Historically, an intensity profile was obtained by simply taking a cut along the major axis of the galaxy or along a few selected axes. With the advent of computers various routines that can average light along isophotes have been devised and used in studying the light distribution.
Throughout, it is important to remember that galaxies are three-dimensional (3D) objects and what is observed is just their 2D projections on the sky. Thus when a profile is obtained, the 2D distribution is being further averaged to a 1D distribution leading to loss of information. Due to the regularities on a large scale in galaxies it is often enough to consider the 1D profile to obtain a fair idea of the light distribution. However, when dealing with particular morphological distortions and localized perturbations, it is better to consider the 2D distributions.
Isophotes of galaxies
are well approximated by
ellipses. One can use this prior knowledge
to obtain a radial brightness profile.
One can thus fit ellipses to the isophotes and obtain mean intensity
as a function of radius.
Besides the mean intensity, one also obtains the radial
profiles of ellipticity, position angle etc., each with its errors.
All these parameters can be used to obtain some idea
regarding the 3D shape of the underlying galaxy. A set of Fourier
components 
(described below)
obtained are indicative of the deviations of the shapes of the
fitted isophote from true ellipses.
These have been used over the years to argue over the
presence of dust, disks etc.
The present section details
the fitting of ellipses, the precautions to be taken, the information
that the resulting parameters reveal and the dependability of the
errors involved.
The 2D CCD image obtained is the projection on to the plane of the sky of a 3D galaxy. As mentioned previously, the major components of a galaxy are the spheroidal bulge and a thin disk. In projection, either component gives rise to elliptical isophotes. The two components taken together will also yield elliptical isophotes, with the isophotes containing information from both the components. Measuring various parameters of these projected ellipses is an important step in understanding the brightness distribution of a galaxy. In this subsection we describe in detail the procedure of fitting ellipses to galaxy isophotes.
Any ellipse
can be fully described by five parameters viz. semi-major axis length,
position angle, ellipticity and the coordinates of the center,
.
The aim here is to fit to the isophotes a series of ellipses with
different semi-major axis lengths.
An ellipse with a given semi-major axis will then have an
ellipticity ( 
), a position angle (PA) and an x- and
y-center. The ellipticity is given 
where a
and b are the major- and minor-axis lengths respectively. PA is the
angle ( 
)
made by the major-axis of the galaxy anticlockwise with the +ve x-axis. The peak
central intensity of the galaxy defines .
For a smooth and featureless spheroidal galaxy the projected
ellipses are concentric with the same ellipticity, PA, 
and
.
A straightforward way of fitting an ellipse to an individual isophote
is to use (x,y) pairs on the isophote and to use some method, like
least squares, to obtain the ellipse parameters.
However, only a finite number of points are used in this procedure.
We have used the IRAF task ellipse which follows the procedure outlined by Jedrzejewski (1987; See also Young, 1976; Kent, 1984). To begin with, one chooses a trial ellipse at a given semi-major axis length. The intensity along the trial ellipse changes as one goes along it. The change in intensity is periodic and can hence be expanded in a Fourier series.
where 
is the mean intensity along the ellipse, E is the
ellipse eccentric anomaly and 
are
harmonic amplitudes.
If the isophotes are ideal ellipses, only the first four coefficients viz.
are non-zero. All the higher coefficients are zero.
For a real galaxy, the isophotes are rarely perfectly elliptical.
One however begins by assuming that they are elliptical and a trial ellipse
is obtained. are expected to be non-zero. The
values are obtained by a least square fit to
Each of the four coefficients is related to one ellipse parameter with the magnitude of each coefficient dependent on the misalignment between the trial ellipse and the actual isophote. Correction factors to the trial ellipse are now derived by changing the initial ellipse parameters. The parameter that corresponds to the largest amplitude is varied, and the image is resampled. The procedure continues iteratively till one of the following criteria is satisfied:
and PA are obtained
for the best fitting ellipse along with the mean intensity, .
After the best fitting parameters have been determined, third and fourth harmonics are obtained using a least-squares fit to
Here 
determine the deviations of the
isophote from
the perfect elliptical shape. For a perfectly elliptical isophote,
the coefficients of these
higher harmonics will all be zero.
When non-zero, the 
and 
components indicate that the isophotes
deviate from ellipses and that the deviation has a 3-fold symmetry.
Similarly, the presence of 
and 
denote 4-fold symmetric
deviations. Typically the deviations are less than 
. Even
a deviation to this extent distorts the ellipse considerably
(see Figure 
). The component is especially
important since it indicates the presence of a disky structure ( 
)
or that of boxiness ( 
).
The procedure is repeated for the next semi-major axis length. For successive ellipses the semi-major axis is increased by, say, 10% over the previous one. One continues in this manner till the desired outer limit is reached. This is determined either by the size of the galaxy (one should typically have more than 60% of the ellipse on the CCD frame) or by the noise levels (with modern CCDs one can reliably go upto a few percent of the sky level).
The ellipticity and position angles (as also other parameters) are not very reliable near the center. This is because the inner light gets distributed to larger radii due to the PSF. Since the PSF is typically azimuthally symmetric, this results in making the isophotes near the center circular.
As far as the complete galaxy image is concerned, it often contains regions which one needs to ignore during the process of fitting. Bad pixels are flagged so that they get a zero weight during fitting. Stars can be masked and interpolated over. Additionally upto 40% of the top pixels on a given isophote can be masked to get rid of faint stars and other spurious features. This ensures that unwanted objects do not affect the fitted ellipses.
The images used for the work in this thesis have been taken in the
B, R and K' bands. The CCD field in case of B and R and
the camera field in
case of K' were large enough to contain the whole galaxy within it.
We stopped the ellipse fit when 
exceeded 0.1,
which corresponds to an error in magnitude of 
.
Often the ellipses were fit well beyond this limit to see how the
profile progressed. However, during all subsequent analyses these
additional outer points were ignored. To look for a possibly varying
center, the center was always left free during ellipse fitting.
However, the typical shifts obtained were
less than half a pixel.
(This is another indication that there was no artificial gradient in
the images. If there is a gradient, the ellipse centers are seen to shift.)
Only when we wanted to look at some particular aspect of an
individual galaxy that needed fixing the ellipse centers did we
actually fix it.
The fitting was begun at a semi-major axis length of five pixels.
Successive ellipses had major-axis length longer by ten percent and the
fitting was continued until the 
error in the mean intensity
( )
was less than 
corresponding to an error of
magnitude. Ellipses were fitted to isophotes smaller than the starting
one at five pixels, by reducing the semi-major axis-length by 10% at
every step. It is necessary to begin at an intermediate semi-major
axis length, and then move inwards, as the parameter values obtained
near the center are not reliable, due to the small number of pixels
involved in their measurement,
as well as the modification produced by the PSF. Using such starting
values can have an adverse effect on parameters derived for the larger
isophotes.
Clipping of 10% was used to
make the program ignore the top 10% pixels during the fit.
In some of the cases a kink was present at 5 pixels from the center.
In such a case we started at a nearby point without the abnormality
and proceeded as above.
