Mass determination of selected galaxies from small group statistics

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Mass determination of selected galaxies from small group statistics
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Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
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Includes bibliographical references (leaves 156-158).
Statement of Responsibility:
by Lance Karl Erickson.
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Typescript.
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Vita.

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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


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
*
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**








*o -' *
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40
8 *m* .
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~ ** ** '



.o .* ..
10 *


.


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


48 47 46


I i


i -iI I I I

"I I 1 .
WO tam/3M

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






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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
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o .
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o .*








*
-*







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.' .. .
o "


- I. .. .
., m'mo













**

*~ 's

S
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*-
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*
-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
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o 4 o ,




.0





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o a







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o 574 km/see
,






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/
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o 0~ 5 o st 91ia/


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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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0


a 0
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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


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.' ISMS ki --


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0

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a o 0 0



S-






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-~ 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



-



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-


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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.