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The cores of galaxies in the Coma cluster
Robert Minchin
April, 1997

Contents
1 Introduction 7
1.1 Historical Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7
1.2 This project : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10
2 Data Reduction 11
2.1 Image processing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11
2.1.1 The images : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11
2.1.2 Cosmic ray removal : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11
2.2 Ellipse fitting : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12
2.2.1 Galphot � ellipfit : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12
2.2.2 IRAF � ellipse : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13
2.2.3 Checking the ellipse fits : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13
2.2.4 Dusty galaxies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14
2.3 Data calibration : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14
2.4 Other data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15
3 Analysis of data 16
3.1 Graphs from the ellipse fits : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16
3.1.1 Isophote shapes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18
1

3.2 Images : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18
3.3 Power laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19
3.3.1 Fitting power laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19
3.3.2 Information from the fit : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19
3.3.3 Graphs from the power�law fits : : : : : : : : : : : : : : : : : : : : : : : : : 21
3.3.4 Statistical analysis of power�law fits : : : : : : : : : : : : : : : : : : : : : : 21
4 Results 22
4.1 The Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22
4.2 Differences between the two regions : : : : : : : : : : : : : : : : : : : : : : : : : : : 22
4.3 M v vs fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27
4.3.1 Central region : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27
4.3.2 Halo region : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29
4.4 r b vs fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29
4.5 r b & r lim vs M v : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29
4.6 Break radius vs Effective radius : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32
4.7 Comparison with previous results : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34
4.7.1 M v vs fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34
4.7.2 r b vs fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34
4.8 Diskiness : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34
4.9 Defining a Core : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 36
5 Conclusions 39
5.1 M v vs fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39
5.2 fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39
5.3 M v : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39
2

5.4 r b vs fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40
5.5 r b & r lim vs M v : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40
5.6 Diskiness : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40
5.7 Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40
A Errors in the ellipses 42
B Notes on Galaxies 44
C Differentials of the Nuker Law 49
D Computer programmes 51
D.1 Numerical Recipes routines : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51
D.2 My routines : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51
D.2.1 Project.c : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51
D.2.2 Funcs.c : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56
3

List of Tables
4.1 Semi�major axis fits : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24
4.2 Average radius fits : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25
4.3 Other galaxy data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 26
4

List of Figures
1.1 Faber et al results for r b vs M v : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9
3.1 Data from ellipse�fitting to NGC4889 : : : : : : : : : : : : : : : : : : : : : : : : : : 17
3.2 Nuker�law fit to NGC4889 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20
4.1 Data from the average�radius fit : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23
4.2 Distribution of fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28
4.3 Nuker�law fit to NGC4886 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30
4.4 Plots of fl against M v for the central and halo regions : : : : : : : : : : : : : : : : 31
4.5 r b vs r e and m v : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 33
4.6 comparison to Faber et al's data : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35
4.7 cos 4` terms for disky galaxies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37
4.8 Nuker�law fit to NGC4886 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38
5

Abstract
I have analyzed HST images of 45 galaxies selected from the central and halo regions of the Coma
cluster. A nuker�law fit to the profiles has been found and the relationships between the parameters
of the nuker�law and between the total luminosity and isophote shape has been investigated.
Problems were encountered with dust in the galaxies, with half of the galaxies in the sample
suffering some degree of dust pollution. Where this was severe enough to affect the profile, the
galaxy was excluded from the data set.
The differences between the central and halo regions have been investigated. It is predicted
that core�type galaxies will form preferrentially in the central region. This predicted relationship
is not proven to exist although there is some evidence for it. Due to a difference in magnitudes
between the two samples it is necessary for a relationship between fl and luminosity to exist in
order to show preferrential core formation in the central region. It is not possible to say that this
relationship is definately seen in the data.
Comparison of the data with previous results shows that the lack of cores seen in less�luminous
galaxies could be an effect of resolution. This would create an artificial trend of fl with luminosity
and an apparent dichotomy as only the steep outer power�law would be resolved in galaxies with
smaller cores, giving a large value of fl. This trend and the values of fl higher than 0.3 could be a
selection effect. Disky galaxies are seen to be able to form cores and a bright (M v ! �22) galaxy
is seen with no resolved core.
6

Chapter 1
Introduction
1.1 Historical Background
Originally, cores were defined as the areas of constant surface brightness in the centre of the King
(1966) model. This was shown by King (1978) to be a reasonable fit to the central regions of
elliptical galaxies. Later CCD imaging showed that the King model did not describe adaquately
the surface brightness profiles of elliptical galaxies � the central regions were not, to the limit of
resolution, areas of constant surface brightness. The regions now known as cores are regions of low
logarithmic slope seperated from the steeper outer logarithmic slope by a sharp break. The inner
logarithmic slope does not go to zero as predicted by the King model (Lauer et al, 1995) at the
limit of resolution.
Imaging of the cores of elliptical galaxies with the Hubble Space Telescope has allowed them to
be examined at previously unattainable resolutions. The galaxies appear to divide into two types
(Jaffe et al, 1994). Larger galaxies generally have profiles with well defined cores, described by a
double power�law, whilst smaller galaxies can be described by a single power�law and have no well
defined core (Bosch et al, 1994; Kormendy et al, 1994).
7

It is known that larger galaxies are triaxial systems supported by random motions and there�
fore having boxy isophotes while smaller galaxies are rotationally supported with disky isophotes
(Davies et al, 1983). This division into two sorts of elliptical galaxies appears to correlate well
with the division into those with and without unresolved cores (Kormendy et al, 1994; Jaffe et al,
1994).
Lauer et al (1995) fitted a 5�parameter double power�law to the inner profiles of galaxies. This
law, known as the `Nuker' law, fits both core�type and power�law galaxies well.
Ir = I b \Theta 2 fi\Gammafl
ff \Theta
i r b
r
j fl
\Theta
`
1 +
`
r
r b
' ff ' fl\Gammafi
ff
The parameters fitted by this law are the inner logarithmic slope, fl, the outer logarithmic
slope, fi, the sharpness of the break between the two slopes, ff, the radius of the break, r b and
the intensity at the break, I b . In terms of defining a core, fl is the most important as it shows
the slope of the surface�brightness profile as it tends towards zero. It can be seen that for r � r b ,
I(r) / r \Gammafl and for r AE r b , I(r) / r \Gammafi .
The results of fitting this law to a number of elliptical galaxies, and to the bulges of spiral
galaxies, are given in Faber et al (1996). This classifies galaxies as having cores if fl ! 0:3 and
if the break is well�resolved. The claim is also made that there is a divide between the core�type
galaxies and the power�law galaxies, which have fl ? 0:5. The core�type galaxies are large, triaxial
systems with M v ! \Gamma20:5 and the power�law galaxies are small, rotationally supported systems
with M v ? \Gamma22.
Faber et al find that there is a clear division between the core and power�law profiles (see fig
1.1, from Faber et al 1996) which can be seen on the graph of r b vs M v and also as a gap in the
distribution of fl between 0.3 and 0.5. Figure 1.1 also shows the division found into boxy and disky
galaxies.
8

Figure 1.1: Faber et al results for r b vs M v
9

1.2 This project
Bender, Burstein and Faber (1992) proposed that the controlling factor in the evolution of a galaxy
is the amount of gas present during formation and mergers. If there is a large amount of gas present
then most of the energy is dissipated and the galaxy forms a small, rotationally supported (disky)
system. Larger galaxies are formed later by mergers of these smaller galaxies when most of the gas
has formed into stars. These stellar mergers do not dissipate energy and so form triaxial systems.
This theorem is called the gas/stellar continuum (GS continuum). Faber et al (1996) puts forward
this theorem as explaining why disky galaxies show power�laws as the gas in dissipative mergers
is carried towards the centre of the galaxies.
The Coma cluster provides a good test area for this theory as the central region consists of a
dynamically relaxed system where the galaxies have interacted frequently and have little gas and
the halo region consists of relatively gas�rich galaxies which have not yet visited the core and have
had little chance to interact. The cluster is rich enough to be able to form a statistically significant
sample of both regions and projection of HST data from the Virgo cluster has shown that it is
possible to distinguish the two kinds of profiles at the distance of the Coma cluster.
If the GS continuum theorem is correct, it should be possible to see preferential formation
of power�law galaxies in the halo region and preferential formation of core�type galaxies in the
central region. This should result in there being a significant difference between the distributions
of fl for the two samples. I will also examine the relationships seen by Faber et al in their survey
to determine if these are seen in the Coma cluster.
There are 45 galaxies in the sample, 23 from the central region of Coma and 22 from the halo.
In chapters 2 and 3 I explain how the images were reduced and the data analysed. In chapter 4
I present my results and in chapter 5 I give my conclusions. The derivation of the error in the
surfact�brightness profile and notes on the galaxies are given in an appendix.
10

Chapter 2
Data Reduction
2.1 Image processing
2.1.1 The images
The Hubble Space Telescope was used to obtain 2 WFPC/2 images of each of the 45 galaxies
in the sample. The Planetary Camera images were extracted and trimmed from [1:800,1:800] to
[55:798,55:798] in order to remove noise around the bottom and left edges of the plates. The images
were examined and those with suspected dust noted.
2.1.2 Cosmic ray removal
Cosmic ray interference was reduced by combining the two images of each galaxy using the IRAF
routine imcombine with the parameters reject=crreject, hsig=2.5, gain=7. These parameters tell
the routine to look for objects that resemble cosmic rays � sharply defined bright (positive) points
� and remove them if they are more than 2.5 times the expected variation from the value on the
the other image. The gain value of 7 was obtained from the image headers and defines how many
photon counts there are for each ADU.
11

In order to ensure the alignment of the two images prior to combining them, the IRAF routine
center was used to find the centre of each image. From this the offset between the centres in the
pair was found and those galaxies with an offset of less that 0.2 pixels were combined, those with
an offset of greater than 0.1 pixels being noted (see Appendix B).
Four galaxies had offsets outside of this range, these were examined using the IRAF routine X�
register. This routine performs a cross�correlation over a wide area rather than finding the centre.
This routine gave two of the galaxies as having shifts less than 0.1 pixels, these were combined.
The remaining two galaxies were shifted using X�shift prior to being combined (see Appendix B).
2.2 Ellipse fitting
A working notation of naming the galaxies c01 to c45 was introduced with c01 to c23 being in
the core (angular distance from the cluster centre less than 0.7 o � 1 Mpc), as defined in Lucey et
al (1991), and c24 to c45 being in the halo. This notation allowed automated routines to be run
easily and the core and halo galaxies to be easily distinguished. Header files were constructed for
each galaxy containing the proper names, exposure times and other data from literature.
2.2.1 Galphot � ellipfit
The Galphot routines by R Peletier were used to fit ellipses to the galaxies. The routines identified
stars and remaining cosmic rays and masked them out. Problems with this masking, such as
unmasked diffraction spikes, were corrected manually and a manual masking routine was also used
to remove dust lanes. A map of bad pixels was also constructed and added to the mask.
An initial fit was made using the harmfit routine. This fits circular isophotes with harmonics.
This produced a residual image which could be used to check the mask.
The final fit was made using the ellipfit routine. This fits elliptical isophotes for a number of
iterations then fits harmonic terms to these isophotes. The routine creates a residual image after
12

each iteration and uses this to correct itself thus giving it good accuracy. The fit is done using
annuli of 5% of the radius at 10% intervals. Outside 20 pixel radius the fit is done to all the pixels
that are inside the annulus, within this radius the width of the annulus is less than 1 pixel so
interpolation is used to calculate the values.
2.2.2 IRAF � ellipse
The IRAF routine ellipse was used as a check to the Galphot fits. This routine fits by sampling
a number of sectors around annuli and using these to construct ellipses. The routine appears less
reliable than the Galphot routines, as it uses a cruder fitting method, but produced comparable
output.
In order to run this routine, two IRAF scripts were used. The first ran an initial fit over 2 to
60 pixels radius and was seeded using 4 points � the ends of the major an minor axes � found from
an isophote contour map of the galaxy. The second script used the largest reliable fit (indicated by
the programme returning a stop�value of 0, meaning that the fit had been trouble free) from the
first fit and ran over the range 2 to 400 pixels radius. For this second fit a minimum value for the
intensity slope was specified in order to stop the ellipses overlapping at low intensities. At these
low intensities (high radii) fitting was done concentrically with the ellipticity and centre�position
held constant.
2.2.3 Checking the ellipse fits
The ellipses used were those produced by the Galphot routines. The fits prduced were consistent
with those from the IRAF routine. The residuals produced by Galphot were also checked, this
often showed dust that had been missed on a prior inspection of the image as the dust led to large
residuals being formed.
13

2.2.4 Dusty galaxies
A number of galaxies are badly affected by dust. If there were serious features caused by dust after
masking of visible dustlanes, an entry was made into the header files of these galaxies and they
were not included in the sample. This excluded 6 galaxies from the central region and 5 from the
halo region.
About half of the sample were affected by dust but half of these were not seriously affected �
the dust lanes were thin enough to be masked out without any major loss of information.
2.3 Data calibration
The data tables were read into Super Mongo. The data was calibrated from ADU to magnitudes
using the equation
m = \Gamma2:5 \Theta log
`
PHOTFLAM \Theta DN
EXPT IME
'
+ PHOTZPT
from the WFPC/2 instrument handbook, where DN is the number of counts, PHOTFLAM and
PHOTZPT are photometric calibration constants (the sensitivity and zero point of the CCD's)
which have the values 1.88 and �21.1 respectively for this data and EXPTIME is the exposure
time. The values for these were obtained from the image header files. EXPTIME was not constant
over the set of readings, being 400 seconds for the fainter galaxies, 200 seconds for the majority
of galaxies in the sample and 160 seconds for the dominant galaxy, NGC4889. The distance scale
was calibrated using 1 pixel = 0.046 arcseconds. This allowed the surface brightness to be found
in terms of magnitude arcsec \Gamma2 .
14

2.4 Other data
The magnitude of the galaxies in the v�band was found from measurements of A e and Sb e carried
by Lucey et al (1991). The distance to Coma was taken from Faber et al (1996) as 7461 km s \Gamma1 .
A value of H 0 = 80 km s \Gamma1 Mpc \Gamma 1 was used to calculate the absolute magnitudes and the angular
scale as this was the value adopted by Faber et al.
The header files eventually contained the proper names, exposure times, distances from the
centre of the Coma cluster (in degrees), effective diameter and average surface brightness within
this diameter and whether the galaxy was seriously affected by dust. The data on the distances
from the centre of Coma, the effective diameter and the effective surface brightness were those from
Lucey et al mentioned above. I made the judgement on whether the galaxies were badly affected
by dust � if there were problems due to dust which prevented ellipses being fitted over an extended
region or if the profile was distorted by dust so it could not be well fitted by a power�law then the
galaxy was defined as being badly affected by dust.
15

Chapter 3
Analysis of data
3.1 Graphs from the ellipse fits
Graphs were made using Super Mongo by reading in the data files produced by the ellipfit routine
in galphot. The data were calibrated so that ADU were converted to magnitudes and distances
were in arcseconds. Data from the IRAF ellipse routine were also read in and graphed. This
provided a check on the galphot data.
The surface brightness per arcsecond squared, position angle, ellipticity, deviation of the centre
in x and y, and the harmonics for 3` and 4` were plotted against the log of a, the semi�major axis.
For data from galphot, a was calculated using a = r
p
1\Gammaffl
where ffl is the ellipticity. This conversion
was unnecessary for ellipse data as it used a rather than r for its scaling.
The ellipse data matched well with the galphot data in most of the graphs. The galphot data
alone was used for analysis.
An example of the graphs obtained is given in figure 3.1, this is of the D�type galaxy NGC4889
(c13 in my numbering scheme). The results from Galphot are shown by crosses and the results
from IRAF are shown by boxes.
16

Figure 3.1: Data from ellipse�fitting to NGC4889
17

3.1.1 Isophote shapes
The range over which the profile appeared reliable was from a radius of 2 pixels to a surface
brightness of 20.5 mag arcsec \Gamma2 . The highest absolute value reached by the cos 4` term in this
region was used to determine whether the galaxy appeared to be boxy or disky.
A Super Mongo routine was written to return the highest absolute value. This value was then
converted to a4/a by dividing by 1.4 (conversion factor from Pelatier, 1989). This figure shows the
distortion of the ellipse into a boxy or disky shape, disky shapes having positive values. For the
purposes of definition, galaxies with a4/a ? 0.02 were said to be disky, if this value was not being
returned due to distortions due to dust.
3.2 Images
Black and white greyscale images were obtained using the print function within SAOimage. They
showed the outer part of the galaxies, scaled logarithmically from 0% to 10% of the luminosity, the
core of the galaxies, scaled logarithmically from 4% to 100% of the luminosity, and the residuals
left by galphot. The outer parts and the residuals were at \Theta1 magnification and the cores were
at \Theta4 magnification with the exception of NGC4889, the cD galaxy, which was shown at \Theta2
magnification.
The greyscales were examined for signs of visible dust and galaxies thus identified were added
to the list of galaxies already noted as containing dust. The residuals were also examined and those
with large residuals in the centre were re�fitted. Most of the large residuals were due to severe dust
contamination and only small improvements in the fit were possible.
18

3.3 Power laws
3.3.1 Fitting power laws
A computer programme was written to fit the nuker law to the surface brightness profile using a
simulated annealing (Levenberg�Marquadt) routine from Numerical Recipes (Press et al, 1992) to
find the five parameters. The fit was made to both semi�major axis and average�radius profiles
between 2 pixels and 20.5 mag arcsec \Gamma1 , the limits that appear to be the limit of resolution and
the point where sky effects become important.
A proportional error in the number of counts (DN) of P!DN? =
p
g\Theta!DN?+oe 2
rn
g\Theta!DN?\Theta
p
2�rffir
was used in
the input data (see Appendix) where ! DN ? is the average number of counts (in ADU) in an
annulus, r is the inner radius of the annulus, ffi r is the width of the annulus, g is the gain and oe rn
is the read noise.
The read noise is taken to be 5
p
2
as each image has a read noise of 5 and two images were
averaged. The gain is taken to be 14 as each image has a gain of 7 so 14 photons are needed for
each ADU.
3.3.2 Information from the fit
The fit returned values and errors for the 5 parameters needed for the nuker law (ff, fi, fl, r b ,
I r ). These were used to construct a model profile which was compared with the original to check
whether the fit was satisfactory. If the errors associated with the constants of the fit were too large
or the fit did not match the original date the fit was rejected as unsatisfactory.
An example graph is given in fig. 3.2 of NGC4889, showing the datapoints with the fit through
them in the top half and the difference between the fit and the datapoints, with the line of error
around the datapoints, in the bottom half.
The fit was unable to fit satisfactorily to some of the profiles, these were removed from the
19

Figure 3.2: Nuker�law fit to NGC4889
20

sample. This removed 4 galaxies, all of them from the central region, on the semi�major axis fits
and 7 galaxies, all but one from the central region, on the average�radius fits. Another galaxy was
removed from the central region in the average�radius fits as it had an inner logarithmic slope, fl,
that indicated a core but its break radius was too close to the limit of resolution to be considered
reliable, as was one from the outer regions on the semi�major axis fit.
For the average�radius fit, this left 10 galaxies in the central region and 16 in the halo. For the
semi�major axis fits, this left 13 galaxies in the central region and 16 in the halo.
The parameters returned for each galaxy are given in tables C�1 (semi�major axis fits) and C�2
(average�radius fits) and the graphs obtained from these are also given.
3.3.3 Graphs from the power�law fits
The values of fl and r b from the nuker�law fits and the values of m v and distance from the centre
of Coma were plotted against each other in 6 graphs of apparent values and 6 graphs of absolute
values. Only those galaxies left in the sample, as detailed above, were graphed. These graphs were
examined for trends and for differences between the samples from the inner and outer regions of
Coma.
3.3.4 Statistical analysis of power�law fits
The Kolomogrov�Smirnov two�sample test was used to examine differences between the samples.
This test gives the confidence that two samples are from the same parent population. Details of
the test were found in Wall (1977). This test was used to look for differences in magnitude and fl.
21

Chapter 4
Results
4.1 The Results
The results are given of fits to both the semi�major axis and the average�radius in tables 4.1 and 4.2.
Additional information is given in table 4.3. Data from the average�radius fit is shown graphically
in figure 4.1 with crosses representing galaxies from the central region and boxes representing
galaxies from the halo region.
4.2 Differences between the two regions
The Kolomogrov�Smirnov test showed that the distributions of fl in the two regions were likely to
be from the same parent distribution, eg. there was no significant difference between them. Using
the same test, the probability that the magnitudes were from the same parent distribution was
tested and was found to be in the 5% tail of the normal distribution, eg. the distributions of the
magnitudes were fairly significantly different.
22

Figure 4.1: Data from the average�radius fit: crosses = central, boxes = halo except for m v vs R
� crosses = fl ! 0.3, open boxes = fl ? 0.5, boxed cross = 0.3 ! fl ! 0.5
23

name ff fi fl r b a ` b b � b c Status d
NGC4886 17.31 1.257 0.8129 139 0.3071 17 0
IC4011 0.0549 1.255 1.219 354.6 0.7834 18.82 2
NGC4876 0.6783 1.852 0.005982 245.5 0.5423 17.72 1
RB43 3 1.5 1.513 427.5 0.9444 18.95 2
IC3959 1.963 1.681 0.06006 304.9 0.6736 17.71 1
IC3957 3 1.461 0.03316 41.19 0.091 15.38 1
RB257 2 7.549 1.102 3157 6.976 23.37 1
RB260 0.4373 2.09 0.03839 225.9 0.4992 18.65 0
NGC4869 1.124 1.687 0.04334 310.6 0.6863 17.69 1
RB6 2.423 1.615 0.5326 913.7 2.019 19.78 0
RB45 3.23 1.413 0.7533 222.2 0.4909 18.09 0
RB18 1.971 1.416 0.4456 277.7 0.6136 18.82 0
NGC4889 2.598 1.318 0.05961 876.8 1.937 17.98 0
IC4021 2 1.711 1.043 483.9 1.069 18.75 2
RB167 1.612 1.455 0.01836 184 0.4066 17.65 1
IC4012 1.523 1.867 0.6039 441.1 0.9746 18.25 0
IC3947 3.181 1.789 0.9645 1226 2.709 19.74 0
RB234 7.854 1.33 0.8507 115.9 0.2561 17.66 0
IC4051 0.563 1.646 0.01028 386.6 0.8542 18.36 0
NGC4860 0.6315 2.473 0.02111 778.6 1.72 18.53 0
RB241 2 1.352 0.9355 223.4 0.4937 17.48 2
NGC4906 0.9332 1.748 0.3634 399 0.8815 18.3 0
NGC4926 1.635 1.566 0.2113 272 0.6009 16.96 0
NGC4816 4.698 1.345 0.1717 215.2 0.4755 16.98 0
Zw160�23 1.292 1.712 0.0567 238.8 0.5276 17.59 0
IC3618 4.343 1.678 0.8824 279.3 0.6172 17.51 0
Zw159�43 0.954 1.694 0.2694 176.5 0.3899 16.71 0
IC3623 2.488 1.433 0.9394 200.9 0.444 17.08 0
NGC4673 0.8752 1.927 0.2442 357.2 0.7892 16.83 0
NGC4692 1.305 1.328 0.8405 717.4 1.585 18.35 0
Zw159�83 3 1.658 0.8535 1130 2.497 19.14 1
Zw159�89 1.123 1.346 0.01302 56.33 0.1244 16.14 3
IC832 0.2229 2.223 0.01205 406.2 0.8974 18.3 1
NGC4789 1.59 1.337 0.003033 180.3 0.3983 16.52 1
NGC4807 2.802 1.521 0.5721 239.6 0.5293 16.78 0
IC834 2 1.638 0.01575 184.6 0.408 16.55 1
NGC4824 2.751 1.629 0.5161 224.9 0.4969 17.6 0
NGC4827 2.464 1.37 0.4234 180.2 0.3981 16.56 0
NGC4839 1.121 1.322 0.05773 346.2 0.7649 17.79 0
NM8603 4.491 1.341 0.6988 130 0.2872 17.15 0
IC4133 3.373 1.601 0.7037 185.3 0.4094 17.03 0
NGC4952 1.043 1.47 0.243 147.6 0.3261 16.1 1
NGC5004 1.24 1.449 0.02914 176.6 0.3902 16.61 0
NGC4957 0.975 1.362 0.002576 122.7 0.2711 16.65 0
Zw160�159 1.836 1.428 1.085 421.8 0.932 18.17 0
a Break radius in parsec
b Break radius in arcsec
c surface brightness in mag arcsec \Gamma2 at the break radius
d 0 = Okay, 1 = Dusty, 2 = Unsatisfactory fit, 3 = Break radius less than limit of resolution (1.6 arcsec)
Table 4.1: Semi�major axis fits
24

name ff fi fl r b a ` b b � b c Status d
NGC4886 2 1.284 0.4267 82.36 0.182 16.53 2
IC4011 1 1.272 0.00273 2.855 0.006307 13.15 2
NGC4876 0.6578 1.892 0.0076 234.8 0.5187 17.79 1
RB43 3.567 1.486 1.539 674.7 1.491 19.82 2
IC3959 2.022 1.623 0.03834 257.7 0.5694 17.65 1
IC3957 2 1.477 0.02171 30.68 0.06779 15.13 1
RB257 2 2.684 1.12 974.9 2.154 20.3 1
RB260 1.509 1.312 0.01441 56.32 0.1244 17.39 3
NGC4869 1.14 1.642 0.06591 261 0.5767 17.63 1
RB6 3.35 1.426 0.5831 514.7 1.137 19.53 0
RB45 3.129 1.411 0.7549 221.5 0.4893 18.1 0
RB18 1.904 1.426 0.4283 210.5 0.465 18.7 0
NGC4889 3.266 1.342 0.07164 759 1.677 17.95 0
IC4021 2 1.711 1.043 459.1 1.014 18.75 2
RB167 1.619 1.455 0.01015 143.5 0.317 17.66 1
IC4012 1.884 1.886 0.6361 403.1 0.8906 18.27 0
IC3947 5.202 1.675 0.9887 727 1.606 19.28 0
RB234 5.209 1.338 0.8142 109.3 0.2415 17.64 0
IC4051 0.4025 2.538 0.000369 2438 5.388 20.51 2
NGC4860 1.124 1.657 0.115 207.9 0.4593 17.13 0
RB241 0.1946 2.523 0.03161 478.1 1.056 18.57 2
NGC4906 0.9681 1.592 0.3301 244.4 0.54 17.88 0
NGC4926 1.746 1.512 0.2227 230.5 0.5094 16.92 0
NGC4816 4.879 1.359 0.1777 197.7 0.4368 16.98 0
Zw160�23 1.093 1.726 0.005016 196.2 0.4336 17.53 0
IC3618 6.347 1.699 0.8842 250.5 0.5534 17.39 0
Zw159�43 0.8081 1.83 0.1348 137.4 0.3037 16.58 0
IC3623 1.78 1.498 0.9522 264.9 0.5852 17.59 0
NGC4673 1.279 1.744 0.3826 285.1 0.6298 16.66 0
NGC4692 1.139 1.419 0.8713 1013 2.239 18.89 0
Zw159�83 2 1.878 0.8326 1294 2.86 19.5 1
Zw159�89 3.552 1.224 0.5308 91.26 0.2016 16.55 0
IC832 2 1.625 0.971 1174 2.593 19.67 1
NGC4789 1.588 1.337 0.006029 160.8 0.3553 16.52 1
NGC4807 2.554 1.555 0.5669 226.7 0.5008 16.83 0
IC834 2 1.639 0.001654 155.3 0.343 16.55 1
NGC4824 2.822 1.595 0.54 206.2 0.4556 17.59 0
NGC4827 1.74 1.398 0.1269 115.1 0.2543 16.25 0
NGC4839 1.156 1.368 0.02589 345.8 0.7641 17.89 0
NM8603 5.912 1.361 0.7665 120.9 0.2671 17.16 0
IC4133 2.964 1.578 0.6853 167.1 0.3693 16.96 0
NGC4952 1.029 1.471 0.2141 118.6 0.262 16.06 1
NGC5004 1.16 1.51 0.01231 160.6 0.3549 16.62 0
NGC4957 1.093 1.355 0.01682 104.5 0.2309 16.58 0
Zw160�159 0.1112 2.621 0.04299 567 1.253 18.74 2
a Break radius in parsec
b Break radius in arcsec
c surface brightness in mag arcsec \Gamma2 at the break radius
d 0 = Okay, 1 = Dusty, 2 = Unsatisfactory fit, 3 = Break radius less than limit of resolution (1.6 arcsec)
Table 4.2: Average radius fits
25

name mag a radius b abs mag c c4 d a4 d
NGC4886 13.93 0.05 �20.92 0.01 0.007143
IC4011 14.87 0.064 �19.98 0.009 0.006429
NGC4876 14.34 0.062 �20.51 �0.62 �0.4429
RB43 15.37 0.056 �19.48 �0.012 �0.008571
IC3959 13.99 0.245 �20.86 0.532 0.38
IC3957 15.2 0.259 �19.65 0.027 0.01929
RB257 15.44 0.209 �19.41 0.014 0.01
RB260 15.75 0.192 �19.1 0.013 0.009286
NGC4869 13.52 0.119 �21.33 0.031 0.02214
RB6 15.64 0.12 �19.21 0.029 0.02071
RB45 15.04 0.04 �19.81 0.015 0.01071
RB18 15.68 0.098 �19.17 0.04 0.02857
NGC4889 11.39 0.06 �23.46 �4.273 �3.052
IC4021 14.78 0.112 �20.07 0.015 0.01071
RB167 14.78 0.212 �20.07 �0.184 �0.1314
IC4012 14.71 0.125 �20.14 �0.091 �0.065
IC3947 14.6 0.287 �20.25 0.032 0.02286
RB234 15.01 0.193 �19.84 0.012 0.008571
IC4051 13.02 0.234 �21.83 0.028 0.02
NGC4860 13.31 0.236 �21.54 0.043 0.03071
RB241 13.81 0.304 �21.04 0.006 0.004286
NGC4906 14.18 0.181 �20.67 0.019 0.01357
NGC4926 12.9 0.565 �21.95 0.018 0.01286
NGC4816 12.79 0.839 �22.06 �0.042 �0.03
Zw160�23 14.51 0.811 �20.34 0.027 0.01929
IC3618 14.3 4.75 �20.55 0.037 0.02643
Zw159�43 13.94 4.545 �20.91 0.011 0.007857
IC3623 13.83 4.604 �21.02 0.015 0.01071
NGC4673 12.9 3.294 �21.95 �0.013 �0.009286
NGC4692 12.61 2.75 �22.24 �0.02 �0.01429
Zw159�83 13.45 2.5 �21.4 �0.082 �0.05857
Zw159�89 13.74 1.986 �21.11 0.039 0.02786
IC832 13.69 2.01 �21.16 �0.063 �0.045
NGC4789 12.39 1.525 �22.46 0.006 0.004286
NGC4807 13.44 1.067 �21.41 �0.014 �0.01
IC834 13.79 1.794 �21.06 0.02 0.01429
NGC4824 14.58 0.846 �20.27 0.014 0.01
NGC4827 13.09 1.053 �21.76 �13.71 �9.793
NGC4839 12.15 0.719 �22.7 0.498 0.3557
NM8603 14.82 1.046 �20.03 0.021 0.015
IC4133 14.23 0.879 �20.62 0.015 0.01071
NGC4952 12.82 1.609 �22.03 0.036 0.02571
NGC5004 12.95 2.959 �21.9 0.042 0.03
NGC4957 12.97 1.246 �21.88 �0.013 �0.009286
Zw160�159 13.92 2.79 �20.93 0.013 0.009286
a Calculted from V�band measurements of Ae and SBe
b Distance from centre of Coma cluster in degrees
c Calcultated from apparent magnitude taking H 0 = 80 Mpc \Gamma1 km s\Gamma1
d cos 4 ` deviations from ellipse and a4/a deviation from ellipse, calculated from c4 by dividing by 1.4
Table 4.3: Other galaxy data
26

The galaxies in the halo region were on the whole a magnitude brighter than the galaxies in
the central region, both regions had the same distribution of fl in their samples. If there is a trend
of fl with M v , this would show that there was a difference in the formation of core�type galaxies
between the halo and central regions.
4.3 M v vs fl
Examination of the graph of M v against fl does not show any definite trend. 3 out of 4 galaxies
brighter than �22 magnitudes show cores, indicating that larger galaxies generally contain cores.
There do not appear to be many smaller galaxies with cores but it is impossible to say whether
this reflects a physical effect or whether this is an artefact of resolution.
There are 3 points in the region 0.3 ! fl ! 0.5, For a uniform distribution between 0 and 1,
which encompasses all the values in the data set, 5.4 points would be expected in this region. The
expected counting error on this is therefore oe =
p
5:4 = 2:3, this means that the number of points
found in this region is barely significantly less than would be expected from a uniform distribution.
It is still the case that this region has a lower number density than the core or power�law regions
either side of.
The frequency distribution of the fl's of galaxies (from the average�radius fit) is shown in fig
4.2. The shaded area represents galaxies in the halo region, the clear area represents galaxies in
the central region.
4.3.1 Central region
There is only one bright galaxy in the sample, NGC4889. This is one of two D�galaxies in the
central region of Coma and shows a definite core. Due to cannibalisation by the two D�galaxies,
there are no other bright galaxies in this region, this is a contributing factor to the difference in
magnitudes between the two regions.
27

Figure 4.2: Binned distribution of fl for the central region (upper graph) and the halo region (lower
graph)
28

The evidence for a trend in fl with M v is fairly good in this region (see fig 4.4), although this
could be a selection effect due to the smaller low�luminosity galaxies. two of the galaxies in the
intermediate zone 0.3 ! fl ! 0.5 are in the central region, exactly the number that would be
expected for a uniform distribution between 0 and 1 and the same number density as is seen in all
bins when fl is binned in tenths (see fit 4.2).
4.3.2 Halo region
There are 3 bright galaxies in the sample (defined as being in the M v ! �22 region where Faber et
al find no cores), two of them show definite cores but the third (NGC4692) has a single power�law
through to the limit of resolution (see fig 4.3). There are few faint galaxies in the sample due to
the lack of comprehensive surveying in the halo region of Coma.
There is only 1 galaxy in the intermediate zone in the halo region, even if this region is extended
to cover 0.2 ! fl ! 0.5. There does appear to be a well�defined gap (see fig 4.2)This region shows
virtually no correlation of fl with M v (see fig 4.4)
4.4 r b vs fl
Taking the values returned by the fit, there is a fairly even distribution of break radii (see fig 4.1).
If it is assumed that the cores are unresolved for fl ? 0.5 then all these galaxies have break radii
smaller than the limit of resolution, r lim = 70 pc (limit of resolution given by log r b (arcsec) =
�0.8, as used in Faber et al).
4.5 r b & r lim vs M v
Faber et al claim values for r lim much smaller than their value used for the limit of resolution
for the cores by examining deconvolved profiles within 0.1 arcsec and comparing these with a
29

Figure 4.3: Nuker�law fit to NGC4886
30

Figure 4.4: Plots of fl against M v for the central and halo regions
31

convolved and deconvolved models with break radii of 0.025, 0.050 and 0.075 arcsec. By saying
that the deconvolved profiles give a lower limit to the surface brightness, they have said that where
the models fall below the deconvolved profile, the break radius for that model gives an upper limit
for the break radius of that galaxy. This has allowed them to set upper limits of 0.05 and 0.08
(rounded from 0.075) on the break radii of some galaxies.
On other galaxies, no upper limit has been found from the models so the upper limit has been
set at 0.1 arcsec, which was the upper limit of the region of the profile they were examining. As
the limit of resolution for cores has been set at 0.16 arcsec, this has led to an apparent gap between
the break radii of the observed cores and the maximum possible break radii of the galaxies without
resolved cores.
I have used r lim � 70 pc (at Coma distance) � 0.16 arcsec. If this is used instead of the 0.05
� 0.10 arcsec range used by Faber et al then no gap is seen between the break radii of resolved
cores and the maximum break radii of unresolved cores, as is seen in the lower graph of figure 4.5
(crosses = central, boxes = halo)
4.6 Break radius vs Effective radius
Faber et al found a linear relationship between effective radius and core radius, r b = 0.03 r e , in
those galaxies with resolved cores. This relationship has been plotted, with my data points, in
the upper graph of figure 4.5. I do not find this relation in the galaxies with resolved cores in our
sample as cam be seen from this graph. Addition of more large galaxies would probably lead to a
relationship being found, as it is in Faber et al, but this relationship does not appear to hold when
only the smaller galaxies are considered (eg. r e ! 10000 pc).
32

Figure 4.5: Upper � plot of r b vs r e for those galaxies with fl ! 0.3, with the fit found by Faber et
al shown by the dotted line.
Lower � Plot of r b vs m v with r lim substituted for galaxies with fl ? 0.3
33

4.7 Comparison with previous results
Plots of my data and Faber et al's data are in figure 4.6 My data is shown by crosses and Faber
et al's by boxes.
4.7.1 M v
vs fl
The results are similar to those obtained by Faber et al (1996) below �19 magnitudes, the cut off
point for our observations (see fig 4.6). I have found significantly more points with 0.3 ! fl ! 0.5
than Faber et al and my results do not show the clear trend given by the low�luminosity tail in
Faber et al's results (see fig 4.1).
4.7.2 r b
vs fl
Once again our results are very similar to Faber et al in the larger galaxies (above our limit of
resolution, log r b � 1.85, r b � 70pc) (see fig 4.6). Below this, all of Faber et al's results have
fl ! 0.5 and have either been classified as cores or as having to small a break radius to classify as
resolved. The 6 galaxies in this section that were classified as having cores would not have had
these cores resolved in our sample.
In the Faber et al sample, M31 has a resolved core at its true distance but this core is unresolved
when it is translated to Virgo. This would appear to support the hypothesis that the core�type /
power�law distinction is an effect of resolution.
4.8 Diskiness
Diskiness is defined, as described in section 3.1.1, as having a maximum value of a4/4 greater than
0.02. Out of the dust�free galaxies, 9 meet this criterion. These are RB6, RB18, IC3947, IC4051
and NGC4860 from the central region and IC3618, Zw159�89, NGC4839 and NGC5004 from the
34

Figure 4.6: Comparison of Faber et al's data (boxes) and my data (crosses)
35

halo region. NGC4839 is a D�type galaxy with dust in the central region that, while it hasn't
affected the fit, has led to distortion of the a4/a term and so can be removed from this list. The
cos 4` deviation profiles of the 8 galaxies which appear to be disky are given in fig 4.6. Galphot
results are shown by crosses and IRAF results are shown by boxes.
The other galaxies appear to have disks on inspection of the graphs. This includes 3 galaxies
(2 in the central region, 1 in the halo) which have resolved cores. All 3 of these galaxies are
fairly bright (between �21.5 and �22 absolute magnitude). This appears to contradict Faber et al's
assertion that galaxies with cores are boxy and galaxies with power�laws are disky.
4.9 Defining a Core
Throughout this report, I have used Faber et al's definition of a core as having fl ! 0.3. There are
galaxies such as NGC4886 (see fig 4.7) with a very well defined break but with a steep inner section.
Galaxies with well�defined breaks are found over a range of values of fl � these would appear to be
more than random flukes and it is possible that the low fl cores are a sub�set of whatever physical
process causes these breaks to lower slopes � there is not particular reason for choosing fl = 0.3 as
the cut�off rather than a higher value except for the apparent dichotomy seen at this point (see
fig4.2). It is also possible that there are a number of these `steps' in some galaxies, leading to an
almost flat central core. This would require a multiple power�law fit to be accurately described.
36

Figure 4.7: cos 4` terms for the 8 galaxies identified as disky. Crosses are Galphot data, boxes are
IRAF data
37

Figure 4.8: Nuker�law fit to NGC4886
38

Chapter 5
Conclusions
5.1 M v vs fl
There is no definite trend in the the data set as a whole or in the halo region sample but there is
an apparent trend in the central region sample.
5.2 fl
It is shown statistically that the values of fl in both regions are probably from the same parent
population. The distribution differs between the regions, with there being two distinct peaks in the
distribution of gamma, as seen by Faber et al, in the halo region while there is a flat distribution
in the central region.
5.3 M v
The populations of M v seen in the two regions are shown to be probably from different parent
populations, the halo population being, on average, a magnitude brighter than the core population.
39

5.4 r b vs fl
There does not appear to be a link between the value for the break�radius returned by the fitting
routine and the value for fl
5.5 r b & r lim vs M v
There is no clear evidence seen for a trend in r b with M v nor is the break seen by Faber et al seen
when the limit of resolution is used as r lim .
5.6 Diskiness
Disky galaxies are seen to be able to form cores as well as power�law profiles, this appears to indicate
that it is not true that rotationally supported galaxies have power�laws and triaxial galaxies have
cores.
5.7 Summary
There is a spread of break radii from the limit of resolution outwards. The gap seen by Faber et
al is not apparent when the maximum possible core radius for galaxies without resolved cores is
set to be the limit of resolution (0.16'').
The drop in the number of galaxies with 0.3 ! fl ! 0.5 is seen only in the halo region, not in
the central region.
It is seen that bright galaxies can form apparent power�laws and that disky galaxies can form
cores.
No definite trend can be seen in the graphs of fl vs M v . It is therefore not possible to use the
difference in magnitudes to say that there would be a difference in the distributions of fl if the two
40

samples were over the same magnitude range. Therefore the difference between the two regions
predicted by the gas�stellar continuum has not been proven.
Acknowledgements
I would like to thank Professor Davies, Dr de Jong and Dr Lucey for their advice and help in doing
this project.
41

Appendix A
Errors in the ellipses
Average number of counts in an annulus = ! DN ?
Gain = g, Average number of photons in annulus = number of electrons = g\Theta ! DN ?
Number of pixels in an annulus � area of annulus = 2�rffir
Read noise per pixel = oe rn (in electrons)
Total read noise in annulus =
p
oe 2
rn \Theta 2�rffir
Average read noise in annulus =
p
oe 2
rn \Theta2�rffir
2�rffir = oe rn
p
2�rffir
Total number of counts in annulus = ! DN ? \Theta2�rffir
Total number of photons in annulus = g\Theta ! DN ? \Theta2�rffir
Counting error in total photons, oe fl = p
g\Theta ! DN ? \Theta2�rffir
Counting error in average photons, oe !fl? =
p
g\Theta!DN?\Theta2�rffir
2�rffir
=
q
g\Theta!DN?
2�rffir
Total error in ellipse:
oe =
r
\Gamma p oe !fl?
\Delta 2
+
i
oe rn
p
2�rffir
j 2
oe =
r i
sqrt g\Theta!DN?
2�rffir
j 2
+
i
oe rn
p
2�rffir
j 2
oe =
q
g\Theta!DN?+oe 2
rn
2�rffir
42

Proportional error in average number of counts in annulus:
P!DN ?=
p
g\Theta!DN?+oe 2
rn
g\Theta!DN?\Theta
p
2�rffir
43

Appendix B
Notes on Galaxies
Those galaxies which had an offset between the centres of the galaxy in the two images of between
0.1 and 0.2 pixels are noted as such below. No note is made if the offset was less than 0.1 pixels.
For the four galaxies with an offset greater than 0.2 pixels it is noted whether they had a shift
less than 0.1 pixels when examined using a cross�correlation routine or whether one of the images
had to be shifted in order to align it. It is also noted what evidence of dust there is in each of the
galaxies and if there are any other features on the image.
ffl NGC4886 had a separation between the two images of between 0.1 and 0.2 pixels when they
were combined. There is no evidence of dust either visibly or in the 3` terms.
ffl IC4011 shows no evidence of dust either visibly or in the 3` terms. There is a small companion
galaxy or a star in the bottom�right of the frame.
ffl NGC4876 had a separation between the two images of between 0.1 and 0.2 pixels when they
were combined. There is visible dust in the core, this also shows up in the 3` terms.
ffl RB43 had a separation between the two images of between 0.1 and 0.2 pixels when they were
combined. There is no evidence of dust either visibly or in the 3` terms.
44

ffl IC3959 has visible dust in the core, this also shows in the 3` terms.
ffl IC3957 shows possible visible evidence for a ring of dust around the core. This does not
show up in the 3` terms but there does appear to be a disruption in the 4` terms. There is
a companion galaxy in the bottom left of the frame.
ffl RB257 shows possible visible evidence for dust. This does not show up in the 3` terms but
there is a discrepancy between the 3` terms measured by ellipfit and ellipse inside of �
0.2 arcsec (4 pixels). There are a couple of stars in the bottom left of the frame. This galaxy
is classified as a possible lenticular.
ffl RB260 shows no visible evidence of dust but there is some evidence of an asymmetry in the
3` terms.
ffl NGC4869 had a separation between the two images of between 0.1 and 0.2 pixels when they
were combined. There is visible dust in the core, this also shows up in the 3` terms. There
is a large star in the bottom left of the frame, there is another object to the top left and two
objects to the top right.
ffl RB6 shows no evidence of dust either visibly or in the 3` terms. There is a star to the top
right of the frame.
ffl RB45 shows no evidence of dust either visibly or in the 3` terms. There are odd `stretch
marks' on the ellipfit residual, lines projecting radially from the centre and symmetric through
the centre.
ffl RB18 shows no visible evidence of dust but there is evidence of an asymmetry in the 3`
terms. The galaxy undergoes a visible rotation between radii of 2 and 4 arcseconds. The 4`
terms are also confused. The core appears disky and this is supported by the cos 4` terms,
this does not extend outwards.
45

ffl NGC4889 had the two images shifted to match them before they were combined. This is the
cD galaxy and is very flat�topped. There is little visible evidence of dust but the 3` and 4`
terms are very confused within 1 arcsecond. This is within the core radius so does not affect
the conclusion that this galaxy has a soft core.
ffl IC4021 has visible dust in the core, this does not show particularly in the 3` terms but there
is a discrepancy between the ellipse and ellipfit terms.
ffl RB167 has visible dust in the core, this also shows up in the 3` terms. The core is visibly
disky. There is a star at the top of the frame. There are odd `stretch marks' on the ellipfit
residual, lines projecting radially from the centre and symmetric through the centre.
ffl IC4012 has visible dust in the core, this also shows in the 3` terms.
ffl IC3947 shows no evidence of dust either visibly or in the 3` terms. There are a number of
small objects on the frame.
ffl RB234 shows no visible evidence of dust, although the 3` terms appear confused.
ffl IC4051 has faintly visible dust in the core, this also shows up in the 3` terms. There are a
large number of small objects on the frame.
ffl NGC4860 appeared to have a separation of greater than 0.2 pixels using center but cross
correlation using X�register gave a smaller shift. There is no visible evidence of dust but
there is evidence of an asymmetry in the 3` terms.
ffl RB241 shows no evidence of dust either visibly or in the 3` terms.
ffl NGC4906 shows no evidence of dust either visibly or in the 3` terms.
ffl NGC4926 had a separation between the two images of between 0.1 and 0.2 pixels when they
were combined. There is no evidence of dust either visibly or in the 3` terms.
46

ffl NGC4816 shows no visible evidence of dust but there is some evidence of an asymmetry in
the 3` terms. The ellipfit routine has not fitted between .5 and 1.5 pixel radius(!).
ffl Zw160�23 shows possible visible evidence of dust and there is evidence of an asymmetry in
the 3` terms. There is an `hump' in the brightness profile at the point where the asymmetry
occurs.
ffl IC3618 had a separation between the two images of between 0.1 and 0.2 pixels when they
were combined. There is no evidence of dust either visibly or in the 3` terms.
ffl Zw159�43 shows no evidence of dust either visibly or in the 3` terms. There is a discrepancy
between the ellipse and ellipfit 3` terms near the core. There is a star in the top right
of the frame.
ffl IC3623 shows no evidence of dust either visibly or in the 3` terms.
ffl NGC4672 shows no evidence of dust either visibly or in the 3` terms. There is a discrepancy
between the ellipse and ellipfit 3` terms within � 1 arcsecond.
ffl NGC4692 shows no evidence of dust either visibly or in the 3` terms.
ffl Zw159�83 shows visible evidence of a dust lane. From the 3` terms, this lane extends from
.5 arcseconds to 4 arcseconds. The fit continues inside .5 arcseconds. The residual is very
confused at the radii of the dust lane.
ffl Zw159�89 shows no evidence of dust. The cos 4` terms show a definite disky component
extending to 2 arcseconds radius.
ffl IC832 shows visible evidence for a dust ring around the core, this also shows faintly in the
3` and 4` terms and in a `hump' on the brightness profile.
47

ffl NGC4789 shows a strong dust lane through the core, this also shows in the 3` and 4` terms
and in a turn�down of the brightness profile near the centre. The ellipfit fit stops at about
0.3 arcseconds radius.
ffl NGC4807 shows no evidence of dust either visibly or in the 3` terms.
ffl IC834 shows visible evidence for a dust lane around the core, this also shows in the 3` terms.
ffl NGC4824 appeared to have a separation of greater than 0.2 pixels using center but cross
correlation using X�register gave a smaller shift. There is no evidence of dust either visibly
or in the 3` terms.
ffl NGC4827 shows visible evidence for a faint dust lane in the core, this also shows in the 3`
terms.
ffl NGC4839 had a separation of between 0.1 and 0.2 pixels between the two images when they
were combined. There is no visible evidence for dust but the 3` and 4` terms are very
confused. It has a very flat profile, similar to that of NGC4889.
ffl NM8603 shows no evidence of dust either visibly or in the 3` terms. There is a discrepancy
between the ellipse and ellipfit 3` terms near the core.
ffl IC4133 shows no evidence of dust either visibly or in the 3` terms.
ffl NGC4952 shows visible evidence for a dust lane in the core, this also shows in the 3` terms.
ffl NGC5004 had the two images shifted to match them before they were combined. There is
no visible evidence for dust but there is evidence for an asymmetry in the 3` terms.
ffl NGC4957 had a separation of between 0.1 and 0.2 pixels between the two images when they
were combined. There is no evidence of dust either visibly or in the 3` terms.
ffl Zw160�159 shows no evidence of dust either visibly or in the 3` terms.
48

Appendix C
Differentials of the Nuker Law
These were calculated using the Maple symbolic processor programme. They are necessary in order
to fit the nuker law to the surfacr�brightness profiles.
I(r) = 2 ( fi\Gammafl
ff ) I b
i r b
r
j fl
`
1 +
`
r
r b
' ff ' ( c\Gammafi
ff )
dI(r)
dff = 2 ( \Gamma fl\Gammafi
ff ) ( fl \Gamma fi ) I r
i r b
r
j fl
`
1 +
`
r
r b
' ff ' ( fl\Gammafi
ff ) `
ln( 2 ) + ln( 2 )
`
r
r b
' ff
\Gamma ln
`
1 +
`
r
r b
' ff '
\Gamma ln
`
1 +
`
r
r b
' ff ' `
r
r b
' ff
+ ff
`
r
r b
' a
ln
`
r
r b
'' OE`
ff 2
`
1 +
`
r
r b
' ff ''
dI(r)
dfi
=
2 ( \Gamma fl\Gammafi
ff ) I b
i r b
r
j fl
`
1 +
`
r
r b
' ff ' ( fl\Gammafi
a ) `
ln( 2 ) \Gamma ln
`
1 +
`
r
r b
' ff ''
ff
dI(r)
dfl = 2 ( \Gamma fl\Gammafi
ff ) I b
i r b
r
j fl
`
1 +
`
r
r b
' ff ' ( fl\Gammafi
ff ) `
\Gammaln( 2 ) + ln
i r b
r
j
ff + ln
`
1 +
`
r
r b
' ff '' ffi
ff
dI(r)
dr b
2 ( \Gamma fl\Gammafi
ff ) I b
i r b
r
j fl
`
1 +
`
r
r b
' ff ' ( fl\Gammafi
ff ) `
fl +
`
r
r b
' ff
fi
'
r b
`
1 +
`
r
r b
' ff '
49

dI(r)
dI b
2 ( \Gamma fl\Gammafi
ff ) i r b
r
j fl
`
1 +
`
r
r b
' a ' ( fl\Gammafi
ff )
50

Appendix D
Computer programmes
D.1 Numerical Recipes routines
The following Numerical Recipes routines were used in the project: covsrt.c, gaussj.c, mrqmin.c
and nrutil.c.
D.2 My routines
Project.c reads the data in from file and calls the fitting programme mrqmin.c. It supplies initial
values and stopping conditions for the fitting.
Funcs.c is called by mrqmin.c and returns the nuker�law and the differentials
D.2.1 Project.c
1 #include ''stdio.h''
#include ''math.h''
#include ''nrutil.h''
struct profile
5 --
float a;
float i;
float di;
char datatype;
10 struct profile *nextaddr;
�;
51

main()
--
15 struct profile *list, *temp;
void display(struct profile *, int, char []);
void fit(int, struct profile *, float *, float **, float **, float *);
int populate(struct profile *, char[]);
float nalamda, nchisq;
20 float *alamda, **covar, **alpha, *chisq;
char galname[10];
int ndata, i;
covar=matrix(1,5,1,5);
25 alpha=matrix(1,5,1,5);
alamda=&nalamda;
chisq=&nchisq;
30 *alamda=�1.0;
*chisq=0.1;
/* Set up the list and then populate it */
list = (struct profile *)malloc(sizeof(struct profile));
35 ndata = populate (list, galname);
/* dump the first (blank) entry */
temp = list;
list = list�?nextaddr;
40 free(temp);
ndata = ndata �1;
/* Display the contents of the list */
display(list, ndata,galname);
45
printf(''``n``t%s``n'', galname);
fit(ndata,list,alamda,covar,alpha,chisq);
�
50 void fit(int ndata, struct profile *list, float *alamda, float **covar, float ``
**alpha, float *chisq)
--
float x[ndata], y[ndata], sig[ndata], a[5];
int ia[5];
55 int iter = 1, zcnt = 0;
void funcs(float,float [],float *,float [],int);
void mrqmin(float [],float [],float [],int,float [],int [],int,float *``
,float **,float *,void (*funcs)(float,float [],float *, float [],int),float *);
int n, m;
60 struct profile *current;
float ochi;
FILE *out�file, *cov�file;
char o�name[10], c�name[10], temp;
int df = ndata;
65
/* Open output data file */
printf(''``nEnter output filename (c??.fit.g or c??.fit.e)``n'');
scanf(''%c'',&temp);
gets(o�name);
70 printf(''``nEnter covariance matrix filename (c??.cov.g or c??.cov.e)``n'');
gets(c�name);
out�file=fopen(o�name,''w'');
cov�file=fopen(c�name,''w'');
fprintf(out�file,''alpha``tbeta``tgamma``tbreak�rad``tbreak�int``tchisq``n'');
75
current = list;
/* Setup an array with the values in */
80 for (n=1; n != ndata; n++)
--
/* printf(''%d '',n); */
x[n] = current�?a;
y[n] = current�?i;
85 current = current�?nextaddr;
52

sig[n] = current�?di * y[n];
�
/* printf(''``n''); */
90
a[1] = 2.0;
a[2] = 1.0;
a[3] = 0.5;
a[4] = 1.0;
95 a[5] = 0.1;
for (n=1 ; n!=5 ; n++)
--
printf(''a[%d] = [%f]. Enter new value: '',n,a[n]);
100 scanf(''%f'',&a[n]);
/* set which values to vary */
printf(''vary a[%d]? (0 = no, 1 = yes): '',n);
fprintf(out�file,''%f``t'',a[n]);
scanf(''%d'',&ia[n]);
105 if (ia[n])
ia[n] = 1;
�
fprintf(out�file,''0``n'');
110 /* Note � a[5] is 1,000,000 times true value */
printf(''Running Iterative Fitting Function``n'');
printf(''Iter. Chisq A[1] A[2] A[3] A[4] A[5]``n'');
mrqmin(x,y,sig,ndata,a,ia,5,covar,alpha,chisq,funcs,alamda);
115 printf(''%4d %12f``t%f %f %f %f %f``n'',iter, *chisq, a[1], a[2], a[3], ``
a[4], a[5] );
do
--
ochi = *chisq;
120 mrqmin(x,y,sig,ndata,a,ia,5,covar,alpha,chisq,funcs,alamda);
iter++;
if(ochi � *chisq ? 0.01)
zcnt = 0;
else
125 zcnt++;
for (n=1 ; n!=5 ; n++)
if (a[n]!0)
a[n]=�a[n];
printf(''%4d %12f``t%f %f %f %f %f``n'',iter, *chisq, a[1], a[2], ``
130 a[3], a[4], a[5] );
�
while ((((ochi � *chisq) ? 0.01) ------ (zcnt != 10)) ------ (*chisq ? ochi));
*alamda=0.0;
mrqmin(x,y,sig,ndata,a,ia,5,covar,alpha,chisq,funcs,alamda);
135
for (n=1; n!=5; n++)
df = df � ia[n];
/* Print out values for the constants */
140 printf(''``nchisq``t%f'',*chisq);
printf(''``nd.f.``t%i'',df);
printf(''``nreduced chi squared``t%f'',*chisq/df);
printf(''``niter``t%d'',iter);
for (n=1; n!=5; n++)
145 printf(''``na[%d]``t%f``tCovar``t%f``tS.E.``t%f'',n,a[n],covar[n][n],``
pow(covar[n][n],0.5));
printf(''``n'');
for (n=1; n!=5; n++)
150 fprintf(out�file,''%f``t'',a[n]);
fprintf(out�file,''%f``n'',*chisq);
for (n=1; n!=5; n++)
--
fprintf(out�file,''%f``t'',pow(covar[n][n],0.5));
155 for (m=1; m!=5; m++)
fprintf(cov�file,''%f``t'',covar[n][m]);
fprintf(cov�file,''``n'');
�
53

160 fprintf(out�file,''%f``n'',*chisq/df);
fclose(out�file);
return;
�
165 int populate(struct profile *current, char galname[10], int *ndata)
--
/* Subroutine to read the data and put it into the list */
FILE *in�file;
170 int n, num;
float exptime, photzpt, photflam, err, pi;
char galnum[4], f�name[11], h�name[8], tmp[161];
float a,r,dr,i,s,x,y,eps,pos,s1,s2,s3,s4,c1,c2,c3,c4,f1,f2,f3,f4;
float meanint,rms,teta,x0,y0,slope,mag,A3,B3,A4,B4,bigA;
175 char choice;
/* Gets a file to work upon */
/* printf(''``nEnter galaxy number: ''); */
/* gets(galnum); */
180
/* Make header file name and data file name from galaxy number */
printf(''``n Enter data file name: '');
gets(f�name);
185 printf(''``n Enter header file name: '');
gets(h�name);
printf(''``n Enter e to fit to ellipse data, g to fit to galphot data: '');
scanf(''%c'', &choice);
190 /* Open header file */
in�file=fopen(h�name,''r'');
fgets(tmp,161,in�file);
/* Set photometric constants and read exptime */
195 photflam = 1.8804679436201;
photzpt = �21.100000381470;
pi = 3.14;
fscanf (in�file,''%f %s'', &exptime, galname);
printf(''exptime = %f``n'',exptime);
200
/* Close header file */
fclose(in�file);
/* Check choice */
205
/* Set num (number of data) to zero */
num = 0;
switch(choice)
210 --
case 'g':
/* Open galphot datafile */
in�file=fopen(f�name,''r'');
215 /* Dump the first two lines */
for (n=0;n!2;n++)
fgets(tmp,161,in�file);
/* Runs through the galphot file reading the data */
220 while (fscanf (in�file,''%f %f %f %f %f %f %f %f %f %f %f %f %f %f ``
%f %f %f %f %f %f'',&r,&dr,&i,&s,&x,&y,&eps,&pos,&s1,&s2,&s3,&s4,&c1,&c2,&c3,``
&c4,&f1,&f2,&f3,&f4) != EOF)
--
225 if ((r ?= 2.0) && (i ?= (exptime/100.0)))
--
/* Set nextaddress */
current�?nextaddr = (struct profile *)malloc(sizeof(struct profile));
current=current�?nextaddr;
230
/* Finds error in i */
if (a ? 20)
err = (pow((14.0*i+12.5)/(pi*2.0*r*(dr�r)),0.5))/(14.0*i);
54

else
235 err = (pow((14.0*i+12.5)/(2.0*pi*(r*2+2)),0.5))/(14.0*i);
/* Finds a */
a = r / sqrt(1.0�eps);
240 /* Calibrate data */
a = a * 0.046;
i = pow(10,(�12+(photzpt/�2.5))) * photflam * i / (exptime * ``
pow(0.046, 2));
245 /* Add data to list and add one to num */
++num;
current�?a = a;
current�?i = i;
current�?di = err;
250 current�?datatype = 'g';
/* printf(''%d '',num); */
�
�
current�?nextaddr = NULL;
255 fclose(in�file);
break;
case 'e':
/* Open ellipse file */
260 in�file=fopen(f�name,''r'');
/* Dump the first two lines */
for (n=0;n!2;n++)
fgets(tmp,161,in�file);
265 /* Runs through the ellipse file reading the data */
while (fscanf (in�file,''%f %f %f %f %f %f %f %f %f %f %f %f %f %f''``
,&a,&meanint,&rms,&eps,&teta,&x0,&y0,&slope,&mag,&A3,&B3,&A4,&B4,&bigA) != EOF)
--
if (i ?=exptime/100.0)
270 --
/* Set nextaddress */
current�?nextaddr = (struct profile *)malloc(sizeof(struct profile));
current=current�?nextaddr;
/* Find the error */
275 r = a * pow(1 � eps, 0.5);
err = (pow((14.0*meanint+12.5)/(2.0*pi*(r*2.0+2.0)),0.5))/(14.0*meanint);
/* Calibrate data */
a = a * 0.046;
280 i = pow(10,(�12+(photzpt/�2.5))) * photflam * meanint / ``
(exptime * pow(0.046, 2));
/* Add data to list and add one to num */
285 ++num;
current�?a = a;
current�?i = i;
current�?di = err;
current�?datatype = 'e';
290 /* printf(''%d '',num); */
�
�
current�?nextaddr = NULL;
fclose(in�file);
295 break;
default:
printf(''No data read'');
�
return(num);
300 �
void display (struct profile *contents, int ndata, char galname[10])
--
305 /* Subroutine to display the a and i values from the list */
int n=0;
55

printf(''``n``t%s'',galname);
for (n=1; n != ndata; n++)
310 --
printf(''``na=%f'',contents�?a);
printf(''``t%d'',n);
printf(''``ti=%f'',contents�?i);
printf(''``tdi=%f'',contents�?di);
315 contents = contents�?nextaddr;
�
return;
�
D.2.2 Funcs.c
1 #include ''math.h''
void funcs(float x, float a[5], float *y, float dyda[5], int na)
--
5 float rad, k, inner, outer;
rad = x / a[4];
k = pow(2, (a[2] � a[3]) / a[1]);
inner = pow(rad, �1 * a[3]);
10 outer = pow(1 + pow(rad, a[1]), (a[3] � a[2]) / a[1]);
*y = a[5] * k * inner * outer;
dyda[1] = *y * (a[3] � a[2]) * (log(2) + log(2) * pow(rad, a[1]) � ``
15 log(1 + pow(rad, a[1])) � log(1 + pow(rad, a[1])) * pow (rad, a[1]) + a[1] * ``
pow (rad, a[1]) * log (rad)) / (pow(a[1], 2) * (1 + pow(rad, a[1])));
dyda[2] = *y * (log(2) � log(1 + pow(rad, a[1]))) / a[1];
dyda[3] = *y * (log(1/rad) * a[1] � log(2) + log(1 + pow(rad, a[1]))) / a[1];
dyda[4] = *y * (a[3] + pow(rad, a[1]) * a[2]) / (a[4] * (1 + pow(rad, a[1])));
20 dyda[5] = *y / a[5];
return;
�
56

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58