Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.naic.edu/~rminchin/download/dissertation.ps.gz Äàòà èçìåíåíèÿ: Thu Sep 1 17:57:57 2005 Äàòà èíäåêñèðîâàíèÿ: Sat Dec 22 03:03:37 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï

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 power­law fits : : : : : : : : : : : : : : : : : : : : : : : : : 21
3.3.4 Statistical analysis of power­law 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 Semi­major 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 ellipse­fitting to NGC4889 : : : : : : : : : : : : : : : : : : : : : : : : : : 17
3.2 Nuker­law fit to NGC4889 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20
4.1 Data from the average­radius fit : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23
4.2 Distribution of fl : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28
4.3 Nuker­law 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 Nuker­law 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 nuker­law fit to the profiles has been found and the relationships between the parameters
of the nuker­law 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 core­type 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 less­luminous
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 power­law 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 power­law, whilst smaller galaxies can be described by a single power­law 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 5­parameter double power­law to the inner profiles of galaxies. This
law, known as the `Nuker' law, fits both core­type and power­law 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 surface­brightness profile as it tends towards zero. It can be seen that for r Ü 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 well­resolved. The claim is also made that there is a divide between the core­type
galaxies and the power­law galaxies, which have fl ? 0:5. The core­type galaxies are large, triaxial
systems with M v ! \Gamma20:5 and the power­law galaxies are small, rotationally supported systems
with M v ? \Gamma22.
Faber et al find that there is a clear division between the core and power­law 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 power­laws 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 gas­rich 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 power­law galaxies in the halo region and preferential formation of core­type 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
surfact­brightness 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 cross­correlation 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 X­shift 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 ú 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 stop­value 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 centre­position
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 v­band 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 power­law 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 semi­major 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 D­type 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 ellipse­fitting 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 re­fitted. 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 (Levenberg­Marquadt) routine from Numerical Recipes (Press et al, 1992) to
find the five parameters. The fit was made to both semi­major axis and average­radius 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
2úrffir
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: Nuker­law fit to NGC4889
20

sample. This removed 4 galaxies, all of them from the central region, on the semi­major axis fits
and 7 galaxies, all but one from the central region, on the average­radius fits. Another galaxy was
removed from the central region in the average­radius 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 semi­major axis fit.
For the average­radius fit, this left 10 galaxies in the central region and 16 in the halo. For the
semi­major 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 C­1 (semi­major axis fits) and C­2
(average­radius fits) and the graphs obtained from these are also given.
3.3.3 Graphs from the power­law fits
The values of fl and r b from the nuker­law 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 power­law fits
The Kolomogrov­Smirnov two­sample 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.
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Chapter 4
Results
4.1 The Results
The results are given of fits to both the semi­major axis and the average­radius in tables 4.1 and 4.2.
Additional information is given in table 4.3. Data from the average­radius 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 Kolomogrov­Smirnov 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.
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Figure 4.1: Data from the average­radius 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
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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