CHAPTER 14 GALAXY PROFILES AND PARAMETERS
14.1 INTRODUCTION
In this chapter we describe how to obtain the surface
brightness distributions and the seeing-free photometric and
structural parameters of elliptical galaxies which may or may not
overlap, from two-dimensional images. The modules can also be
used to model star images which may or may not overlap, as well.
The surface brightness distributions are averages around
elliptical contours expressed as a function of semimajor axis.
The central location, inclination, and eccentricity of each
elliptical contour are allowed to be independent of each other
and can be solved for as a function of semimajor axis by the
method of Fourier coefficients if desired. Thus one can look for
twisting of isophotes and variations of ellipticity.
Alternately, these quantities can be held constant and simple
contour averages made. This approach is less desirable, but is
sometimes necessary if the center of the galaxy lies off the
frame. The photometric and structural parameters, e.g., central
surface brightness, effective surface brightness, Hubble radius
parameter, effective radius parameter, etc., are derived from
nonlinear least squares fits (with image smearing taken into
account) of one-dimensional files representing the contour
averaged surface brightness.
The surface brightness distribution of an isolated galaxy is
easily solved for, however the images of most galaxies are
contaminated by overlapping images of nearby stars and other
galaxies, especially if one is working in a rich cluster of
galaxies. Bad pixels and ghost images must also be dealt with.
Briefly, one proceeds with the following recipe, which is
explained in more detail in the later sections of this chapter.
The editing options EC (EDITCIRCLE), EP (EDITPOINT), and ER
(EDITRECT) allow one to remove bad areas of the 2-dimensional
image (SAD file) when starting the analysis. We use these
options to edit out everything but the brightest galaxy, or the
galaxy of interest. Then by using the options EA (ANNULUS), PR
(PROFILE), and/or SG (SMOOTHGAL) (described in more detail
below), we arrive at a first approximation to the two-dimensional
surface brightness distribution of that galaxy. By subtracting
this first approximation model with option FG (FAKGAL), other
objects which were contaminated by the light from this galaxy can
now more accurately be modeled by further use of these same
options. Stars and ghost images are similarly modeled and
subtracted. A helpful option at this stage is SA (SADMASK),
which operates with with a mask file to quickly remove areas know
to be bad or which contain objects in the frame, too small to be
modeled. In general, one proceeds from the brightest object to
the fainter ones. You may not wish to model the very faintest
galaxies since less accurate surface brightness parameters and
radius parameters will result. However, these fainter objects
are sometimes located where their modeling and subtraction will
improve the results for another more desirable galaxy. The
fainter objects are still useful for obtaining aperture derived
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integrated magnitudes with option BB (BBEN). Once the initial
models for the objects have been determined and subtracted, the
frame should look reasonably clear except for the unmodeled
fainter and less offensive objects. The first object can be
remodeled in a second iteration. Severely overlapping images may
require several iterations before a final model is obtained.
The surface brightness distribution as derived from
iterative modeling is still affected by atmospheric seeing. In
order to derive the actual galaxian surface brightness parameter
and radius parameter, this smearing must be accounted for. In
the nonlinear least squares fitting routine, RG (RADGAL), a
stellar profile is used as a smearing function to convolve the
fitting function before the actual fit is made. Finally, the
photometric parameters are calibrated by placing them on some
photometric system, and the structural parameters are transformed
to units of arc seconds. These conversions operations are done
with options ED (EDITDATA1) and E3 (EDITDATA3).
Galaxies' models too faint to be accurately fit with RG can
be integrated to yield magnitudes with option E3, command I.
Galaxies too faint to be modeled can be photometered with option
BB.
14.2 CONSTRUCTING A FINDING CHART
Before any modeling work on a frame can begin, it is helpful
to have a finding chart. Option AC can be used here to produce a
contour plot. However, if the flat fielded exposure of the frame
of interest contains an enormous galaxy which overlaps other
galaxies, it may be necessary to subtract out this galaxy before
constructing the contour plot. Once the frame has been selected,
use option GA to determine the sky level and the sigma of the
exposure in a blank area. The lowest contour intensity level
should be set at approximately the sky plus 7 sigma. The
hightest contour level should be set equal to the center of the
brightest interesting object. Intermediate contour levels, say 4
of them, should be spaced in a geometric progression between the
lowest and highest levels. Some trial and error may be necessary
before finding the right intensity levels. Experiment on a small
area of the map, as this option is slow. Write the plot to a
file but do not put a frame around the plot. The plot file can
then be sent to one of the supported plot devices. Observe the
offsets, scale, and rotations recommended in Chapter 15 of this
volume. The option FI (FIND) can be used to automatically create
a list of objects and their positions in the frame, or one can do
this manually with option CT (CENTR).
14.3 CONSTRUCTING A MASK FILE
When operating on the galaxy images in a frame, one will
find that certain areas need to be repeatedly excluded at each
stage of the iterative process. These areas might be stars that
have saturated, bad pixels, cosmic ray events, and faint galaxies
that are not to be modeled and subtracted. It is thus convenient
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to have a mask file which can quickly set those areas to zero in
the working frame. At some point, one can edit all such areas in
a frame and set them equal to zero with the options EC
(EDITCIRCL), EP (EDITPOINT), and ER (EDITRECT), for circles,
single pixels, and rectangles, respectively. This is then used
repeatedly as a mask frame in the option SA.
14.4 MODELING THE GALAXIES AND STARS
One begins this iterative procedure by making a first
approximation to the brightest object in the field. In order to
do this in a way which is reasonably free of contamination, one
should edit out areas of the less brighter objects which overlap
the program object. The editing options EC, EP, and ER can be
used for this purpose. If a mask file has been constructed, that
can be used with option SA. Keep a copy of the original file of
course. The preferred way to make the model is with option PR
(PROFILE). See the file PR.DOC for an example of how to do this.
Option PR will yield a character file of type 1 with columns of
values for distance along the major axis, x and y central
location of each elliptical contour, inclination, eccentricity,
and surface brightness. It also integrates the surface
brightness, but option RG should be used for the final integrated
value. An alternative way to obtain some of the information that
option PR provides, is available through the option EA (ANNULUS).
This option simply uses fixed values of central location,
inclination, and eccentricity for each contour. It does not
solve for these quantities; they must be input by the user.
Option EA is useful when an object lies off the edge of the frame
and no detailed solution for the model is possible. For each
model, one should estimate a sky level which needs to be
subtracted from the file when writing data back to the frame at
latter stages in the modeling. To do this, display the surface
brightness values in a plot on a graphics terminal. One is
prompted for this plot in both option PR and EA. Press R and the
crosshairs will appear. Position the crosshairs at the left and
right boundaries of the section of the plot to be replotted,
pressing the <CR> after each position. A sky value can be read
by pressing C to turn on the crosshairs, and then the <CR> to
read a value. Press Q to turn off the crosshairs. The plot will
also indicate if certain values at large distances from the
center, are bad. If this is the case, use the EDT editor to
delete those outer contours. Option SG (SMOOTHGAL) is useful for
recomputing the run of surface brightness with distance from the
center of an object. The operation which it provides is
available in option PR, but it is possible that the user may wish
to smooth the outer profile parameters by polynomial fitting at
some later time, without going throught the entire solution
again. This option operates on the 2-dimensional SAD file and
the character file containing the previous solution for a given
object. Another use of the option is to recompute the surface
brightness distribution after some change has been made to the
SAD file, such as removing addditional contamination. Here
again, a saving results by not having to run option PR over
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again. Although options PR, EA, and SG provide plotting of all
profile parameters against distance from the center of the
object, it is possible to request only plotting of a particular
data file by using option DA (DISPANN). One simply enters the
name of the file in response to the prompt and selects what set
of parameters is to be plotted.
14.5 GHOSTS, THE LED EFFECT, AND THE LEFT SIDE OF THE CCD
Occasionally, a CCD frame with suffer from one or more
artifacts which produce unwanted images in the area to be
analyzed. Ghost images, resulting from reflections of bright
stars from the dewar window or other optical surfaces, can be
modeled in a way similar to galaxies and stars. A ghost image
will usually have a donut appearance, circular symmetry, a
central hole, and uniform surface brightness out to some limiting
radius. In this case, option PR will not work; the derivative
will be positive and the central region will not offer a
solution. Here we must use option EA, assuming an eccentricity
of 0.0, and an estimated central location. If the ghost image is
free of ovelapping objects, the first model will be sufficient.
Usually, however, more than one iteration is required. On some
frames, an LED effect will be evident. This is caused by
excessive current throught the on-chip amplifier on the CCD. For
long exposures, some CCD systems turn off the CCD amplifier to
avoid this problem. If it is present, there will be a bright
radiating source in one of the corners of the CCD (upper right on
the RCA CCD). This source cannot be modeled with either options
PR or EA. One must model the LED effect by selecting several
dark frames of the same exposure as the program frame, averaging
them, and then subtracting a constant background level. This
model should be set to zero everywhere except in the region of
the LED effect, and subtracted from each program frame as
required. Since the effect is repeatable, only one model need be
kept. The leftmost columns on some CCDs have a higher background
level due to some unknown cause. To model this effect, which may
be slightly different on each exposure, model and subtract stars
and galaxies near the left edge of the frame. If the effect is
strong, these models will be only a first approximation. Next
use option AV to compute average values for each column. The
averaging process will neglect edited areas which were set to
zero. These average values can be written to a frame. At some
point, say column 20 or so, the effect fades into the background.
Subtract this background from the frame and set the columns to
the right of that point, exactly equal to zero. This frame is
then subtracted from the program frame to rid it of the
troublesome columns on the left side.
14.6 THE SECOND AND HIGHER ITERATIONS
Let us assume that you have modeled a few of the brighter
objects (galaxies and stars) and have data files of type 1 for
each of these objects. These models were created by editing out
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contamination from overlapping objects. The files themselves may
have required additional editing to remove the outer contours
which may have been too noisy or contaminated. In this stage you
will obtain a better approximation to each object by subtracting
away contamination, rather than editing it out. With option FG
(FAKGAL) one can subtract an image from the working SAD file. To
start this process, the SAD file must be exactly the same as the
original frame (no edited areas). Option FG will prompt for the
sky, which was determined with the cursor, and for a scale
factor. Enter -1 for the scale factor since we must subtract.
Do this operation for the second brightest object and then
observe the frame on the TV display. There will be less
contamination from this object. Repeat this for all the other
objects fainter than the second, and then run option PR on the
brightest object once more. There should be an improvement. If
the brightest object is now subtracted, and the second brightest
written back to the frame (set the scale factor equal to +1 in
option FG), the second brightest object can be more accurately
modeled. This process is continued until all of the brigher
objects in the frame have been modeled a few times. You are done
when, by subtracting all objects, a uniform field results.
Several iterations may be required for rich fields. Some of the
processing can be done by batch; simply create a command file
with all of the information required by the option prompts,
contained in the file.
14.7 THE STELLAR POINT SPREAD FUNCTION
In order to fit surface brightness formulas to the surface
brightness data, a proper accounting of atmospheric seeing must
be made. It is thus necessary to model the stellar point spread
function. While the original exposures are two-dimensional
files, the surface brightness data is in the form of
one-dimensional files. The point spread function will also be in
this form. In modeling the objects in the field, one will have
modeled several stars as well as the galaxies. Usually, the
stars in the center of the frame will not be different than those
at the edges, but one should verify this before adding the
stellar profiles together. Since the stellar profiles, as
produced by option PR, already exist, one only has to average a
few of these profiles in order to obtain a good point spread
function. Option ED allows one to perform mathematical
operations on profile data. With this option, we first add the
surface brightness, inclination, and eccentricity values of all
the stellar files, and then divide each of these parameters by
the number of files. Two files may be operating on each time,
with several parameters being operated on at once. The positions
of the stars are unimportant for deconvolution, hence no
operation is performed on x and y. Likewise, the integrated
surface brightness is also ignored. The averaged files will
still have a sky background which must be removed. The sky is
estimated in the same way as described above in section 14.4.
Here however, access to the plotting prompt is obtained by use of
option DA. This sky value is then subtracted from the file with
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option ED. See the file [-.MIIPS.SAMPLES]POINT_SPREAD.DAT for an
example of how a stellar point spread function looks, and plot it
with option DA.
14.8 STRUCTURAL AND PHOTOMETRIC PARAMETERS
Once an object has been modeled with options PR or EA, one
will have a data file of type 1 where the surface brightness
averaged around elliptical contours is presented as a function of
distance outward along the major axis of each contour. This
one-dimensional file can be used to obtain the photometric and
structural parameters of the particular object. It is necessary
to take account of the blurring of the light passing through the
earth's atmosphere and the imaging optics which make the galaxy
image appear to be larger than its true size, and to possess a
dimmer central surface brightness than it actually has. The data
file for the stellar point spread function, derived as indicated
above, is used in conjuction with a nonlinear least squares
fitting routine to derive the galaxy's parameters.
Parameters from several different surface brightness
formulas can be solved for with the option RG, e.g., the modified
Reynolds Hubble formula, the de Vaucouleurs formula, etc. The
command HE RG will provide online help in the form of an actual
example solution. Alternately, one could TYPE the file RG.DOC in
the [-.MIIPS.DOC] directory, to see this. A sample file of the
data type needed for using option RG (data file type 1) is given
in the [-.MIIPS.SAMPLES] directory. This file,
GALAXY_PROFILES.DAT, contains a large galaxy, ID=70, extending
out to 243 pixels, and a small galaxy, ID=170, extending out to
50 pixels. A solution should be tried with the point spread
functions given in the sample file POINT_SPREAD.DAT in the same
directory. This file contains three objects, with ID=STARS, 138,
and STARS2, which extend out as far as 17 pixels. The object
STARS represent an average of severage stars whose point spread
functions were measured by option PR.
14.9 CALIBRATING THE FILES
For the files of surface brightness distribution (type 1),
the following calibrations will be necessary. 1) The surface
brightness values will be noisy at large distances from the
center of the object, and some smoothing by binning will be
necessary if plots are to be made. The binning is not necessary
in order to use option RG. 2) Since the inclination angle is
measured on the frame, and the frame may be tilted relative to
north on the sky, a constant may have to be added. 3) The radius
is in pixels and should be converted to arc seconds. 4) The
surface brightness values are in ADU and should be converted to
magnitudes for plots and tables.
These calibrations are carried out with option ED in
successive steps. The binning can be done with a step size
proportional to the radius (command B1) and in the log domain.
Before converting to magnitude units, the sky (as determined from
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option RG) must be subtracted. It is most convenient to do all
the computations in batch mode where the command file contains
the necessary answers to the prompts and the calibration
constants. Start with a small amount of binning and then plot
the file with option DA to see the effect. The file of galaxy
parameters (type 3) also needs to be calibrated. Here a
conversion from ADU to magnitudes, and a conversion from pixels
to arc seconds must be made. The conversion to magnitudes is
given simply by m=-2.5log(ADU)+constant. Certain options, e.g.
CG, will ask for the constant when plotting error bars. Option
E3 (EDITDATA3) is designed to files of galaxy parameters.
Similar operations, ie., addition, multiplication, etc., can be
done on several columns simultaneously. In this regard, it is
similar to option ED.
14.10 PLOTS OF THE SURFACE BRIGHTNESS DISTRIBUTIONS AND
PARAMETERS
Option DA can be used to plot the surface brightness
distrbution in the units which exist in a single file of type 1.
No error bars can be included with this option however. This
option is most useful for looking at the raw data, before any
calibrations or binning have been done. When it is desirable to
include a fitting function formula and error bars in a plot of
measured surface brightness in magnitude units, one should use
option CG. The data file should have already been converted to
magnitude units, and have been binned (see above). The fitting
function must exist as a data file of type 1 in units of
magnitudes and should have been convolved with the appropriate
point spread function. The files
[-.MIIPS.SAMPLES]GALAXYPROFILESMAG.DAT and
[-.MIIPS.SAMPLES]GALAXYPROFILESMAGDEVAUC.DAT are examples of data
and reconstructed profiles using the de Vaucouluers formula, of
two galaxies, ID 70 and 170. The command file
[-.MIIPS.SAMPLES]GALAXYPROFILESMAG.COM illustrates how the
binning and calibration were carried out for ID 170 under batch
operation. Option FG is used to create the reconstructed, or
artificial, data file, making use of the parameters which were
solved for with option RG, and the point spread function. After
this stage, option ED is used to convert to magnitudes and arc
seconds. The command file
[-.MIIPS.SAMPLES]GALAXYPROFILESDEVAUC.COM illustrates how the
reconstruction was accomplished in batch mode for the galaxy with
ID=170. Try plotting the data for these galaxies with option CG,
command TD. Answer the prompts with the following parameters:
readout noise and gain = default, sky and sigma of sky = 491 0.05
for ID 70 (sky and sigma of sky = 491 0.1 for ID 170), and
calibration constant = 27.4212.
Option CG can also be used to plot color as a function of
radius, along with the predictions of two fitting formulas, the
de Vaucouleurs formula and the modified Hubble formula (Abell and
Mihalas 1966, A.J.). Here one requires several files, two files
for the blue and red data, two files for the modified Hubble
formula at blue and red wavelengths, and two files for the de
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Vaucouleurs formula at two wavelengths. An example of this usage
is given in CG.DOC.
Option DG (DISPGAL) is used to plot the photometric and
structural parameters (character file of type 3) and perform
various kinds of fits. One could, for example, plot log(radius
parameter) against magnitude, fitting two intersecting straight
lines to the bright and faint ranges. One can also plot
luminosity functions, surface brightness parameter against
magnitude, color delta against magnitude, etc. In certain
instances, a histogram of the residuals about a fitted line can
be made. The file DG.DOC gives two examples of the usage of
option DG. Also, the directory [-.MIIPS.SAMPLES] contains the
files GALAXY_PARAMETERS_BLUE.DAT and GALAXY_PARAMETERS_RED.DAT
which can be used to get familiar with option DG.