The number counts 
given by every pixel of the CCD are now
proportional to the apparent surface flux coming from the sky (
) and
from the objects (
) projecting onto these pixels:
where t is the exposure time.
The programs you will have to use during the exercise ( dodel and
docog, see Section 3) allow you to derive the
circularized surface brightness and integrated magnitude profiles of
the galaxies on the frames from the matrix of 
values. These
profiles are defined as the circularly averaged surface brightnesses
(in the R or B band) at a distance R from the center of the galaxy
under study and the magnitudes (in the R or B band) inside R.
The programs operate as follows. The first one ( dodel) allows
you to determine in an interactive way the value of the sky background
sky (a surface flux per pixel) and the position of the center
(coordinates 
) of the
galaxy you want to analyse. You can then ``delete'' all of the
foreground stars or galaxies that are superimposed to your object. The
second one ( docog) performs the azimuthal average of the
not-deleted galaxy number counts 
in radial steps of one
pixel, starting from the center of the galaxy. At a radius of N
pixels, the azimuthal average of the surface flux per pixel is
computed over approximately 2
N points uniformly distributed in
azimuth. This means that the positions with
coordinates:
where 
, are considered. If they correspond to regions that have
not been deleted, than their surface fluxes per pixel (minus the sky value)
are averaged.
The resulting circularized instrumental surface luminosity is converted to instrumental surface brightness and integrated to give the instrumental magnitudes inside the given aperture.
Last step involves the conversion of these instrumental values 
into
apparent magnitudes. In general, this is done by observing stars of known
apparent magnitude, having different colours (for example, Landolt stars) and
different air masses (this means observed at different zenith angles).
The shifts between observed instrumental magnitudes
and apparent ones are then used to determine the parameters
and 
of the calibration equation:
where t is the exposure time. Eq. 12 has been used to
transform instrumental magnitudes measured from the CCD frames of
A2593 into apparent ones. You will not have to repeat the procedure,
but just use the fact that Eq. 12 has allowed us to determine
the surface brightness of the sky, and that, given the lineary of the
CCD detector, every measured flux can be expressed in units of
the sky. If 
is this surface brightness value in a given band and
you have
measured 
counts per arcsec
(see Sect. 3), then
counts per arcsec
measuread on the same frame will
correspond to a calibrated surface brightness 
of:
Note that more information is contained in the CCD frames than it is in the circularized profiles. For example, one can try to parametrize the shapes of the isophotes (lines of constant surface brightness) of galaxies and derive ellipticity profiles, twisting of isophotes, deviations from ellipses. But this is beyond the goals of this exercise.
