The cores of galaxies in the Coma cluster
Rob ert Minchin
April
Contents
Intro duction
Historical Background
This pro ject
Data Reduction
Image pro cessing
The images
ray removal Cosmic
Ellipse tting
Galphot ellip t
IRAF ellipse
Checking the ellipse ts
Dusty galaxies
Data calibration
Other data
Analysis of data
Graphs from the ellipse ts
Isophote shap es
Images
Power laws
Fitting p ower laws
Information from the t
Graphs from the p ower law ts
analysis of p ower law ts Statistical
Results
The Results
Di erences b etween the two regions
vs M
v
Central region
Halo region
r vs
b
r r vs M
b lim v
Break radius vs E ective radius
Comparison with previous results
M vs
v
r vs
b
Diskiness
De ning a Core
Conclusions
vs M
v
M
v
vs r
b
r r vs M
b lim v
Diskiness
Summary
A Errors in the ellipses
Notes on Galaxies B
C Di eren tials of the Nuker Law
programmes D Computer
Recip es routines D Numerical
D My routines
D Pro ject c
D Funcs c
List of Tables
Semi ma jor axis ts
Average radius ts
Other galaxy data
List of Figures
Fab er et al results for r vs M
b v
Data from ellipse tting to NGC
Nuker law t to NGC
Data from the average radius t
Distribution of
Nuker law t to NGC
Plots of against M for the central and halo regions
v
and m r vs r
v b e
comparison to Fab er et al s data
cos terms for disky galaxies
Nuker law t to NGC
Abstract
I have analyzed HST images of galaxies selected from the central and halo regions of the Coma
cluster A nuker law t to the pro les has b een found and the relationships b etween the parameters
of the nuker law and b etween the total luminosity and isophote shap e has b een investigated
with half of the galaxies in the sample Problems were encountered with dust in the galaxies
su ering some degree of dust p ollution Where this was severe enough to a ect the pro le the
galaxy was excluded from the data set
The di erences b etween the central and halo regions have b een investigated It is predicted
that core typ e 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 di erence in magnitudes
b etween the two samples it is necessary for a relationship b etween and luminosity to exist in
order to show preferrential core formation in the central region It is not p ossible to say that this
relationship is de nately seen in the data
Comparison of the data with previous results shows that the lack of cores seen in less luminous
galaxies could b e an e ect of resolution This would create an arti cial trend of with luminosity
and an apparent dichotomy as only the steep outer p ower law would b e resolved in galaxies with
smaller cores giving a large value of This trend and the values of higher than could b e a
M galaxy selection e ect Disky galaxies are seen to b e able to form cores and a bright
v
is seen with no resolved core
Chapter
Intro duction
Historical Background
Originally cores were de ned as the areas of constant surface brightness in the centre of the King
mo del This was shown by King to b e a reasonable t to the central regions of
elliptical galaxies Later CCD imaging showed that the King mo del did not describ e adaquately
the surface brightness pro les 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 slop e sep erated from the steep er outer logarithmic slop e by a sharp break The inner
logarithmic slop e do es not go to zero as predicted by the King mo del Lauer et al at the
limit of resolution
Imaging of the cores of elliptical galaxies with the Hubble Space Telescop e has allowed them to
b e examined at previously unattainable resolutions The galaxies app ear to divide into two typ es
Ja e et al Larger galaxies generally have pro les with well de ned cores describ ed by a
double p ower law whilst smaller galaxies can b e describ ed by a single p ower law and have no well
Kormendy et al de ned core Bosch et al
It is known that larger galaxies are triaxial systems supp orted by random motions and there
fore having b oxy isophotes while smaller galaxies are rotationally supp orted with disky isophotes
Davies et al This division into two sorts of elliptical galaxies app ears to correlate well
with the division into those with and without unresolved cores Kormendy et al Ja e et al
Lauer et al tted a parameter double p ower law to the inner pro les of galaxies This
er law ts b oth core typ e and p ower law galaxies well law known as the Nuk
r r
b
I r I
b
r r
b
The parameters tted by this law are the inner logarithmic slop e the outer logarithmic
the sharpness of the break b etween the two slop es the radius of the break r and slop e
b
the intensity at the break I In terms of de ning a core is the most imp ortant as it shows
b
the slop e of the surface brightness pro le as it tends towards zero It can b e seen that for r r
b
I r r r I r r and for r
b
The results of tting this law to a numb er of elliptical galaxies and to the bulges of spiral
galaxies are given in Fab er et al This classi es galaxies as having cores if and
if the break is well resolved The claim is also made that there is a divide b etween the core typ e
galaxies and the p ower law galaxies which have The core typ e galaxies are large triaxial
systems with M and the p ower law galaxies are small rotationally supp orted systems
v
with M
v
Fab er et al nd that there is a clear division b etween the core and p ower law pro les see g
from Fab er et al which can b e seen on the graph of r vs M and also as a gap in the
b v
distribution of b etween and Figure also shows the division found into b oxy and disky
galaxies
Figure Fab er et al results for r vs M
b v
This pro ject
Bender Burstein and Fab er prop osed 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 supp orted 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
continuum GS continuum Fab er et al puts forward This theorem is called the gas stellar
this theorem as explaining why disky galaxies show p ower laws as the gas in dissipative mergers
is carried towards the centre of the galaxies
The Coma cluster provides a go o d 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 b e able to form a statistically signi cant
cluster has shown that it is sample of b oth regions and pro jection of HST data from the Virgo
p ossible to distinguish the two kinds of pro les at the distance of the Coma cluster
If the GS continuum theorem is correct it should b e p ossible to see preferential formation
of p ower law galaxies in the halo region and preferential formation of core typ e galaxies in the
central region This should result in there b eing a signi cant di erence b etween the distributions
of for the two samples I will also examine the relationships seen by Fab er et al in their survey
to determine if these are seen in the Coma cluster
There are galaxies in the sample from the central region of Coma and from the halo
In chapters and I explain how the images were reduced and the data analysed In chapter
I present my results and in chapter I give my conclusions The derivation of the error in the
surfact brightness pro le and notes on the galaxies are given in an app endix
Chapter
Data Reduction
Image pro cessing
The images
The Hubble Space Telescop e was used to obtain WFPC images of each of the galaxies
to in the sample The Planetary Camera images were extracted and trimmed from
in order to remove noise around the b ottom and left edges of the plates The images
were examined and those with susp ected dust noted
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 crre ject hsig gain These parameters tell
the routine to lo ok for ob jects that resemble cosmic rays sharply de ned bright p ositive p oints
and remove them if they are more than times the exp ected variation from the value on the
the other image The gain value of was obtained from the image headers and de nes how many
photon counts there are for each ADU
In order to ensure the alignment of the two images prior to combining them the IRAF routine
center was used to nd the centre of each image From this the o set b etween the centres in the
pair was found and those galaxies with an o set of less that pixels were combined those with
an o set of greater than pixels b eing noted see App endix B
Four galaxies had o sets outside of this range these were examined using the IRAF routine X
register This routine p erforms a cross correlation over a wide area rather than nding the centre
This routine gave two of the galaxies as having shifts less than pixels these were combined
The remaining two galaxies were shifted using X shift prior to b eing combined see App endix B
Ellipse tting
A working notation of naming the galaxies c to c was intro duced with c to c b eing in
o
the core angular distance from the cluster centre less than Mp c as de ned in Lucey et
al and c to c b eing in the halo This notation allowed automated routines to b e run
easily and the core and halo galaxies to b e easily distinguished Header les were constructed for
each galaxy containing the prop er names exp osure times and other data from literature
ellip t Galphot
The Galphot routines by R Peletier were used to t ellipses to the galaxies The routines identi ed
stars and remaining cosmic rays and masked them out Problems with this masking such as
unmasked di raction 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 t was made using the harm t routine This ts circular isophotes with harmonics
This pro duced a residual image which could b e used to check the mask
The nal t was made using the ellip t routine This ts elliptical isophotes for a numb er of
iterations then ts harmonic terms to these isophotes The routine creates a residual image after
each iteration and uses this to correct itself thus giving it go o d accuracy The t is done using
annuli of of the radius at intervals Outside pixel radius the t is done to all the pixels
that are inside the annulus within this radius the width of the annulus is less than pixel so
interp olation is used to calculate the values
ellipse IRAF
The IRAF routine ellipse was used as a check to the Galphot ts This routine ts by sampling
a numb er of sectors around annuli and using these to construct ellipses The routine app ears less
reliable than the Galphot routines as it uses a cruder tting metho d but pro duced comparable
output
In order to run this routine two IRAF scripts were used The rst ran an initial t over to
pixels radius and was seeded using p oints the ends of the ma jor an minor axes found from
an isophote contour map of the galaxy The second script used the largest reliable t indicated by
the programme returning a stop value of meaning that the t had b een trouble free from the
rst t and ran over the range to pixels radius For this second t a minimum value for the
intensity slop e was sp eci ed in order to stop the ellipses overlapping at low intensities At these
low intensities high radii tting was done concentrically with the ellipticity and centre p osition
held constant
Checking the ellipse ts
The ellipses used were those pro duced by the Galphot routines The ts prduced were consistent
with those from the IRAF routine The residuals pro duced by Galphot were also checked this
often showed dust that had b een missed on a prior insp ection of the image as the dust led to large
residuals b eing formed
Dusty galaxies
A numb er of galaxies are badly a ected by dust If there were serious features caused by dust after
masking of visible dustlanes an entry was made into the header les of these galaxies and they
were not included in the sample This excluded galaxies from the central region and from the
halo region
Ab out half of the sample were a ected by dust but half of these were not seriously a ected
the dust lanes were thin enough to b e masked out without any ma jor loss of information
Data calibration
The data tables were read into Sup er Mongo The data was calibrated from ADU to magnitudes
using the equation
P H O T F LAM D N
PT m log PHOTZ
E X PTIME
from the WFPC instrument handb o ok where DN is the numb er of counts PHOTFLAM and
PHOTZPT are photometric calibration constants the sensitivity and zero p oint of the CCD s
which have the values and resp ectively for this data and EXPTIME is the exp osure
time The values for these were obtained from the image header les EXPTIME was not constant
over the set of readings b eing seconds for the fainter galaxies seconds for the ma jority
of galaxies in the sample and seconds for the dominant galaxy NGC The distance scale
was calibrated using pixel arcseconds This allowed the surface brightness to b e found
2
in terms of magnitude arcsec
Other data
b carried The magnitude of the galaxies in the v band was found from measurements of A and S
e e
1
by Lucey et al The distance to Coma was taken from Fab er et al as km s
1
A value of H km s Mp c was used to calculate the absolute magnitudes and the angular
0
scale as this was the value adopted by Fab er et al
The header les eventually contained the prop er names exp osure times distances from the
centre of the Coma cluster in degrees e ective diameter and average surface brightness within
this diameter and whether the galaxy was seriously a ected by dust The data on the distances
from the centre of Coma the e ective diameter and the e ective surface brightness were those from
Lucey et al mentioned ab ove I made the judgement on whether the galaxies were badly a ected
by dust if there were problems due to dust which prevented ellipses b eing tted over an extended
region or if the pro le was distorted by dust so it could not b e well tted by a p ower law then the
galaxy was de ned as b eing badly a ected by dust
Chapter
Analysis of data
from the ellipse ts Graphs
Graphs were made using Sup er Mongo by reading in the data les pro duced by the ellip t 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 p er arcsecond squared p osition angle ellipticity deviation of the centre
in x and y and the harmonics for and were plotted against the log of a the semi ma jor axis
r
p
For data from galphot a was calculated using a where is the ellipticity This conversion
1
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 gure this is of the D typ e galaxy NGC
c in my numb ering scheme The results from Galphot are shown by crosses and the results
from IRAF are shown by b oxes
Figure Data from ellipse tting to NGC
Isophote shap es
The range over which the pro le app eared reliable was from a radius of pixels to a surface
2
brightness of mag arcsec The highest absolute value reached by the cos term in this
region was used to determine whether the galaxy app eared to b e b oxy or disky
A Sup er Mongo routine was written to return the highest absolute value This value was then
converted to a a by dividing by conversion factor from Pelatier This gure shows the
distortion of the ellipse into a b oxy or disky shap e disky shap es having p ositive values For the
purp oses of de nition galaxies with a a were said to b e disky if this value was not b eing
returned due to distortions due to dust
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 to of the luminosity the
core of the galaxies scaled logarithmically from to of the luminosity and the residuals
left by galphot The outer parts and the residuals were at magni cation and the cores were
at magni cation with the exception of NGC the cD galaxy which was shown at
magni cation
The greyscales were examined for signs of visible dust and galaxies thus identi ed 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 tted Most of the large residuals were due to severe dust
contamination and only small improvements in the t were p ossible
Power laws
er laws Fitting p ow
A computer programme was written to t the nuker law to the surface brightness pro le using a
simulated annealing Levenb erg Marquadt routine from Numerical Recip es Press et al to
nd the ve parameters The t was made to b oth semi ma jor axis and average radius pro les
1
b etween pixels and mag arcsec the limits that app ear to b e the limit of resolution and
the p oint where sky e ects b ecome imp ortant
p
2
g D N +
r n
p
was used in A prop ortional error in the numb er of counts DN of P
N D
g D N 2 r r
the input data see App endix where D N is the average numb er of counts in ADU in an
r is the width of the annulus g is the gain and annulus r is the inner radius of the annulus
r n
is the read noise
5
p
The read noise is taken to b e as each image has a read noise of and two images were
2
averaged The gain is taken to b e as each image has a gain of so photons are needed for
each ADU
from the t Information
The t returned values and errors for the parameters needed for the nuker law r
b
I These were used to construct a mo del pro le which was compared with the original to check
r
whether the t was satisfactory If the errors asso ciated with the constants of the t were to o large
or the t did not match the original date the t was rejected as unsatisfactory
An example graph is given in g of NGC showing the datap oints with the t through
them in the top half and the di erence b etween the t and the datap oints with the line of error
around the datap oints in the b ottom half
The t was unable to t satisfactorily to some of the pro les these were removed from the
Figure Nuker law t to NGC
sample This removed galaxies all of them from the central region on the semi ma jor axis ts
and galaxies all but one from the central region on the average radius ts Another galaxy was
removed from the central region in the average radius ts as it had an inner logarithmic slop e
that indicated a core but its break radius was to o close to the limit of resolution to b e considered
reliable as was one from the outer regions on the semi ma jor axis t
For the average radius t this left galaxies in the central region and in the halo For the
semi ma jor axis ts this left galaxies in the central region and in the halo
The parameters returned for each galaxy are given in tables C semi ma jor axis ts and C
average radius ts and the graphs obtained from these are also given
Graphs from the p ower law ts
The values of and r from the nuker law ts and the values of m and distance from the centre
b v
of Coma were plotted against each other in graphs of apparent values and graphs of absolute
values Only those galaxies left in the sample as detailed ab ove were graphed These graphs were
examined for trends and for di erences b etween the samples from the inner and outer regions of
Coma
analysis of p ower law ts Statistical
The Kolomogrov Smirnov two sample test was used to examine di erences b etween the samples
This test gives the con dence that two samples are from the same parent p opulation Details of
the test were found in Wall This test was used to lo ok for di erences in magnitude and
Chapter
Results
The Results
The results are given of ts to b oth the semi ma jor axis and the average radius in tables and
Additional information is given in table Data from the average radius t is shown graphically
in gure with crosses representing galaxies from the central region and b oxes representing
galaxies from the halo region
b etween the two regions Di erences
The Kolomogrov Smirnov test showed that the distributions of in the two regions were likely to
b e from the same parent distribution eg there was no signi cant di erence b etween them Using
the same test the probability that the magnitudes were from the same parent distribution was
tested and was found to b e in the tail of the normal distribution eg the distributions of the
magnitudes were fairly signi cantly di erent
Figure Data from the average radius t crosses central b oxes halo except for m vs R
v
crosses op en b oxes b oxed cross
c d a b
name r Status
b b b
NGC
IC
NGC
RB
IC
IC
RB
RB
NGC
RB
RB
RB
NGC
IC
RB
IC
IC
RB
IC
NGC
RB
NGC
NGC
NGC
Zw
IC
Zw
IC
NGC
NGC
Zw
Zw
IC
NGC
NGC
IC
NGC
NGC
NGC
NM
IC
NGC
NGC
NGC
Zw
a
Break radius in parsec
b
Break radius in arcsec
c 2
surface brightness in mag arcsec at the break radius
d
Okay Dusty Unsatisfactory t Break radius less than limit of resolution arcsec
Table Semi ma jor axis ts
a b c d
name r Status
b b b
NGC
IC
NGC
RB
IC
IC
RB
RB
NGC
RB
RB
RB
NGC
IC
RB
IC
IC
RB
IC
NGC
RB
NGC
NGC
NGC
Zw
IC
Zw
IC
NGC
NGC
Zw
Zw
IC
NGC
NGC
IC
NGC
NGC
NGC
NM
IC
NGC
NGC
NGC
Zw
a
Break radius in parsec
b
Break radius in arcsec
c 2
surface brightness in mag arcsec at the break radius
d
Okay Dusty Unsatisfactory t Break radius less than limit of resolution arcsec
Table Average radius ts
a b c d d
name mag radius abs mag c a
NGC
IC
NGC
RB
IC
IC
RB
RB
NGC
RB
RB
RB
NGC
IC
RB
IC
IC
RB
IC
NGC
RB
NGC
NGC
NGC
Zw
IC
Zw
IC
NGC
NGC
Zw
Zw
IC
NGC
NGC
IC
NGC
NGC
NGC
NM
IC
NGC
NGC
NGC
Zw
a
from V band measurements of A and SB Calculted
e e
b
Distance from centre of Coma cluster in degrees
c 1
Calcultated from apparent magnitude taking H Mp c km s
0
d
deviations from ellipse and a a deviation from ellipse calculated from c by dividing by cos
Table Other galaxy data
The galaxies in the halo region were on the whole a magnitude brighter than the galaxies in
the central region b oth regions had the same distribution of in their samples If there is a trend
of with M this would show that there was a di erence in the formation of core typ e galaxies
v
b etween the halo and central regions
vs M
v
Examination of the graph of M against do es not show any de nite trend out of galaxies
v
brighter than magnitudes show cores indicating that larger galaxies generally contain cores
There do not app ear to b e many smaller galaxies with cores but it is imp ossible to say whether
a physical e ect or whether this is an artefact of resolution this re ects
There are p oints in the region For a uniform distribution b etween and
which encompasses all the values in the data set p oints would b e exp ected in this region The
p
exp ected counting error on this is therefore this means that the numb er of p oints
found in this region is barely signi cantly less than would b e exp ected from a uniform distribution
It is still the case that this region has a lower numb er density than the core or p ower law regions
either side of
The frequency distribution of the s of galaxies from the average radius t is shown in g
The shaded area represents galaxies in the halo region the clear area represents galaxies in
the central region
Central region
There is only one bright galaxy in the sample NGC This is one of two D galaxies in the
central region of Coma and shows a de nite 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 di erence in
magnitudes b etween the two regions
Figure Binned distribution of for the central region upp er graph and the halo region lower
graph
The evidence for a trend in with M is fairly go o d in this region see g although this
v
could b e a selection e ect due to the smaller low luminosity galaxies two of the galaxies in the
intermediate zone are in the central region exactly the numb er that would b e
exp ected for a uniform distribution b etween and and the same numb er density as is seen in all
bins when is binned in tenths see t
region Halo
There are bright galaxies in the sample de ned as b eing in the M region where Fab er et
v
al nd no cores two of them show de nite cores but the third NGC has a single p ower law
through to the limit of resolution see g There are few faint galaxies in the sample due to
the lack of comprehensive surveying in the halo region of Coma
There is only galaxy in the intermediate zone in the halo region even if this region is extended
to cover There do es app ear to b e a well de ned gap see g This region shows
virtually no correlation of with M see g
v
r vs
b
Taking the values returned by the t there is a fairly even distribution of break radii see g
If it is assumed that the cores are unresolved for then all these galaxies have break radii
smaller than the limit of resolution r p c limit of resolution given by log r arcsec
lim b
as used in Fab er et al
r r vs M
b l v im
Fab er et al claim values for r much smaller than their value used for the limit of resolution
lim
for the cores by examining deconvolved pro les within arcsec and comparing these with a
Figure Nuker law t to NGC
Figure Plots of against M for the central and halo regions
v
convolved and deconvolved mo dels with break radii of and arcsec By saying
that the deconvolved pro les give a lower limit to the surface brightness they have said that where
the mo dels fall b elow the deconvolved pro le the break radius for that mo del gives an upp er limit
for the break radius of that galaxy This has allowed them to set upp er limits of and
rounded from on the break radii of some galaxies
On other galaxies no upp er limit has b een found from the mo dels so the upp er limit has b een
set at arcsec which was the upp er limit of the region of the pro le they were examining As
the limit of resolution for cores has b een set at arcsec this has led to an apparent gap b etween
the break radii of the observed cores and the maximum p ossible break radii of the galaxies without
resolved cores
I have used r p c at Coma distance arcsec If this is used instead of the
lim
arcsec range used by Fab er et al then no gap is seen b etween the break radii of resolved
cores and the maximum break radii of unresolved cores as is seen in the lower graph of gure
crosses central b oxes halo
Break radius vs E ective radius
Fab er et al found a linear relationship b etween e ective radius and core radius r r in
b e
those galaxies with resolved cores This relationship has b een plotted with my data p oints in
the upp er graph of gure I do not nd this relation in the galaxies with resolved cores in our
sample as cam b e seen from this graph Addition of more large galaxies would probably lead to a
relationship b eing found as it is in Fab er et al but this relationship do es not app ear to hold when
only the smaller galaxies are considered eg r p c
e
Figure Upp er plot of r vs r for those galaxies with with the t found by Fab er et
b e
al shown by the dotted line
Lower Plot of r vs m with r substituted for galaxies with
b v lim
Comparison with previous results
Plots of my data and Fab er et al s data are in gure My data is shown by crosses and Fab er
et al s by b oxes
M vs
v
The results are similar to those obtained by Fab er et al b elow magnitudes the cut o
p oint for our observations see g I have found signi cantly more p oints with
than Fab er et al and my results do not show the clear trend given by the low luminosity tail in
Fab er et al s results see g
vs r
b
Once again our results are very similar to Fab er et al in the larger galaxies ab ove our limit of
resolution log r r p c see g Below this all of Fab er et al s results have
b b
and have either b een classi ed as cores or as having to small a break radius to classify as
resolved The galaxies in this section that were classi ed as having cores would not have had
these cores resolved in our sample
In the Fab er et al sample M has a resolved core at its true distance but this core is unresolved
when it is translated to Virgo This would app ear to supp ort the hyp othesis that the core typ e
p ower law distinction is an e ect of resolution
Diskiness
Diskiness is de ned as describ ed in section as having a maximum value of a greater than
Out of the dust free galaxies meet this criterion These are RB RB IC IC
and NGC from the central region and IC Zw NGC and NGC from the
Figure Comparison of Fab er et al s data b oxes and my data crosses
halo region NGC is a D typ e galaxy with dust in the central region that while it hasn t
a ected the t has led to distortion of the a a term and so can b e removed from this list The
cos deviation pro les of the galaxies which app ear to b e disky are given in g Galphot
results are shown by crosses and IRAF results are shown by b oxes
The other galaxies app ear to have disks on insp ection of the graphs This includes galaxies
in the central region in the halo which have resolved cores All of these galaxies are
fairly bright b etween and absolute magnitude This app ears to contradict Fab er et al s
assertion that galaxies with cores are b oxy and galaxies with p ower laws are disky
De ning a Core
Throughout this rep ort I have used Fab er et al s de nition of a core as having There are
galaxies such as NGC see g with a very well de ned break but with a steep inner section
Galaxies with well de ned breaks are found over a range of values of these would app ear to b e
more than random ukes and it is p ossible that the low cores are a sub set of whatever physical
pro cess causes these breaks to lower slop es there is not particular reason for cho osing as
the cut o rather than a higher value except for the apparent dichotomy seen at this p oint see
g It is also p ossible that there are a numb er of these steps in some galaxies leading to an
almost at central core This would require a multiple p ower law t to b e accurately describ ed
Figure cos terms for the galaxies identi ed as disky Crosses are Galphot data b oxes are
IRAF data
Figure Nuker law t to NGC
Chapter
Conclusions
M vs
v
There is no de nite 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
It is shown statistically that the values of in b oth regions are probably from the same parent
p opulation The distribution di ers b etween the regions with there b eing two distinct p eaks in the
distribution of gamma as seen by Fab er et al in the halo region while there is a at distribution
in the central region
M
v
The p opulations of M seen in the two regions are shown to b e probably from di erent parent
v
p opulations the halo p opulation b eing on average a magnitude brighter than the core p opulation
vs r
b
There do es not app ear to b e a link b etween the value for the break radius returned by the tting
routine and the value for
r r vs M
b l im v
There is no clear evidence seen for a trend in r with M nor is the break seen by Fab er et al seen
b v
when the limit of resolution is used as r
lim
Diskiness
Disky galaxies are seen to b e able to form cores as well as p ower law pro les this app ears to indicate
that it is not true that rotationally supp orted galaxies have p ower laws and triaxial galaxies have
cores
Summary
There is a spread of break radii from the limit of resolution outwards The gap seen by Fab er et
al is not apparent when the maximum p ossible core radius for galaxies without resolved cores is
set to b e the limit of resolution
The drop in the numb er of galaxies with is seen only in the halo region not in
the central region
It is seen that bright galaxies can form apparent p ower laws and that disky galaxies can form
cores
No de nite trend can b e seen in the graphs of vs M It is therefore not p ossible to use the
v
di erence in magnitudes to say that there would b e a di erence in the distributions of if the two
samples were over the same magnitude range Therefore the di erence b etween the two regions
predicted by the gas stellar continuum has not b een proven
knowledgements Ac
I would like to thank Professor Davies Dr de Jong and Dr Lucey for their advice and help in doing
this pro ject
App endix A
in the ellipses Errors
Average numb er of counts in an annulus D N
Gain g Average numb er of photons in annulus numb er of electrons g D N
Numb er of pixels in an annulus area of annulus r r
Read noise p er pixel in electrons
r n
p
2
r r Total read noise in annulus
r n
p
2
2 r r
r n
r n
p
Average read noise in annulus
2 r r
2 r r
Total numb er of counts in annulus D N r r
Total numb er of photons in annulus g D N r r
p
Counting error in total photons g D N r r
p
q
g D N 2 r r
g D N
Counting error in average photons
2 r r 2 r r
Total error in ellipse
r
2
2
p
r n
p
2 r r
r
2 2
g D N
r n
p
sq r t
2 r r
2 r r
q
2
g D N +
r n
2 r r
Prop ortional error in average numb er of counts in annulus
p
2
g D N +
r n
p
P DN
g D N 2 r r
App endix B
Notes on Galaxies
Those galaxies which had an o set b etween the centres of the galaxy in the two images of b etween
and pixels are noted as such b elow No note is made if the o set was less than pixels
For the four galaxies with an o set greater than pixels it is noted whether they had a shift
less than pixels when examined using a cross correlation routine or whether one of the images
had to b e 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
NGC had a separation b etween the two images of b etween and pixels when they
were combined There is no evidence of dust either visibly or in the terms
IC shows no evidence of dust either visibly or in the terms There is a small companion
galaxy or a star in the b ottom right of the frame
NGC had a separation b etween the two images of b etween and pixels when they
were combined There is visible dust in the core this also shows up in the terms
RB had a separation b etween the two images of b etween and pixels when they were
combined There is no evidence of dust either visibly or in the terms
IC has visible dust in the core this also shows in the terms
IC shows p ossible visible evidence for a ring of dust around the core This do es not
show up in the terms but there do es app ear to b e a disruption in the terms There is
a companion galaxy in the b ottom left of the frame
RB shows p ossible visible evidence for dust This do es not show up in the terms but
there is a discrepancy b etween the terms measured by ellipfit and ellipse inside of
arcsec pixels There are a couple of stars in the b ottom left of the frame This galaxy
is classi ed as a p ossible lenticular
RB shows no visible evidence of dust but there is some evidence of an asymmetry in the
terms
NGC had a separation b etween the two images of b etween and pixels when they
were combined There is visible dust in the core this also shows up in the terms There
is a large star in the b ottom left of the frame there is another ob ject to the top left and two
ob jects to the top right
RB shows no evidence of dust either visibly or in the terms There is a star to the top
right of the frame
RB shows no evidence of dust either visibly or in the terms There are o dd stretch
marks on the ellip t residual lines pro jecting radially from the centre and symmetric through
the centre
RB shows no visible evidence of dust but there is evidence of an asymmetry in the
terms The galaxy undergo es a visible rotation b etween radii of and arcseconds The
terms are also confused The core app ears disky and this is supp orted by the cos terms
this do es not extend outwards
NGC had the two images shifted to match them b efore they were combined This is the
cD galaxy and is very at topp ed There is little visible evidence of dust but the and
terms are very confused within arcsecond This is within the core radius so do es not a ect
the conclusion that this galaxy has a soft core
IC has visible dust in the core this do es not show particularly in the terms but there
is a discrepancy b etween the ellipse and ellipfit terms
RB has visible dust in the core this also shows up in the terms The core is visibly
disky There is a star at the top of the frame There are o dd stretch marks on the ellip t
residual lines pro jecting radially from the centre and symmetric through the centre
IC has visible dust in the core this also shows in the terms
IC shows no evidence of dust either visibly or in the terms There are a numb er of
small ob jects on the frame
RB shows no visible evidence of dust although the terms app ear confused
IC has faintly visible dust in the core this also shows up in the terms There are a
large numb er of small ob jects on the frame
but cross NGC app eared to have a separation of greater than pixels using center
correlation using X register gave a smaller shift There is no visible evidence of dust but
there is evidence of an asymmetry in the terms
RB shows no evidence of dust either visibly or in the terms
NGC shows no evidence of dust either visibly or in the terms
NGC had a separation b etween the two images of b etween and pixels when they
were combined There is no evidence of dust either visibly or in the terms
NGC shows no visible evidence of dust but there is some evidence of an asymmetry in
the terms The ellipfit routine has not tted b etween and pixel radius
Zw shows p ossible visible evidence of dust and there is evidence of an asymmetry in
the terms There is an hump in the brightness pro le at the p oint where the asymmetry
o ccurs
IC had a separation b etween the two images of b etween and pixels when they
were combined There is no evidence of dust either visibly or in the terms
Zw shows no evidence of dust either visibly or in the terms There is a discrepancy
b etween the ellipse and ellipfit terms near the core There is a star in the top right
of the frame
IC shows no evidence of dust either visibly or in the terms
NGC shows no evidence of dust either visibly or in the terms There is a discrepancy
b etween the ellipse and ellipfit terms within arcsecond
NGC shows no evidence of dust either visibly or in the terms
Zw shows visible evidence of a dust lane From the terms this lane extends from
arcseconds to arcseconds The t continues inside arcseconds The residual is very
confused at the radii of the dust lane
Zw shows no evidence of dust The cos terms show a de nite disky comp onent
extending to arcseconds radius
IC shows visible evidence for a dust ring around the core this also shows faintly in the
and terms and in a hump on the brightness pro le
NGC shows a strong dust lane through the core this also shows in the and terms
and in a turn down of the brightness pro le near the centre The ellipfit t stops at ab out
arcseconds radius
NGC shows no evidence of dust either visibly or in the terms
IC shows visible evidence for a dust lane around the core this also shows in the terms
NGC app eared to have a separation of greater than pixels using center but cross
correlation using X register gave a smaller shift There is no evidence of dust either visibly
or in the terms
NGC shows visible evidence for a faint dust lane in the core this also shows in the
terms
NGC had a separation of b etween and pixels b etween the two images when they
were combined There is no visible evidence for dust but the and terms are very
confused It has a very at pro le similar to that of NGC
NM shows no evidence of dust either visibly or in the terms There is a discrepancy
b etween the ellipse and ellipfit terms near the core
IC shows no evidence of dust either visibly or in the terms
NGC shows visible evidence for a dust lane in the core this also shows in the terms
NGC had the two images shifted to match them b efore they were combined There is
no visible evidence for dust but there is evidence for an asymmetry in the terms
NGC had a separation of b etween and pixels b etween the two images when they
were combined There is no evidence of dust either visibly or in the terms
Zw shows no evidence of dust either visibly or in the terms
App endix C
tials of the Nuker Law Di eren
These were calculated using the Maple symb olic pro cessor programme They are necessary in order
to t the nuker law to the surfacr brightness pro les
c
r r
b
I r I
b
r r
b
r r r dI r
b
I ln ln
r
d r r r
b b
a
r r r r r
2
ln ln ln
r r r r r
b b b b b
r
r
b
a
r r r
b
I ln ln
b
dI r
r r r
b b
d
r r r r dI r
b b
I ln ln ln
b
d r r r r
b b
r r r
b
I
b
dI r
r r r
b b
dr
r
b
r
b
r
b
a
dI r r r
b
dI r r
b b
App endix D
Computer programmes
Recip es routines D Numerical
The following Numerical Recip es routines were used in the pro ject covsrt c gaussj c mrqmin c
and nrutil c
D My routines
Pro ject c reads the data in from le and calls the tting programme mrqmin c It supplies initial
values and stopping conditions for the tting
Funcs c is called by mrqmin c and returns the nuker law and the di erentials
D Pro ject 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 nextadd r
main
15 struct profile list temp
display s tr uc t profile int char void
void fit int struct profile float float float float
int populate s tr uc t profile char
float nalamda nchisq
20 float alamda covar alpha chisq
char galname
int ndata i
ri x covar mat
25 alpha mat ri x
al amd a alamda n
chisq nc hi sq
alamda 30
chisq
Set up the list and then populate it
of st ru ct profile list struct profile malloc si ze
35 ndata populate list galname
dump the first blank entry
temp list
ta dd r list list nex
40 free temp
ndata ndata
the contents of the list Display
display l is t ndata gal na me
45
n t s n galname printf
fit ndata l ist a la md a c ov ar al ph a ch isq
50 void fit int ndata struct profile list float alamda float covar float
alpha float chisq
float x ndata y ndata sig ndat a a
int ia
55 int iter zcnt
void funcs flo at f lo at float float int
void mrqmin fl oa t float float int flo at int int f lo at
float float void funcs f lo at fl oa t float float int fl oa t
int n m
60 struct profile current
float ochi
out file cov file FILE
char o name c name temp
int df ndata
65
output data file Open
printf nE nte r output filename c fit g or c fit e n
scanf c te mp
gets o na me
printf nE nte r covarianc e matrix filename c cov g or c cov e n 70
gets c na me
out file fo pen o n am e w
cov file fo pen c n am e w
fprintf o ut fi le al pha t be ta tg am ma tb re ak ra d tb re ak in t tch is q n
75
current list
Setup an array with the values in
80 for n n ndata n
printf d n
x n current a
y n current i
85 current current ne xt add r
sig n current d i y n
printf n
90
a
a
a
a
95 a
for n n n
printf a d f Enter new value n a n
100 scanf f a n
set which values to vary
printf v ar y a d no yes n
fprintf o ut f il e f t a n
scanf d i a n
105 if ia n
ia n
fprintf o ut fi le n
110 Note a is times true value
un nin g Iterative Fitting Function n printf R
printf I te r Chisq A A A A A n
mrqmin x y sig n da ta a ia co va r al pha c hi sq fu nc s ala md a
115 printf d f t f f f f f n ite r chisq a a a
a a
do
ochi chisq
120 mrqmin x y s ig nd at a a i a cov ar a lph a ch is q f un cs al am da
iter
if ochi chisq
zcnt
else
125 zcnt
for n n n
if a n
a n a n
printf d f t f f f f f n it er chisq a a
130 a a a
zcnt chisq ochi while ochi chisq
alamda
mrqmin x y sig n da ta a ia co va r al pha c hi sq fu nc s ala md a
135
for n n n
df df ia n
Print out values for the constants
140 printf nc his q t f c hi sq
printf nd f t i df
printf nr edu ce d chi squared t f chi sq d f
printf ni ter t d ite r
for n n n
145 printf n a d t f tC ov ar t f tS E t f n a n co var n n
pow cova r n n
printf n
for n n n
150 fprintf ou t fil e f t a n
fprintf o ut fi le f n ch isq
for n n n
fprintf ou t fil e f t po w cov ar n n
155 for m m m
fprintf c ov fi le f t c ov ar n m
fprintf co v fil e n
160 fprintf o ut fi le f n ch isq d f
fclose ou t fil e
return
165 int populate st ru ct profile current char galname int ndata
Subroutin e to read the data and put it into the list
FILE in file
170 int n num
float exptime photzpt photflam err pi
tmp char galnum f name h name
float a r dr i s x y ep s po s s s s s c c c c f f f f
float meanint rm s te ta x y s lo pe m ag A B A B b igA
175 char choice
a file to work upon Gets
printf n En ter galaxy number
gets gal nu m
180
header file name and data file name from galaxy number Make
printf n Enter data file name
gets f na me
185 printf n Enter header file name
gets h na me
printf n Enter e to fit to ellipse data g to fit to galphot data
scanf c choice
190 Open header file
in file f op en h na me r
fgets tmp in f il e
Set photomet ric constant s and read exptime
195 photflam
photzpt
pi
fscanf in file f s exptime galname
printf e xp tim e f n exp tim e
200
Close header file
fclose in f ile
Check choice
205
Set num number of data to zero
num
switch ch oi ce
210
case g
Open galphot datafile
in file fo pe n f nam e r
215 Dump the first two lines
for n n n
fgets tm p i n fi le
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 d r i s x y e ps pos s s s s c c c
EOF c f f f f
225 if r i exptime
Set nextaddr ess
current n ex tad dr struct profile malloc si ze of st ru ct profile
current cu rr ent ne xta dd r
230
Finds error in i
if a
err pow i pi r d r r i
else
235 err pow i pi r i
Finds a
a r sqrt ep s
240 Calibrat e data
a a
i pow ph otz pt photflam i exptime
pow
245 Add data to list and add one to num
num
current a a
current i i
current d i err
250 current d at aty pe g
printf d num
current n ex ta dd r NULL
255 fclose in fi le
break
case e
Open ellipse file
260 in file fo pe n f nam e r
Dump the first two lines
for n n n
fgets tm p i n fi le
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 mea ni nt r ms eps te ta x y sl op e ma g A B A B b igA EOF
if i exptim e
270
Set nextaddr ess
current n ex tad dr struct profile malloc si ze of st ru ct profile
current cu rr ent ne xta dd r
Find the error
275 r a pow eps
err pow me ani nt pi r m ea nin t
Calibrat e data
a a
280 i pow ph otz pt photflam meanint
exptime pow
Add data to list and add one to num
285 num
current a a
current i i
current d i err
current d at aty pe e
290 printf d num
current n ex ta dd r NULL
fclose in fi le
295 break
default
printf No data read
return nu m
300
void display struct profile contents int ndata char galname
305 Subroutin e to display the a and i values from the list
int n
printf n t s ga ln ame
for n n ndata n
310
printf n a f c on ten ts a
printf t d n
printf t i f c on ten ts i
printf t di f co nte nt s d i
315 contents contents n ex ta dd r
return
D Funcs c
1 include math h
void funcs f loa t x float a float y float dyda int na
5 float rad k inner outer
rad x a
k pow a a a
inner pow rad a
10 outer pow pow rad a a a a
y a k inner outer
dyda y a a log log pow rad a
15 log pow rad a log pow rad a pow rad a a
pow rad a log rad pow a pow rad a
dyda y log log pow rad a a
dyda y log r ad a log log pow rad a a
dyda y a pow rad a a a pow rad a
20 dyda y a
return
References
Bender R Burstein D Fab er S M Ap J
van den Bosch F C Ferrarese L Ja e W Ford H C O Connell R W AJ
I Grillmair C J Fab er S M Ajhar E A Dressler A Kormendy J Lauer T Byun Y
R Richstone D Tremaine S AJ
Davies R L Efstathiou G Fall S M Illingworth G Schechter P L Ap J
Fab er S M Tremaine Ajhar E A Byun I A Dressler A Gebhardt K Grillmair C
Kormendy J Preprint
Ferrarese L van den Bosch F C Ford H C Ja e W O Connell R W AJ
Gebhardt K Richstone D Ajhar A E Lauer T R Byun Y I Kormendy J Dressler A
Fab er S M Grillmair C Tremaine S Preprint
Ja e W Ford H C O Connell R W van den Bosch F C Ferrarese L AJ
King I R AJ
King I R ApJ
Kormendy J Dressler A Byun Y I Fab er S M Grillmair C Lauer T R Richstone D
Tremaine S ESO OHP Workshop on Dwarf Galaxies ed Meylan G
Lauer T R Ajhar E A Byun Y I Dressler A Fab er S M Grillmair C Kormendy J
Richstone D Tremaine S AJ
Lucey J R Guzman R Carter D Terlevich R J M N R A S
Peletier R Elliptical Galaxies Structure and Stellar Content PhD Thesis
Press W H Teukolsky S A Vetterling W T Flannery B P Numerical Recip es in
C Second Edition
Wall J V Practical Statistics for Astronomers