of
uncertainty of the overall rotation curve and also for the
uncertainty in the linear and angular variables.
Tests with the first order least squares fitting
routine using synthesized velocity fields show a strong
dependence of the minimum inclination angle on either non¬
circular or random noise velocities in the velocity field.

94
although the product of the maximum rotational velocity and
the sine of the inclination angle remains constant. A number
of tests were performed with random velocities or non-
symmetrical motion to the velocities or as an offset in the
actual position in the velocity field center. Sample tests
show solutions for shallow inclination angles in a velocity
field of approximately 50 pixel diameter whose center is
offset by 2 pixels, will increase the inclination angle
solution by 10 to 15 degrees. Position angle and systemic
velocity are not appreciably affected by an offset in the
center, although the solution to the scale length b of the
Toomre n=0 model shows increasing variations with increasing
random noise or with center position off-set. Similar tests
on synthesized velocity fields with an added random
(uniformly distributed, not Gaussian) velocity component
increase the inclination angle solution for a shallow
velocity field by the ratio of the non-circular velocity to
the maximum rotational velocity to a maximum inclination of
approxiamtely 50°. The inclination values listed in Table
3.2 are assumed to be upper limits because of the noise
dependence of the inclination angle solution.
The rotation curves and the results from the least
squares fitting procedure to the four primary galaxies are
shown in Figures 4.10 to 4.13. The calculated disk mass for
each of the primary galaxies is taken from each of these de-
pro jected rotation curves at the most distant point with a

95
reasonable signal to noise ratio for the rotational
velocity. The linear scale of each rotation plot is in
angular units of arc minutes. The adopted distance of each
of the galaxy groups is shown in Table 3.2.

NGC3893
Figure 4.10 Rotation curve of the primary galaxy
NGC 3893 showing the asymmetrical rotational
velocities between the positive (+) axis with
respect to the systemic velocity and the negative
axis. The Toomre n=0 model curve is also shown.
Error bars display the 2x standard deviation in the
velocity averhged of the radial interval.

97
NGC4111
Figure 4.12 Rotation curve for the primary galaxy
UGC 7089 showing the positive and negative axis
averages and the Tooinre n=0 model fit.

98
NGC4258
Figure 4.13 Rotation curve for NGC 4258 for the
positive axis only. The rotational velocities
were angle averaged within 45° of the major
axis.

99
Figure 4.14 Rotation curve for NGC 4303 with both
positive and negative axis averages and the Toomre
n=0 model curves.

CHAPTER V
SUPPLEMENTAL GROUPS
A number of galaxies which met the selection criteria
listed in Chapter 2 were not selected for observation, but
are included in the study because the primary and satellite
data were available the studies of other researchers. Other
galaxies which satisfied most of the selection criteria were
included in the data base if the observations suggested that
the group members were isolated, bound, and were dominated
by one member only. This is a list of the small groups that
are used to supplement the orbital-disk mass ratio data.
NGC 224 (M31)
NGC 224 was chosen for inclusion in this study of mass
distribution because the massive primary galaxy, M31, and
four observed dwarf satellites are well studied. The
distance to the the primary is approximately 700 kPc which
was assumed to be sufficient to provide the same distant
reference for the satellite positions as the more distant
galaxy groups. One satellite, NGC 221 (M32), was not
included because of the tidal interaction with the disk of
NGC 224 (Byrd,1983). The suspected tidal interaction of NGC
100

101
205 with the disk of NGC 224 is not as obvious as M32 and
the pair will be retained. However, most of the
distribution tests will be made on the observed data with
and without NGC 205. The disk mass calculation for M31 is
taken from the HI rotation curve of Roberts (Roberts et al.,
1978) shown in Table 3.2. The velocities of each object are
from Einasto and Lynden-Bell (1982) and are in ASR, not the
more commonly used heliocentric velocity rest frame.
NGC 1023
This peculiar system of a hydrogen rich type SB0-
primary, and two dwarf satellites, was observed with the
Westerbork Synthesis Radio Telescope by Sancici et al.
(1984). The disk mass is calculated from the rotation curve
of Dressier and Sandage (1983), which was measured
optically. Positions and velocities of the members are from
the Sancisi paper (1984) and listed in Table 3.2. The
hydrogen rich character of NGC 1023 is confirmed in the SO
HI study of Wardle and Knapp (1986). The accretion of HI
from the smaller surrounding features suggests tidal
interaction with these objects. The relatively large
velocity differences in the primary and two satellites (well
separated from the two tidal objects) also indicates a
large, non-luminous mass surrounding the primary galaxy, or, ,
perhaps as likely, a dynamical interaction of the satellites

102
with the primary. Since the HI interaction is primarily
between the dominant galaxy and two non-satellite
components, refered to as the blue (shift) and red (shift)
components, this system was included in this study.
NGC 1961
This galaxy group is dominated by the NGC 1961, a
massive Sb type, observed by Shostak et al. (1982) with the
Westerbork Synthesis Radio Telescope. Single dish
observations by Shostak (1978) reveal a similar total mass
calculation from the global profile as that calculated from
the rotation curve, approximateley 1x10^ MQ. The most
unusual feature of this system is the apparent HI stripping
of the primary galaxy by the intragalactic medium. Merging
and/or tidal interaction among the members is not suspected
according to Shostak et al. However, the unusual properties
of the HI emission from NGC 1961 remains to be exaplined.
The entries of Table 3.2 are from Shostak et al. (1978) and
from Gottesman et al. (1983).
NGC 3359
This system was studied by Ball (1984) in his 1984
Ph.D. dissertation. The primary galaxy is an SBc without
visible satellites in the surrounding region. However, a

103
dwarf satellite was discovered by Ball in his VLA HI
observations. The disk mass and radial values are shown in
Table 3.2 along with the associated observed chi mass ratio.
NGC 3992
This galaxy system was studied by Gottesman et al.
(1984) using VLA HI observations. The truncated disk feature
for the primary galaxy indicates no significant halo mass
beyond the HI edge of the galaxy. Three dwarf satellites in
this system were observed in these HI measurements for which
the orbital masses calculated from the velocity differences
and separations from the primary. The primary galaxy mass
value from Gottesman et al. (1978) differs fom the simple
rotation curve disk mass calculated in Table 3.2 because of
the Toomre model mass calculation used in the reference and
the rotational mass used for Table 3.1.
NGC 4731
The NGC 4731 barred spiral galaxy was found to have a
dwarf satellite in the observations of Gottesman et al.
(1984). This is the same paper that includes the study of
NGC 3992. A Toomre n=0 mass model (Hunter et al., 1984) is
used to analyze the irregular disk and bar structure of this
primary galaxy. There is again a discrepancy in the disk
mass calculation for NGC 4731 between the published model

104
mass and the disk mass used for this analysis, owing to the
Toomre model mass calculation for the published mass.
NGC 5084
This massive SO galaxy has been observed optically and
in HI by several authors, including van Worden et al.
(1976), and Gottesman and Harwarden (1985). The large
maximum rotational velocity (approximately 328 km/sec) and
maximum radius (approximately 34 kpc) indicate the large
mass of this galaxy. The HI study of Wardle and Knapp (1986)
also shows this to have a large HI content. The rotational
velocity of the Gottesman and Harwarden paper shows no
significant pecularities, but does show the flat rotation
curve of a typical massive early-type spiral.
Satellite Characteristics
The dwarf satellite galaxies are intended to act as
test probes in the potential field of the primary galaxy.
Because of this, the satellites require an estimate of the
accuracy of this assumption. These total mass estimates for
the satellites are made using the global profile and the
mass estimate developed by Casertano and Shostak (1980).
Only the velocity profiles are available for the satellites
observed in this study. The equation for the total mass
calculation from Casertano and Shostak (1980) is
Mtotal-2-xlo5iVoDDo'
(5-8)

105
In this equation the profile width at the 20% peak level is
AVq/ while the diameter of the object in kpc is DQ. D(Mpc)
is the distance to the object in Mpc, which was taken as the
distance to the primary galaxy.
The average satellite mass, as shown in Table 5.1, is
5.3% of the primary mass, with the largest mass percentage
of 21% for the UGC 7098-UGC 7094 pair. However, these
profile widths are not corrected for inclination of an
assumed disk structure. If such a disk structure is assumed
for these dwarfs, the increase in the satellite to primary
mass ratio is increased by the inverse of the sine of the
inclination angle. Since the inclination angle is not
easily determined for these dwarfs an average inclianation
may substitute for the individual inclinations. The average
value of the sine function between 0 and 90 degrees is 2/n
or approxiamtely 1.6. With this correction the average mass
ratio becomes 8.1%, or, without the 21% value, 1.8%.
The satellite velocities and the velocities of the
satellites relative to the primary galaxy are measured with
the same integrated moment calculations as the velocity
profiles and HI masses. For consistency, the velocities for
the satellites quoted from other references are taken from
the same reference work that lists the primary galaxy
velocity. Seperations between the satellite and primary
galaxies are measured on the emission maps for the observed
groups, and from the reference papers for the supplemental

106
groups. A Hubble constant of 100 km Mpc”^ sec"^ is used for
distance and separation calculations.
Table 3.2 shows the separation and velocity data for
each of the satellites as well as the calculated mass ratio,
*obs‘ Tab^-e 5.1 displays the measured characteristics for
the satellites observed for this study. The X0^s values for
all primary-satellite pairs are shown in distribution of
Figure 5.1, with an e random distribution plotted with the
same normalized area as the xobs distribution.

107
TABLE 5.1
Observed Satellite Characteristics
vela
km/sec
FIb
Jykm/sec
HI massc
xl0° Mq
profile
width
km/sec
dia.
d Total
masse
xlO8 Mq
%f
mass
UGC
6797
963
5.1
1.20
67
1.7
3.9
.4
UGC
6834
987
2.3
0.55
134
1.2
9.3
1.2
dwarfg
1102
1.0
0.24
93
-
-
-
UGC
7094
787
1.1
0.25
198
1.2
19.7 21.9
UGC
7335
479
1.8
0.41
136
3.2
15.4
0.8
UGC
7439
1276
18.3
4.31
179
1.4
28.9
2.1
Average of satellite/primary mass ratios = 5.3%
Total number of satellites with redshift wrt primary = 13
Total number of satellites with blueshift wrt primary = 10
a. Heliocentric velocity reference.
b. Flux integral in Jansky km/sec.
c. Integrated HI mass / D(Mpc) .
d. Total mass from (Casertano and Shostak,1980)
2.35x10^ VQ d('), not corrected
for inclination.
e. Optical diameter (blue) from Nilsen (1973).
f. Satellite to primary mass ratio xlOO.
g. Satellite of NGC 3893. Optical diameter not available.

108
theoretical vs. observed distribution for chi
0 12 3 4 5 6 7 0 9
X
Figure 5.1
Model and observed chi values plotted with
a random e2 eccentricity model curve for
comparison. Both curves are normalized to the
same area.

CHAPTER VI
MASS DISTRIBUTIONS
Chi Distribution
The number distribution of the observed chi values
taken from all of the primary-satellite pairs used for this
study resembles the chi model distributions shown at the end
of the previous chapter (Fig. 5.1). The greater values
within the observed distribution imply that the mass
distribution which represents the primary galaxies is
slightly greater than the interior mass of each primary as
shown by the observed values exceeding the chi model maximum
of 2. However, the resemblance between the chi model and
the observed distributions can only be stated qualitatively
at this point. A quantitative evaluation of the observed and
model chi distributions is necessary to evaluate the
similarities in the distributions and to evaluate the total
mass distribution which represents the small, compact galaxy
groups in the form of the observed chi distribution. In
addition, the different characteristics of the primary
galaxies, the individual groups and satellites, and the
satellite environment, need to be examined in order to
support the conclusions concerning the estimate of the
representative mass distribution.
109

110
The first step in the assessment of the similarities in
the chi model and the observed data is a comparison of the
two distributions. An assumed normal density distribution
for the observed data samples would suggest a chi squared
test (unrelated to the chi model) for a "goodness of fit"
analysis or a similar test which could establish a
quantitative comparison of the two distributions. However, a
normal distribution of sampled data cannot be assumed,
either for the relatively few observed chi values, or for
the unknown character of the distribution of the small group
data. A method of comparison which does not assume any
particular distribution form and does not assume any other
specific parameters, is necessary for checking similarities
in these data. The Kolmogorov-Smnirov (K-S) test satisfies
both this distribution-free requirement and the continuous
distribution limitation requirement for the model (Gibbons,
1971). Hence, the K-S test was chosen for the purpose of
investigating the similarities in the model and observed
distributions.
A negative hypothesis is used for comparing the model
and observed distributions for this parameter free test,
although two tests of significance are used. The
significance is associated with either an assumption of a
"poor" fit or a "good" fit. The null hypothesis is assumed
when an acceptable fit is expected between the model and
observed distributions and, in addition, has an acceptance

Ill
level at a calculated probability of 15 to 20 %. In
contrast, the research hypothesis is assumed when a poor fit
of the two distributions is likely and has an associated
level of acceptance of 5% (i.e. a 5% or greater error is
probable if the poor fit hypothesis is rejected). The chi
model used in this analysis is tested with the K-S
stastistic to determine the eccentricity distribution for
the model which could be accomodated by the observed data
(x0bs) using both hypotheses. Although the significance
levels for the tests are somewhat arbitrary, the research
hypothesis will tested at 5% and the null hypothesis will be
tested at 20%. A test at this level will allow a
conservative margin for an assumed "good" fit.
The cumulative character of the Kolmogorov-Smirnov test
allows a comparison of the model and the sample over the 1 +
e limit of the model chi values, as well as the observed chi
values which lie outside this chi model limit for bound
orbits. The method of the K-S statistical test is derived
from the maximum difference in the accumulated distribution
for the model and the observed values, which are first
normalized to unity. The tabulated probabilities used for
these tests incorporate the number of samples in the
calculation for the critical value of the statistic as an
uncertainty.
The insensitivity of the chi model to a semi-major axis
distribution is verified in section 6.2. However, the

112
distribution of chi values is dependent on the
eccentricities used as seen in Figure 2.2. Since the
character of the eccentricities for the % ^ distribution
cannot be determined because of the unknown orbital
parameters, separate x model distributions with various
eccentricities are tested for a corresponding fit to the
*obs distribution. K-S tests are made with model data, N(x),
and observed data, N(x0ks)/ calculated with eccentricities
between 0 and 0.9 for the range of values for both the 1 + e
maximum and the maximum range of the x0bs values. These
tests are summarized in Table 6.1 for the observed data set
and chi models. The statistics and probabilities indicate a
less than acceptable fit for all eccentricities (including
random eccentricities) if 20% is assumed for the null
hypothesis using the entire data set. However, the tests
show an acceptable fit to the point mass (chi) model if only
the values less than 2 are used. The table indicates that,
for either the data set (with or without NGC 205), the more
circular orbits are not accepted as well as the most
eccentric (0.9) model tested againt the data without NGC
205. The highly eccentric orbit model does not support a
halo mass model from simple dynamical friction calculations.
If a halo mass is assumed to surround the primary galaxy to
a radius of 100 Kpc with a mass of one or two times the
primary mass, the energy loss of an orbiting satellite by
dynamical friction is a significant fraction of the total

113
orbital energy (although the available dissipation
mechanisms are unlikely to allow the entire energy loss to
go into orbital energy loss). The friction coeffecient from
Tremaine et al. (1975) shows the mass density dependence of
the energy loss, making a smaller diameter halo or a more
massive halo have an even more pronounced effect. Since the
orbital period at 100 Kpc from a 2x10^ MQ galaxy is of the
order of a Hubble time, the single orbit encounters serve as
an upper limit to the friction effects, but do indicate that
halos are not suggested by the distribution of x0bs values.
The x model does not account for the observed values
which are greater than the theoretical limit of 2, therefore
it is necessary to test the chi model for biases which could
be introduced by the selection process.
Selection Bias
Several tests for selection effects in the mass
distribution analysis for binary galaxies are described in
detail by van Moorsel. Although the selection criteria for
the compact groups used in this study galaxies are
substantially different from those used for binary galaxies,
the selection bias can be tested with similar procedures.
This will provide a verification of the bias effects found
by van Moorsel and allow a comparison between those effects
on the binary sample and on the small group sample.

114
TABLE 6.1
K-S Tests for Observed and Model chi
with NGC 205
without NGC 205
eccen.
values < 2
all values
values < 2
all values
0.0
0.321
0.261
0.291
0.273
<5%
>5%
10%
>5%
0.1
0.309
0.252
0.291
0.267
>5%
>5%
10%
<10%
0.2
0.313
0.247
0.284
0.260
>5%
10%
>10%
<10%
0.3
0.270
0.241
0.244
0.253
>10%
>10%
>20%
10%
0.4
0.234
0.236
0.226
0.250
>20%
>10%
>20%
10%
0.5
0.231
0.251
0.221
0.270
>20%
<10%
>20%
<10%
0.6
0.220
0.276
0.191
0.292
>20%
<5%
>20%
<5%
0.7
0.182
0.289
0.163
0.311
>20%
>2%
>20%
2%
0.8
0.200
0.336
0.218
0.352
>20%
<1%
>20%
<1%
0.9
0.246
0.372
0.268
0.406
20%
<1%
<10%
<1%
o
random e
0.181
0.277
0.153
0.294
>20%
<5%
>20%
<5%
K-S statistic given with
approximate
probability.
Acceptance
level is 20% <
or greater.
Rejection
level is 5% or
less.

115
These selection tests are made primarily with
simulations of galaxy catalogs which provide random model
inputs for the chi distribution tests (tests which determine
the effects of the selection rules on chi (see equation
(2-7)).The simulations can also provide a means of checking
the self-consistency of the samples taken from the Uppsala
and the simulated catalogs. Because of the linear dependence
of the orbital mass on the separation limit which has been
established as a part of the selection criteria, a second
set of tests which are described at the end of this section
are used to check for the selection effects of this limit.
The chi model is of particular interest in the analysis
of bias effects because of the fundamental role that it
plays in the final mass distribution estimate. In order to
test for these effects, however, simulated galaxy catalogs
must be created to generate galaxies with specific physical
characteristics. A comparison between the x values before
and after the selection process can then be made with the
simulated catalog entries. The physical characteristics of
the galaxies used in this study (magnitude, diameter, and
spatial distribution) are compiled from the Uppsala catalog,
and then used to produce the simulated catalogs. Random
values taken from the compiled distributions for the
luminosities and diameters, and from chosen separation
parameters for eccentricity and semi-major axis, determine
the makeup of the simulated galaxy population before, and
after, the selection process.

116
An important test for self-consistency in the catalog
populations comes from a comparison of the magnitude
distributions of the galaxies satisfying the preliminary
selection criteria (i.e. for the primary galaxy in the small
group study) from both the simulated and actual catalogs.
The distribution of magnitudes should be the same for both
catalogs if galaxy selections from the simulated catalog are
representative of either sample. In both the binary study of
van Moorsel and this small group study, the catalog
simulations were made using the Schechter relation for the
luminosity function (Schechter, 1976) which is written as
with L representing the selected random luminosity, L* the
characteristic luminosity, and (L) the luminosity
distribution. The binary and group simulations both use the
Turner and Gott values (Turner and Gott, 1976) of a=-0.83
and M*=-19.1 (the absolute magnitude corresponding to L*)
Results of a comparison between the simulated and
actual magnitude distributions, using the selection rules
established for this study, are similar to those of the van
Moorsel tests, except for the abrupt termination of the two
distributions at 13m which is shown in Figure 6.1. The
obvious truncation for both of these magnitude distributions
can be attributed to the magnitude limit used in the group
selection process. The simulated distribution (bold) shows

117
little apparent difference from the Uppsala catalog
distribution of galaxies satisfying the selection criteria.
The magnitude distributions for the groups are in better
agreement than those for the binary galaxies. As van Moorsel
suggests, this is likley to be attributed to the fainter
magnitudes which are allowed in the binary samples, and the
innacuracy of the luminosity function in representating the
fainter objects.
If a background density of objects is used to calculate
a probability of bound membership, as van Moorsel does, the
representation of the fainter objects in the catalog
(completeness) could have a noticeable effect on the
selection simulations. Van Moorsel uses a Poisson
distribution for each of 11 magnitude intervals to determine
the probability that a pair is bound. Since most of the
selected binary galaxies were larger than the 1' diameter
limit with a brightness greater than the 15m limit (for
galaxies less than 1' in diameter) the fainter objects did
not influence the preliminary pair selection directly, but
did affect the final selection which was based on the
probability of membership. The small group selection
simulations were not noticeably affected by the faint
members included in the Uppsala Catalog. Simulations of
group selection which were similar to the "tenth nearest
neighbor" calculations of van Moorsel did not reduce the
number of selected groups owing to the isolation criteria
established for the small group selection.

5 10 15 20
APPARENT MAGNITUDE DISTRTPt ITTON
Figure 6.1 Distribution of the Uppsala primary galaxy
magnitudes and the simulated (bold) primary
magnitudes.

119
The effects of the unknown separation parameters on the
biased and unbiased chi distribution were tested by van
Moorsel by specifying various semi-major axes and
eccentricities for the simulated pairs. The chi values (eg.
(2-7)) were calculated from simulated galaxy pairs and then
accumulated in an unbiased distribution. If the various chi
parameters allowed the simulated pair to satisfy the
selection criteria the chi values were then accumulated in a
biased distribution. Verification of van Moorsel tests was
accomplished by using a Gaussian distribution for each of
the 11 magnitude intervals instead of a least squares
exponential fit. However, the same weak or nonexistent
dependence of the biased chi distribution on semi-major axis
and eccentricity was noted. These tests of chi were then
applied to the group selection criteria. The results
indicate equivalent distributions for the biased and
unbiased chi values. Differences in the mean of these biased
and unbiased chi values were on the order of 5% to 20% for
both the verification tests and the group tests.
The effect of the projected separation as a selection
bias of Xqj-,3 is important, and can be tested by several
methods. The first test is accomplished by calculating the
linear correlation coeffecient for the x0b values and
projected separation of each primary-satellite pair. This
should provide a reasonable estimate of the dependence of
the projected separation limits on the observed chi values,

120
and indicate any possible problems with the somewhat
artifical separation limit on the orbital masses. The
coefficient of correlation for the 23 pairs used in this
study is 0.46 (see Table 3.2 for Xot)S and separation values)
and 0.32 if NGC 205 is not included. This is not considered
significant since the square of the correletion coefficient,
r, is a measure of the percentage of correlated values, and
O
1-r is a measure of the uncorrelated values. A Z test
applied to these data produces a 2.2 o value for the
correlation in Z (which assumes a normal distribution) and
an expected range of r (assumed to be a good measure of
p) as 0.04 significant at 1% and is not sufficiently well bounded to be
considered significant. The correlation of X0j-,s with
relative separation has a correlation coeffecient of 0.47
and, by the same arguments, is not significanlty correlated
with separation. The same calculations without the M31-NGC
205 pair results in a correlation coeffecient of 0.46.
A second test for bias associated with the separation
criteria is shown in Figure 6.2. The number of satellites
decreases with an increase in satellite separation. This
relationship indicates that the separation values for any
possible remaining satellites are probably not significant.
However, this conclusion is complicated by the decreasing
number of larger separations for random orbits. This would
not be insignificant if the distribution were flat or

121
increasing for increasing separation, however, implying that
the arbitrary separation cut-off value was chosen too small
to include most satellites.

1Y6 N
Figure 6.2
Small group satellite separation distribution.

CHAPTER VII
DISCUSSION
Halo Model
A second model is introduced at this point to account
for the observed chi values beyond the chi model maximum of
2. Simulations using n-body integrations with three
satellites and a logarithmic potential for the halo mass
extending to a specified radius are used to develop a halo
model distribution for comparison with the xqks
distribution. The radial extent of the halo is varied
between 20 and 100 Kpc from the center of the primary
galaxy. The range of the halo mass is from 1 to 10 times the
primary mass. The potential of the halo mass reverts to a
1/r potential outside of the maximum halo radius in order to
O
simulate a 1/r force and to reduce the large 1/r force on
the satellites at large separations. The tests on resulting
simulations are shown in Table 7.1, with the K-S stastic and
associated probability. The model values which are
considered acceptable (as a model) are shown by a
probability at or above 20%, while the model values with
probabilities below 5% are rejected.
123

124
TABLE 7.1
K-S Tests for Halo Model
Radius (kpc)
Mass
(x primary)
20
40
60
80
100
1
0.166
>20%
0.148
>20%
0.137
>20%
0.154
>20%
0.240
>10%
2
0.419
<1%
0.158
>20%
0.209
>20%
0.191
>20%
-
3
0.183
>20%
0.230
<20%
0.370
<1%
-
-
4
0.165
>20%
0.170
>20%
-
-
5
0.161
>20%
-
-
-
“
6
0.085
>20%
-
-
-
-
7
0.163
>20%
-
“
-
-
8
“
The K-S stastictic is given for the 23 observed data
when normalized to the simulated distribution (unity).
The probability of the statistic is shown as a percentage.

125
Simulations
N-body simulations for three different mass
configurations of small galaxy groups were performed with an
integration code based on the interpolation method of
Aarseth (1976). These simulations were made on a VAX 11/750
using double precision code with an exponential range limit
*)Q _ O Q
of approximately 10 to 10 . Each group simulation was
given from 1 to 5 satellites, each with a point mass
potential including the primary galaxy. A softening
parameter was not used in the potential which required more
simulations to accumulate the same number of x values due
to the increase in the number of escaping members. An
integration was halted if the separation between the
satellites was less than 1 Kpc, if the separation between a
satellite and the primary galaxy was less than 10 Kpc, or if
an escape occurred. xot)S values were accumulated for the
simulations with the same number of satellites with a
maximum integration period of 2x1o11 years (approxiamtely 10
Hubble times). If, however, a close approach or escape
occurred after 2xl010 years, the x0jjS values were kept for
the preceeding measurements (taken at the approximate
crossing period of the innermost satellite) and the
integration was halted.
With the assistance of Dr. Haywood Smith at the
University of Florida, initial tests of accuracy of the
source code were made with two body simulations, and with

126
tests of the accumulated error in total energy incurred
after integrations of 5x1o11 years. For simulations with 2
to 4 members and without a halo mass the typical difference
in the total energy between the beginning and the final
integrations were from 0.2% to 1.5%, the greatest difference
was as large as 6%. The code was further tested for
pathological behavior by increasing and decreasing the time
step increment (a variable step size based on the
acceleration derivitaves is used in the Aarseth
interpolations, although a scale factor can be used to
adjust for optimum time step size) until the integrations
became unstable. The accumulated roundoff error in the
decreased time step size produced relatively constant energy
differences to approximately l/20th of the optimum step
size, when the integrations exceeded 30 hours of cpu time.
The truncation error associated with increasing step size
produced an increase in the energy difference values with
approximately 10 times the optimum step size. At step sizes
this large or larger the integrations would encounter fatal
errors.
The first type of simulations were made using a point
mass primary galaxy of 95% of the total group mass. The
accumulated x0bs values for 2 to 5 satellite integrations
show no significant difference between the N(x0bs)
O
distribution and the N(x) distribution for random e
eccentricity values as shown in Figures 7.3a and 7.3 b.

Chi distribution
127
CHt
CHI DISTRIBUTION
Figure 7.3 The distribution of observed chi values
are shown with a primary mass of 95% total group
mass. The chi model distribution is shown for comparison
using a random eccentricity squared (bold). Both
distributions are normalized to the same area.
a) Simulation with 3 satellites.
b) Simulation with 5 satellites.

128
The second set of simulations were made with groups of
point mass galaxies with approximately equal masses which
consisted of from 3 to 6 members per group. The accumulated
XQbs distribution was the same as the x distribution for
these simulations, provided that the velocity reference was
fixed. Velocity difference values used to compute x0£S which
are computed from the velocity difference between members
produces a distribution of X0bs which is substantially
different from the chi model x and with values greater than
1 + e. This distribution difference is also shown in van
Moorsel's simulations with 4 galaxies. The difference in the
chi distributions that occurs from the different velocity
references occurs in the projected mass expression for the
equivalent masses (eq. (2.10)) which measures a potential
and kinetic energy sum between members and not from a fixed
reference. These simulations are shown in Figures 7.4a and
7.4b.
The third set of simulations was made for the halo mass
model in which an exterior mass was added to the primary
galaxy mass using a logarithmic potential. The assumed
isothermal character of the halo mass is described by a
relationship between density and separatiion as
R
M(R) = J p(r)r dV.
0
(7-1)

129
CHI DISTRIBUTION
CHI
CHI
Figure 7.4 The distribution for chi observed
values is shown for 4 orbiting masses of equivalent
mass, a) The velocity reference for calculating chi
observed is taken from each member with respect to
each other member, b) The same simulations with
the barycenter as the velocity reference.
ruL

130
where p(r) is the density, and dV is the volume element.
Hence, the potential and force are described as,
M^Idr
oc log r
(7-2)
F(r) = -grad Vp
GmM(r)
r
(7-3)
The disk mass used for these simulations was the
primary mass plus the halo mass interior to the disk radius.
This is expressed as
R
M = M + M
aisk primary naloR.
di sk
halo
(7-4)
The halo model was simulated with a mass from 1 to 8
times the primary mass and with a maximum radius of 20 to
100 Kpc. The simulations were continued for a minimum of
2x10^® years and a maximum of 2x10^ years. N-body
simulations of the halo model show increasing occurances of
escaping members or eccentric orbits allowing the satellites
within 10 Kpc of the primary galaxy. This occurs for both
larger halo radii and for larger masses because of the
greater influence of the logarithmic force at larger
separations. These more massive ( > 5 Mprimary or larger)
halos failed to produce chi values within 2x10^ years

131
integration, indicating an unbound nature for these halo
models. Successful simulations were limited to halos of
approximately 4 primary masses at 40 Kpc radii with almost
linearly increasing radii allowed with decreasing halo mass,
as shown in Table 7.1. Simulations with halo masses of up to
7 primary masses were successful if the maximum radius of
the halo was restricted to 20 Kpc. The xobs distribution is
shown for four of these simulations in Figures 7.5 and 7.6.
Since bound orbits are assumed, the flat rotation curves can
limit the halo radius and mass found by these simulations.
If a 4 primary mass halo were assumed, then the mass which
scales as the radial distance, implies a halo radius of 4
times the disk radius. This was the maximum halo mass and
radius allowed by the n-body simulations, and is assumed to
be the maximum halo values acceptable using the K-S tests
and the halo simulations.
Membership
The importance of the galaxy group members being bound
cannot be overemphasized since the assumptions which are
used to infer the final mass distribution from the masured
orbital masses are invalid without bound orbits. Efforts to
avoid non-members in the selected groups are made by
applying specific selection criteria to the list of galaxies
in the Uppsala catalog and to each of the prospective
supplemental groups. However, there is still a need for a

132
CHI DISTRIBUTION
CHI DISTRIBUTION
Figure 7.5 Halo simulation with a halo mass
of 1 primary mass with a random eccentricity
model distribution for comparison. The area
under both curves are normalized the the
same area. The total number of chi obs
values in each simulation plot is approximately
1000. a) Simulation with the halo mass
extending to 20 Kpc. b) The same simulations
with a halo radius of 100 Kpc.

CHI DISTRIBUTION
CHI DISTRIBUTION
Figure 7.6 Simulation of halo mass equivalent to 3
times primary mass shown with the ez model
normalized to the same area, a) The halo radius
is 20 kpc for this simulation, b) The same
simulation with a 50 Kpc halo radius.

134
more definitive test to establish the bound membership of
the satellites used in this study. An evaluation of the
bound membership of a sample of galaxies is described in a
recent paper by Valtonen and Byrd (1986). The basic argument
in the evaluation is that, within a given volume, there
should be a symmetric distribution of red and blue shifted
members of bound pairs or groups after adjusting for the
larger volume at greater distance. An excess of red or of
blue shifted members significantly above the calculated
excess could be an indication of optical members.
The "redshift" excess described by Byrd and Valtonen is
calculated as a number density of galaxies for a specified
sampling region. This expected excess is calculated by
integrating the luminosity function which characterizes the
galaxies from a specified catalog. The number density can be
computed from the following equation:
n(L)dLdV = n_L_adLdV. (7-1)
Here, L is the absolute luminosity of the galaxy, n(L) is
the number density of galaxies in the luminosity interval L
and L+dL within volume interval dV. The exponent a is
approximately 1.75 for the faint galaxies, and 3.0 for the
bright galaxies (Abell, 1975). The resulting expression for
the theoretical redshift excess, Z, is
l+p(5“2a)-2x(5_2a)
l_p(5-2a)
Z
(7-2)

135
where p is r-^ / ^ and x is rQ / r2, r-^ is the minimum
radial separation in the sample volume, ^ is the maximum
radial separation, and rQ is the distance to the primary.
The actual redshift excess (which is called a redshift
excess for both redshift or blueshift excesses) which is
significantly above the expected excesses indicates that
optical companions may be present in the sample. This actual
excess is calculated using the formula
N4 — N~
N+ + N~
(7-3)
where Z_ is the actual excess, N+ is the number of red
d
shifted objects, and N“ is the number of blue shifted
obj ects.
The average distance of the 23 primary-satellite pairs
used in this study is approximately 13 Mpc with the number
of red shifted pairs (N+) equal to 15 and the number of blue
shifted pairs (N“) equal to 8. The luminosity model predicts
an excess (Z) of -0.25 for the bright pairs, and 0.04 for
the faint pairs if all the pairs are optical and not bound.
The actual excess, Z_ for the bright pairs (sum of
luminosities is less than 22m) is 1.0 because no bright blue
shifted pairs were detected, while the actual excess for the
faint pairs is 0.33±.23. True bound pairs should give an
excess of 0.0 for a relaitvely random sampling. The actual
excess could indicate optical pairs, however, the relatively

136
small sample size and large standard deviation is an
indication that the sample is not large enough to assess the
bound membership with reasonable certainty.
The van Moorsel sample contains binary pairs, with an
approximate optical model excess of -0.12 for the 9 bright
members and 0.03 for the 7 faint members with an average
distance to the pairs assumed to be 25 Mpc. The number of
red shifted galaxies is 9 for the faint galaxies and 0 for
the bright galaxies. The corresponding number of blue
shifted galaxies are 7 and 0 respectively. This gives an
actual excess of 0.13±0.25 for the faint galaxy pairs and 0
for the actual bright excess. The small number uncertainty
is even more pronounced for this binary sample where the
actual excess is within la of the theoretical bound excess
and the optical excess. Therefore, the certainty in the
bound membership of this sample is no greater than that in
the group sample, although the possibility of bound
membership appears greater.

CHAPTER VIII
CONCLUSIONS
A mass distribution could not be inferred from a
comparison of the X0bs distribution with the x model because
of the XQbs values greater than 2. In order account for the
large x0bs values, a halo mass model was developed using n-
body simulations of the halo mass surrounding the primary
galaxy at specified maximum radii. These simulations were
used to find the the acceptable limits of the halo mass and
radius assuming bound orbits for the satellites within a
specified integration time. The stastical tests which were
used to compare the simulated distributions, x0bs(halo)' and
the observed distribution, X0bs/ although not actually
correct because of the discontinuous character of the
simulated model halo data, are acceptable at the 20%
confidence level for most of the halo simulations. The
limitation on the halo parameters arise from the success or
failure of the simulations to maintain bound orbits for a
minimum of 2xl010 years in order to provide a realistic time
frame to measure the effects of the halo model parameters.
The logarithmic potential used to describe the halo model
mass surrounding a point mass galaxy is suggested by the
137

138
flat rotation curves of most spiral and barred spiral
galaxies. The allowed halo radii are limited by the observed
rotation curves since the total galaxy mass is assumed to
increase as the radial distance from the center. The halo
mass limit was determined by the halo simulations to be on
the order of 4 times the primary mass at a maximum radius of
40 Kpc for a 10 Kpc radius galaxy disk.
Alternately, the xQbs were greater than 2 were
associated with the more massive, peculiar galaxies,
indicating that dark matter may be associated only with the
largest galaxies or that the peculiar galaxies do not
represent the simple models presented here. Certainly not
all spiral or barred spiral galaxies suggest exterior halos
as in the case of NGC 3992 (Gottesman et al., 1984).
The distribution of x0bs values for the van Moorsel
data were tested with the x model distributions by the K-S
statistic. Tests were also made on the extended mass model
with circular orbits (the mass of each galaxy extending to
halfway between the galaxy pair). These tests of a poor fit
were confirmed using the research hypothesis at the 5%
level. The galaxy group data were also tested with the
extended mass model of van Moorsel and confirmed at 5%
significance. The significant difference in the statistical
tests between the data from the binary samples and the
group samples tells us that the data are not the same, but
does not tell us how the distributions differ. The careful

139
sample selection of this study (although a more conservative
rejection of the peculiar groups could eliminate nearly all
of the X0bs values greater than 2) has several important
differences from the sample selection procedures of previous
binary studies and also has several similarities. However,
the halo model proposed in this investigation can reconcile
the difference in the binary and small group distributions
since the average radial separation in the van Moorsel
sample is less than twice the halo radius maximum. The
circular model of van Moorsel with the truncated logarithmic
potential halo can produce the observed distribution for
both samples, although the halo may be limited to the
largest galaxies measured.

APPENDIX
INTERFEROMETRY AND IMAGING
It follows from the nature of electrmagnetic radiation
that radio frequency emissions measured by an radio
interferometer exhibit the same characteristics the same as
optical interferometer measurements, but with geometries and
wavelengths scaled to radio frequencies. However, the
approximations which are available to radio interferometry,
owing to longer wavelengths and typically large source
distances, can simplify the methods used for imaging. The
assumed spatial incoherence and the nearly flat wavefronts
from distant radio sources allow a relatively simple
expression to be written for the electric field at two
separate positions. The positions for measuring the field
are assumed to be spaced more closely than the source
distance. This spatial coherence function or autocorrelation
function can be measured by a two element interferometer and
expressed according to the formula given by Clark as (Clark,
1985)
V(u,v) = jj Iv(l,m)e"27rl(ul+Vm)dl dm. (A-l)
In this equation, the two position vectors, u and v, are
perpendicular to the direction of the observed source, with
140

141
positive u in the direction of North and positive v in the
direction of West. The spatial coordinates of the source
are represented by the vector components in the same
coordinate system, but are denoted as 1 and m. Also, the
observed intensity is given as Iv, and the measured
coherence value which is usually the voltage from the
recieved source signal is denoted as V(u,v).
Since actual measurements are made with an antenna
response pattern (beam) which has a finite width the
coherence function must be expressed as the convolution of
the antenna pattern and the observed intensity. If Au
represents the antenna response pattern, the coherence
(cross correlation) function can be written as
Vv(u,v) = JJ Av(l,m)Iv(i,m)e"27T1(ul+mv)dl dm. (A-2)
This cross correlation function represents the response of
the received source emission and appears in the form of a
Fourier transform. The inverse Fourier transform can be used
to produce an image of the observed intensity, Iv, by
several inversion techniques.
The signal received at a particular point with respect
to the source is measured by the two antenna elements and is
described by the visibility function. In image space, the
received intensity, convolved with the beam response, can be
obtained by a Fourier transform of the visibility value.
Seperate visibility values can be produced by varying the u

142
and/or v values (the antenna baseline projections as seen
from the source) and transformed to produce a likeness of
the source of emission. The rotation of the Earth can be
used to provide a range of u and v values which correspond
to a range of Fourier transformed positions and intensities
in the image (map) plane. Since the visibility is a function
of the separation only and not a function of position,
redundant u-v plane coverage is possible with an array of
antennae. Hence, the spatial configuration of the antennae
plays an important role in producing optimum coverage of
spatial measurements for the source image reconstruction.
The visibilities measured by the receiving system are
produced from the cross correlation of the antenna voltage
values for each pair of antennas used. A visibility value,
which is actually cross correlated power at this point, is
produced by each of the N(N-l)/2 independent antenna pairs.
These cross correlation calculations are produced by an
integration over a specified time period for the multipled
values of the antenna voltages and a selected band of
frequencies. The spectral cross correlations used in HI line
emission measurements are produced with a network of digital
delay elements acting as narrowband filters. The digital
circuitry produces separate phase delays for each of the
input values which are then multiplied and integrated at
each frequency band. Thus, correlated values are available
for each of the frequency bands selected for spectral line

143
observations. In the case of continuum observations, the
narrowband correlations can be averaged together. The
complex cross correlation values produced by the correlator
can now be calibrated in order to produce the visibilities
used for astronomical imaging.
The calibration process is invoked to provide each of
the correlated output values with a magnitude and phase
which accurately corresponds to the true received source
emission. Calibration of the correlated amplitude is
accomplished by observing a strong, unresolved source whose
flux strength is well known. Phase calibration of the
correlated values is accomplished by observing a strong,
unresolved source close to the observed object. This
secondary calibration source is used to provide a stable
phase reference for the program object observations. The
source which is used to calibrate the visibility amplitudes
is chosen to be strong emission source which does not
exhibit significant variations in flux strength or spectral
index. In this study, 3c286 was used as the primary
(amplitude) calibration source for the HI observations made
at the VLA. This amplitude calibration process ultimately
provides associated gain values for each antenna which is
based on the response of the individual receiver systems to
the primary calibrator. The observations for amplitude
calibration were made at the beginning and end of the
observations of the program object (galaxy group). The

144
calibration observations were scheduled for an integration
period to attain an rms noise significantly less than the
expected flux from the program object.
To calculate the rms noise for an interferometer pair
which is assumed to have the same receiver frequency
response, system temperature, and antenna characteristics,
the correlator output is given as (Crane and Napier, 1985)
= GkSKri dv (A-3)
o 'c v '
where is the D.C. voltage level at the correlator
output, G is the gain at this stage, k is Boltzman's
constant, S is the flux density (assumed to be a point
source), dv is the bandwidth, ti is the correlator
effeciency, and K is defined as
(A-4)
Here, ti represents the antenna aperture effeciency and A is
the geometrical area of the antenna. The rms power
fluctuation, Pnoise/ is given according to
T^
P . =G2k2-^(S2K2 + SKT + 1^-). (A-5)
noise dt sys 2
In this equation, dv is the bandwidth and dt is the
integration time.

145
The rms noise of the flux density is proportional to
the square root of the noise power and inversely
proportional to the correlator scaling value and is
expressed as
AS =
VP •
noise
dV /dS
o
ST
n Vdvdt
c
(S'
'sys
K
+
sys j 1/2
2K
(A-6)
for a source smaller than the beam width. The a weak source
approximation is appropriate for these observations, where S
<< Tsys/K* Hence'
AS =
V 2kT
sys
Ticr|AV dvdt
(A-7)
An array of N antennas has N(N-l)/2 baseline pairs and so,
the rms sensitivity is given by
2kT
AS =
' sys
TicnAVN(N-l)dvdt
(A-8)
where the frequency band is expressed in KHz and the
integration time is in hours. For Hanning smoothing, the
noise is reduced by a factor of approximately V2 because of
the averaging of two separate receiver systems (IF's). If
single dish observations are made the same equation applies,

146
but with improved sensitivity over the interferometer pair
because the interferometer does not use the autocorrelation
of the signals from each antenna separately. For the single
dish we find
kT
AS = (A-9 )
tiAV dvdt
The characteristics of the VLA 25 m antennas are given
for the 21 cm band in the spectral line mode as
ti=.51, ti = .77, A= 490m2' and T =25°K.
1 ‘c sys
The flux sensitivity expression can then be written more
simply as
AS = 450mJy (A-10)
VN(N-l)dvdt
For every six hours of program galaxy observations at
the expected flux levels of 70-200 mJy, approximately 20
minutes were spent on the primary calibration source. The
calibrated flux for 3c286 was 12.7 Jy for these 1.41 GHz
spectral observations.
Phase calibration is provided by observations of the
secondary calibration source. The phase angles between each
antenna pair are determined by the secondary calibration
observations in order to provide a stable phase reference
for the visibility values from the program observations. The

147
phase calibration source is selected to be near the program
object in order to measure the effects of the same
atmospheric conditions and provide smaller closure errors.
The primary calibration source can also be used for
secondary calibration, if the primary calibration source is
sufficiently close to the program object to provide accurate
phase values for the program observations.
The secondary calibration observations were made using
1225+368, an unresolved source near three of the four galaxy
groups observed at the VLA. The source 1219+285 was used as
the secondary calibrator for the NGC 4303 group
observations. The calibration observations of the secondary
reference were made approximately once per hour because of
the relative stability of the 21 cm receiver system and the
2 Jy flux level of the calibrators. The secondary
calibration observation time was based on the same
observation and flux criteria as that used for the primary
calibration observations. Approximately 8 minutes of
secondary calibration observation were dedicated per hour of
program observation time. The flux strength of 1225+368 and
1219+285 were 2.1 Jy and 1.6 Jy respectively.
The amplitudes and phases of the calibraton
observations (visibilities) are now used for a linear least
squares fit to provide a representative gain and relative
phase values for each antenna. Obviously discrepant values
for the calibration gain or phase values can be removed in

148
the data editing process. For VLA data, each visibility
contains a flag record which can be used to accept or reject
the individual visibilities at a later stage in the data
reduction. The calibration data which exhibited large
variance in the least squares solution can be removed with
this procedure and the fit can be recalculated. The program
object visibilities between the intervals which were flagged
in the calibration data are also flagged and then checked
for consistency by the same least squares procedure. The
consistency of the visibility data can also be verified by
plotting the amplitude versus observation time for each
visibility in order to identify interference, antenna gain
or phase problems, or other irregularities. These
visibilities can now be used for mapping the emission
features using Fourier transforms.
Since the u-v coordinates which are measured from the
projected baseline in wavelengths and the arc distance
coordinates in image space, 1 and m, are both related to the
frequency of observation, the Fourier transformation of the
visibilities used to make an image can be made directly with
equation (A-2). However, a 10 hour observation with 20
antennas and an integration period on the order of 30
seconds can produce approximately 10~* visibilities. A data
base of this size suggests the use of Fast Fourier Transform
(FFT) technique over the direct Fourier transform,
especially for images larger than 256x256 pixels. For an NxN

149
O
image, execution for the FFT scales as N log2N while the
execution time for the direct transform scales as N^.
Nonetheless, the FFT's require an interpolation of the data
for a uniform grid spacing which can introduce image
artifacts such as aliasing (reflecting source emissions
outside the mapped region into the image area).
After interpolation by convolution, the data are
resampled at the uniformly spaced intervals required by the
FFT. The problem of aliasing can be avoided by mapping a
larger region than is required for the actual image in order
to check for strong sources in the adjoining areas. Often
this is not a serious problem since the sources outside the
intended image region are normally supressed by the
convolution function used. The convolution function is
chosen so as to drop off rapidly at the image boundary. The
convolution function used in the VLA imaging steps for these
observations was a truncated spheroidal function (see Sramek
and Schwab, 1985).
Before data can be transformed into the emission source
image, the pixel (cell) size must be specified along with
the weighting and tapering parameters. The pixel size and
number represents the image scale but, more importantly,
should represent the visibility values from the largest to
the smallest u-v separations. The product of pixels in one
dimension with the pixel size should be greater than or
equal to the inverse of the minimum corresponding u-v value

150
so as to represent the smallest u-v separations (largest
emission features). Also, in order to represent the
unresolved sources accurately, the pixel size must be less
than or equal to the inverse of the maximum u-v separations.
From a sampling argument, this must be the inverse of twice
the maximum u-v values. This sampling requirement can be
visualized by comparing the synthesized beam size to the
pixel size i.e. the synthesized beam must contain at least
two pixels to satisfy the Nyquist sampling requirement.
Choosing a pixel size too small (oversampling) can produce
an inaccurate representation of the visibilities in each
pixel, while the choice of a pixel that is too large
(undersampling) can produce intensities that are not
representative of unresolved sources.
Weighting the visibility data in each pixel determines
the significance of each of the visibilities within that
cell. Weighting can be chosen as either natural weighting or
uniform weighting. Natural weighting gives the greatest
detection sensitivity by assigning each visibility a weight
of unity. Uniform weighting, on the other hand, assigns each
visibility in a specified cell a weight which is given as
the inverse the number of visibilities in that cell. In
general, this is a less sensitive weighting scheme because
the smaller u-v values, corresponding to smaller baselines,
are weighted equally with the larger values. For natural
weighting, the higher resolution data is downweighted,

151
producing a somewhat smoother and brighter image. Natural
weighting would be the optimum choice for these detection
measurements, but several test maps which were made to
examine the effects of such a weighting scheme produced
anomalous features which could not removed by the cleaning
process (sidelobe removal). This effect was more noticable
in the observations at lower declinations, where the u-v
coverage is less evenly distributed. The increase in
sensitivity associated with natural weighting for images
consisting of 512x512 pixels is not as obvious as in smaller
maps produced with the same weighting scheme. Therefore,
uniform weighting was chosen for all four of the galaxy
group map sets.
The tapering parameters of the uniformly weighted data
determine the resolution of the image by reducing the
weighting of the data at the outer edges of the u-v coverage
(Sramek, Schwab, 1985). This not only reduces the sidelobe
response of the synthesized beam, but it also widens the
beam. Tapering the visibility data also reduces the
sensitivity by removing the larger spatial frequencies (u-v
values), producing a reduced signal to noise.
Combining visibility data from several observations or
from separate array configurations can be time consuming if
the visibilities for each data set must be sorted
separately. A time savings in image construction of a factor
of from six to ten is possible for map averaging with the

152
VLA data reduction equipment and software because of a
dedicated griding system at the VLA (Pipeine) which was
available for this averaging technique (van Gorkum, private
communication). The equivalance of a convolution and a
Fourier transformed product allows these data sets to be
combined in the image plane. A weighting scheme for the
visibility data can be duplicated for two or more sets of
visibilities by first making a u-v coverage map (a map of
the u-v plane coverage by the data) for each visibility data
sets using natural weighting. The natural weighting will
retain the individual u-v values. Each of these visibilities
is mapped on the u-v plane and reproduces the u-v coverage
of the baselines and orientations for the observations.
These u-v coverage maps are combined, by averaging, while
the visibilities are added in the u-v plane. The gridding
(interpolation) and mapping processes can then use the u-v
coverage map to weight the visibilities in each of the
separate data sets. The images are then produced with
uniform weighting, without tapering, in order to retain the
data from all baselines. All map sets which are produced
with the combined u-v coverage maps have the same weighting
and gridding interpolation and hence, can be averaged in the
image plane to produce the same images as those which are
made by combining the visibility data sets with uniform
weighting and without tapering.

153
The maps made by this averaging procedure are convolved
with a Gaussian beam to obtain the same resolution, signal
to noise ratio, and sensitivity as those produced by summing
the visibilities, provided the Gaussian beam shape was the
same or similar to the uniformly weighted, tapered maps. The
convolution of the map with the beam is equivalent to a
multiplication of the data by a tapering function with the
same resolution as that of the convolving beam. The maps of
the galaxy groups associated with NGC 3893 and UGC 7089 were
made with this process. Because of relatively high continuum
emission in the regions around the groups associated with
NGC 4258 and NGC 4303, the imaging was completed after the
stronger continuum was removed from the visibility data
bases.
The resulting channel maps contain both continuum
emission and spectral line emission. The continuum emission
can be removed efficiently by averaging the channel maps
which have no line emission, typically near the end
channels. The resulting averaged continuum map is used to
subtract out the continuum from the line emission channels.
For instances of strong continuum, removal of this emission
prior to mapping is useful in reducing large sidelobe
effects from the continuum emission. This is accomplished by
making maps of the continuum channels and then combining
these maps to produce an averaged continuum map. The
continuum sources are then cleaned to approximately ten

154
times the rms noise, producing a list of the visibilities
responsible for the strong continuum levels. These strong
continuum visibilities are then subtracted from the original
visibility data base and maps are produced from the
remaining visibilities. The mapping process produces a set
of images without the strong continuum, but which still
requires removal of the lower level continuum emission. The
same line free channels are again averaged and then
subtracted from the remaining channel maps. The removal of
continuum for the NGC 4258 and NGC 4303 images was
accomplished with this technique in order to reduce the
effects of moderate continuum emission in the field of view.
The sidelobe features of the channel maps which contain
significant line emission features must be removed or
reduced. This sidelobe removal and a smoothing/interpolation
process were made with an algorithm which was available at
NRAO (Clark, 1980; Cornwell, 1985). The "clean" procedure
iteratively selects and removes the strongest feature in an
image until a specified level is reached. The source
emission components are modeled as point sources which can
then be reconstructed as sidelobe-free "clean" beams,
normally Gaussian in shape. The removal of the peak features
as a "dirty" beam structure also removes the sidelobe
features of the stronger emissions. The reconstruction of
the image, a convolution of the clean beam with each of the
peak positions, is completed with the addition of the

155
residual values in the original image. This convolution
also serves to smooth the image.
The spectral maps for the four galaxy groups shown in
Chapter III were cleaned to the 3a level in order to remove
the sidelobe features as completely as possible without
operating on the noise. Continuum maps were also cleaned to
the 3a level (measured on the continuum map) and plotted for
each group in order to show the relative positions and
strengths of the background and primary galaxy continuum
emissions.

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BIOGRAPHICAL SKETCH
Lance Erickson was born in Denver, Colorado on June 26,
1946. Mr. Erickson graduated from Wheatridge High School in
1965. Soon afterwards he was drafted into the U. S. Army and
served as a Lieutenant in the Vietnam War. After his
discharge in 1969, Mr. Erickson proceeded with his interest
in flying and in 1974 obtained his pilot's and flight
instructor's licenses. At this time, he chose to continue
his formal education at Sonoma State University where he
earned a B.S. in physics in 1980.
Mr. Erickson began his graduate studies at the
University of Florida, pursuing his long-standing interests
in Extra-galactic radio astronomy. On August 8, 1987, Lance
Erickson was awarded a PhD in astronomy with a minor in
physics.
159

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
vv, ‘ y.1 -fyt/K
Stephen T. Gottesman, Chairman
Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
tí1 Afwi-vl
ér
Jaiies H. Hunter Ji1. , Cochairman
Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
JáátseL / i mñh. A
Hay^ood'C. Smith Jrij
Associate Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
( Oc—
David B. Tanner
Professor of Physics
This dissertation was submitted to the Graduate Faculty of
the Department of Astronomy in the College of Liberal Arts
and Sciences and to the Graduate School and was accepted as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
August 1987
Dean, Graduate School

UNIVERSE OF FLORIDA
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