In the absence of a point source, the profile can be viewed as a sum of the bulge and disk components.
In order to fit models to the observed profile, use has been made of a program written by Ajit Kembhavi in standard Fortran 77. The program makes use of the Minuit 94.1 (James, 1994) minimization package from CERN and routines from Numerical Recipes (Press et al., 1986). A detailed discussion of the program follows.
The program fits a de Vaucouleurs profile to the bulge and an exponential to the disk. The parameters used are:
: The bulge scale length.
: The disk scale length.
: The central magnitude.
: The bulge ellipticity.
: The disk ellipticity.
The bulge and disk ellipticities are invisible parameters and the D/B ratio appears as the relative luminosity of the disk and the bulge.
One can specify the range within which a specific parameter is to vary and the step size. Alternately, one can hold a parameter fixed at a particular value if that value is known. If, for instance, one knows that a galaxy does not have a disk component, D/B for that galaxy can be fixed at zero.
The brightness profile for which a best fit is sought forms the main
input for the decomposition program.
The profile contains the surface brightness along with 
errors
on it as a function of r.
The trial values viz.
are put together in a file which we call the
minimization file.
The FWHM of the PSF, as extracted from the input galaxy
image is also included in the minimization file. Finally, the file also
contains information as to which points from the input profile are to be
actually used during a given run. This facility helps us exclude
pathological parts of an image. It is also desirable to leave
out certain inner and outer points from the profile during the
process of fitting since it is possible that de Vaucouleurs' law does not hold
right upto the center (inner points) and since the signal-to-noise ratio
decreases away from the center (outer points).
The minimization process involves the convolution of the model galaxy with a PSF. The FWHM of the PSF, is obtained by fitting a Gaussian to a number of stars in the object frame using psfstar. The convolution step is crucial since the light near the center of the galaxy is highly affected by the seeing and it is not possible to carry out a satisfactory deconvolution of the image.
The decomposition of the surface brightness profile is done in an
iterative fashion. At each stage a model profile is constructed from
the current guess parameters.
At initialization, the fitting program reads in the intensity profile.
The minimization file is also read in. Using Equation 
a model galaxy is built. It is then convolved
with a Gaussian PSF with FWHM as read in from the input.
The generated galaxy has a fixed position angle
viz. zero since the position angle is not being fit. A profile is now
obtained along the major-axis of the model at the points
corresponding to the input profile. This profile is then compared with
the observed galaxy profile, the guess parameters adjusted and the
model generation repeated until the best fit is obtained.
The best fit model is obtained by minimizing the value of the reduced
chi-squared function 
, given by
where,
and 
are the observed and model surface brightness at the
point and 
is the standard deviation in the
surface brightness.
Here 
is the number of degrees of freedom and is given by
where, f is the number of free parameters of the fit, and n is the number of data points used.
The
minimization and error evaluation is handled by the powerful
package minuit (described below).
Very rarely, for pathological profiles, the program may terminate
with incorrect values if the maximum number of iterations is exceeded before
reaching convergence. In such a case a flag in the output marks
the reason for the error exit.
The information on the goodness of fit is output along with the error matrix,
warnings, if any, arising during fitting and an optional set of
contours for parameter pairs.
The program ends by creating several files.
The various files contain the best fit model profile, a summary of parameters
being fit, their initial values and the best fit values
along with the error bars.
The output profile is in the form of a fitted bulge profile, fitted disk
profile and a summed profile.
The value is output as well.
Minuit: Function Minimization and Error Analysis is a package from CERN, Geneva, Switzerland, offering a host of minimization routines. Minuit is a tool to find the minimum value of a multi-parameter function and analyze the shape of the function around the minimum. Minuit acts on a user defined Fortran function containing the different parameters.
Minuit is a smart program and hence rather than plodding along a regular grid in a multidimensional space, it is capable of making sudden jumps to different parts of the space to check if it can find a better minimum. A flag, called the strategy flag, can be set within minuit. This flag decides whether a very careful search is to be carried out around the minimum that has been located. If the strategy is thus set, one can avoid obtaining only a local minimum rather than a global minimum. The cost is several additional function calls.
The shape of the function close to the minima decides the errors on the
best fit parameters. Normally these are errors. However,
one can set this number to whatever one desires, say, 
.
In the profile fitting program, use has been made of three minuit
processors viz. migrad, hesse and minos.
Migrad and hesse produce an error matrix which is the inverse
of the matrix of second derivatives of the function being minimized. The minuit
error matrix takes into account all parameter correlations, but not
the non-linearities. It is assumed that the problem was of a linear
nature and that the error
bars are symmetric. Thus, errors are always twice the errors.
On the other hand minos cam handle asymmetric errors in case of a
non-linear problem. Minos starts with the error matrix generated by
a method like migrad. It then actually traverses along a parameter to see
when a error bar is reached. It does this on both sides and for each parameter. Minos is thus essential to estimate the correct parameter errors when
the problem is significantly non-linear.
To sum up, at the end of the day you have your extracted parameters along with their errors.
The region over which the fitting is actually carried out can turn out to be crucial for the accuracy of the extracted parameters. In the literature one finds that a number of different methods are used in selecting the data for fitting. Once the data has been fit, several additional parameters are obtained for specific purposes.
The profile can be fit with de Vaucouleurs' law, either separately or in combination with an exponential disk and/or a point source. In case of nearby galaxies that subtend large angles on the CCD, one has the luxury of being able to go to several tens of arcsec from the center. Sometimes a metric radius of, say, 13 kpc is chosen and the fit done up to that point ( Gonzalez-Serrano et al., 1992). However, such a radius may cover the whole galaxies in the case of dwarf galaxies, but would cover only a part of cD galaxies.
However, in case of the radio galaxies of the sample the angle subtended on the CCD is small. Therefore one has to stop at only several arcsec when the background noise starts being dominant. The outermost points we used were those with an error of 0.1 magnitude in the surface brightness.
The innermost point which can be included in the profile fit has also been widely debated. In the inner region the effect of seeing is very dominant and the validity of de Vaucouleurs' law close to the center is also suspect. For the above reason, the inner few points have to be disregarded during the fit even if the convolution of the model profile with the PSF is carried out. The decision on the innermost point to be included was taken following a series of tests described below.
During the program runs the
ellipticity was fitted for (a constant rather than as a
function of r), but the position angle was not. This is
because in case of a galaxy showing isophotal twist the profile that is output by
ellipse is along the twisting
major axis. If the ellipticity
and position angle
are not being fit, the value of PA and 
at 
can be used. Values at a fiducial magnitude (e.g. 
Owen and Laing, 1989) have been used too.
In our work, the profile was always obtained along the major axis.
However, since we are using a single value for the
ellipticity, the corresponding profiles
along the minor-axis, or at any other PA will provide similar results
since these profiles are related to the major axis profile through
the ellipticity.
