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4 The catalogue of Abell 496

4.1 Isophotal magnitude definition

To define properly an isophotal magnitude we first need to consider some definitions and correlations (see also Trentham 1997).

4.1.1 Isophotal versus total magnitude

The difference between total and isophotal magnitude is the difference between the total flux, extrapolation of the curve of growth, and the flux integrated within a fixed SB value. To simulate such difference, we extract from the frames some bright sources ($\sim $ magnitude 16) of different morphological types and integrate the total flux on an extrapolated model. We then increase the magnitude up to our frame limits by dividing the original flux by a numerical coefficient. In this way, we obtain a list of expected total magnitudes in the range of interest. We compare these values with the isophotal magnitudes as measured by the analysis routine with the threshold listed below. The amplitude of the differences is dependent on the source profile. In our data at $r \sim 24$ the differences range from 0.1 mag for point like sources to few tenth of mag for E0/E6 galaxies and, little more than a magnitude for disk dominated objects (Fig. 6 shows the case of an elliptical-r1/4-galaxy). The difference is seeing dependent. To show this dependence we convolve the original frames (seeing $\simeq$ 1.3 arcsec) with a Gaussian point spread function to simulate worse seeing (1.6 arcsec). The effect is illustrated in Fig. 6.
  \begin{figure} {\psfig{figure=8680_f6.ps,width=9cm,angle=270} } \end{figure} Figure 6: The differences between the isophotal magnitude and the total magnitude of an elliptical galaxy (seeing = 1.3 arcsec) are plotted (filled squares) versus the total magnitude. Dashed line and crosses show the feature of the same elliptical galaxy with an artificially degraded seeing (1.6 arcsec). Open squares show the seeing-degraded galaxy after the correction performed according to the relationship seeing-threshold

4.1.2 Dependence on the seeing

To reach internal consistency on frames obtained with different seeing we must correct the isophotal magnitudes for the seeing of each frame. Our approach is as follows. We choose not to apply directly any correction to the isophotal magnitude, but, varying the value of the SB of the last isophote as a function of the seeing of the frame, we ensure that the isophotal magnitude value of a fixed morphological type always corresponds to the same fraction of the total flux of the source. The procedure is easily justified. Consider, for simplicity, a source with a Gaussian spatial brightness profile: in this case different seeing levels correspond to different values of the standard deviation $\sigma $ (Fig. 7) and the problem has a simple analytical solution. Let us consider a bidimensional symmetric Gaussian profile I1 with $\sigma = \sigma_1$; given the threshold $ \Sigma_1$ we have to consider the flux $\cal F$ subtended by I1from 0 to r1, such that $ I_1(r_1) = \Sigma_1$:

\begin{displaymath}{\cal F} = {1 \over 2\pi \sigma^2} \int_0^{r_1} {\rm e}^{{-r^2\over 2\sigma^2}} 2 \pi r {\rm d}r. \end{displaymath}

After the integration, we can write it as function of $ \Sigma_1$

\begin{displaymath}{\cal F} = 1 - 2 \pi \sigma_1^2 \Sigma_1.\end{displaymath}

Therefore given a different $\sigma = \sigma_2$ (and the same normalization), the same isophotal flux $\cal F$is obtained using the threshold $\Sigma _2$ such that

\begin{displaymath}\Sigma_2 = \left({\sigma_1 \over \sigma_2}\right)^2~\Sigma_1.~~~~~ (1) \end{displaymath}

We conventionally assume a limit surface brightness value as threshold in a frame with a certain seeing value and we use the relationship (1) to find the correct threshold in the other frames. The reference values of the limiting isophote SB are 25.5,  25.5,  25.0 mag/arcsec2 for g, r, i filters, respectively with PSF = 1.3 arcsec.
  \begin{figure} {\psfig{figure=8680_f7.ps,width=8cm} } \end{figure} Figure 7: The two Gaussian profiles simulate the same object observed with different PSF. The profiles are the projections of two bidimensional profiles with the same normalization and different FWHM. The marked areas represent the same quantity of flux. They represent the isophotal fluxes with different thresholds at two different seeing levels. According to Eq. (1), the second threshold $\Sigma _2$ is chosen in a such a way that the isophotal flux of the left profile is kept constant

The relation (1) has been deduced in the case of Gaussian profile source. We find that the corrections drawn from (1) give good results also for different morphological types as shown in Figs. 6 and 8. In Fig. 6 we show, in the case of an elliptical-r1/4 galaxy, the difference between isophotal and true magnitude at two different seeing levels (one artificially degraded), and the difference after the correction. At low luminosity the correction substantially removes the seeing dependence.

The quality of the correction discussed above can be tested in the intersection regions of two overlapping frames, which have been obtained in different seeing conditions. In this region we have 2 different measures performed with different seeing of a list of sources of random magnitude and morphological type. For the differences between the 2 independent measures, we expect a symmetric distribution with a dispersion exponentially increasing with the magnitude due to the Poissonian uncertainty. If we remove this dependence by normalizing by an exponential factor, we expect a Gaussian distribution. In Fig. 8 we can observe that the distribution of the measures performed with the same threshold is slightly asymmetric; after adopting the threshold corrected according the relation (1) we find that the distribution of differences is perfectly symmetric as a test of reliability of the method described.

  \begin{figure} {\psfig{figure=8680_f8.ps,width=9cm,angle=270} } \end{figure} Figure 8: The distribution of the differences between 2 measures with different seeing (1.3 arcsec vs. 1.4 arcsec) of 75 sources after the correction. The distribution of the differences of the measures before the correction is shown with the solid line and it is slightly asymmetric. The dashed line shows the symmetric distribution after the correction

4.2 Sample completeness

Background statistical variations and source crowding may affect the accuracy of the automatic detection routine and the completeness of the photometric catalogue. We use a bootstrapping technique to test the sensitivity of our results to both factors. First, we extract the image of a giant elliptical galaxy from one of the frames. Then, dividing by a numerical coefficient, we generate a set of more than 30 different images for each filter in the relevant range of isophotal magnitudes: $ 16.07 \leq r \leq 24.56,~ 15.86 \leq g \leq 24.85,~15.87 \leq i \leq 24.01 $. The test images are added to the observed frames positioned at 25 subsequent distances from the centre of the cluster (assumed to be in the centre of cD galaxy). For each value of the distance from the centre and magnitude, we repeat this procedure 100 times in each filter, randomly changing the angular coordinate of the added test image. These 100 repetitions are divided in small groups in different runs to avoid bias due to artificial additional crowding. This allows us to estimate the probability of detecting a galaxy of magnitude m at distance r from the centre of the cluster P(r,m). For each P(r,m) we estimate the uncertainty by the binomial distribution PB[x,100,P(r,m)], which gives the probability of observing x successes on 100 attempts given a probability P(r,m) for a single success. At a fixed distance r from the centre we find a 100% detection rate for bright galaxies, and a drop in the rate at characteristic magnitude $ \sim m_0$ (Fig. 9).
  \begin{figure}{\psfig{figure=8680_f9.ps,width=8cm} } \end{figure} Figure 9: Bootstrap results at two different distances from the centre of the cD galaxy are shown. The two different curves are fitted by Fermi-Dirac function with different value of the characteristic magnitude m0. Going off centre m0 increases: at fixed magnitude, finding a faint galaxy is easier. The uncertainty of the test results is estimate by binomial statistic and $1~\sigma$ level is shown in the figure

The analytical formula of this function, given by a fit performed with a Fermi function is:

\begin{displaymath}P(r,m) = {1 \over {\rm e}^{{m-m_0\over c}} + 1}~. \end{displaymath}

We also find that m0 depends on the distance r. Smaller radii are associated with brighter m0. The relationship can be parameterized by an hyperbole

\begin{displaymath}m_0 = m_0(r) = A-{B \over r}~~, \end{displaymath}

where A and B are slightly different for the 3 filters. As we expect, this relation is affected by background statistical variation and sources crowding. The first steep increase of m0 is due to crowding effect of the central part of the cluster and to the cD halo. The flat shape near an asymptotic value is due to the statistical variations in the background noise. The asymptotic value of m0 corresponds to 50% detection probability independently of any crowding effect and for each filter we assume it as the limiting magnitude value of the catalogue (24.14, 24.46, 23.75, for filter g,  r,  irespectively).

The test is performed on the raw image, without the exclusion of the bright, extended objects. Indeed, we stress that subtracting the signal from extended sources (see previous section) does not substantially improve the automatic routine detection capability of faint galaxies.


   
Table 5: By using function P(r,m), we can estimate our sample completeness. Here we give the completeness value of the last three-1magnitude bin for each filter
  App. mag. bin Abs. mag. bin Compl. (%)
filter g [23.14, 24.14] [-12.39, -11.39] 67+4.5-4.8
  [22.14, 23.14] [-13.39, -12.39] 95+1.7-2.4
  [21.14, 22.14] [-14.39, -13.39] 99+0.9-0.6
filter r [23.46, 24.46] [-12.07, -11.07] 68+4.5-4.7
  [22.46, 23.46] [-13.07, -12.07] 96+1.5-2.2
  [21.46, 22.46] [-14.07, -13.07] 99+0.9-0.6
filter i [22.75, 23.75] [-12.79, -11.79] 71+4.3-4.6
  [21.75, 22.75] [-13.79, -12.79] 97+1.2-2.0
  [20.75, 21.75] [-14.79, -13.79] 99+0.9-0.6

4.3 Stars and galaxies

We identify and remove foreground bright stars from the catalogue by using the isophotal magnitude-isophotal radius plane (Fig. 10). In this plane there is a clear distinction between two different populations of sources up to the magnitude $r~\simeq$ 20.75: within this range stars have smaller isophotal radius than galaxies at any given magnitude.
  \begin{figure}{\psfig{figure=8680_f10.ps,width=8cm} } \end{figure} Figure 10: The isophotal magnitude-isophotal radius plane: we can easily identify 279 bright stars up to r=20.75. At fainter magnitudes our data do not allow us to classify morphologically the sources of our sample. The continuous line mark the separation between the star and the galaxy fields

We cannot classify fainter stars morphologically. Their identification from our data can be achieved only in a statistical way by estimating the contamination level of our sample. In the bright part of the catalogue ( $14.0 < {\rm mag} < 20.75$) we identify 279 stars. In the remaining part of the catalogue, we expect to have 290 foreground faint stars (Table 6), about $15\%$ of the total of faint sources (Robin et al. 1995). The star contamination level falls under $10\%$ if we limit our analysis to the "sequence'' galaxies sample (see next section).
   
Table 6: Number of faint, unclassified, stars expected within our catalogue, divided into 1 magnitude bins. First row refers to the whole sample, second row refers to "sequence'' colours. The last column reports ratios between stars and galaxies: contamination level of the whole sample is about $15\%$, whereas sequence galaxies contamination is under $10\%$
MAG 21.25 22.25 23.25 24.25 TOTAL
stars 56 75 94 65 290/1867
stars 13 7 13 14 47/530

4.4 Error estimate

The Poissonian uncertainty is the largest source of error in our photometric measurements and can be estimated by comparing independent magnitude measurements of the same objects. In our sample, we have independent photometric measurements of the objects belonging to the intersections of two adjoining fields. As shown in Table 7, they represent a statistically significant subsample.


   
Table 7: Number of objects belonging to the intersection of different fields
Filter g r i
Field 1 $\cap$ Field 0 91 141 128
Field 1 $\cap$ Field 2 95 126 107


   
Table 8: A subsample of the photometric catalogue (http://www.merate.mi.astro.it/$\sim $molinari/A496-cat.dat) reporting the 40 brightest objects in the complete list is presented. The luminosity sorting has been made in the r filter. The ID number refers to the position in the whole catalogue
ID x y g r i Rg Rr Ri g-r g-i note

1429

 -652.6 419.8 15.22 14.70 14.54 32.0 35.0 31.0 0.50 0.68 star
661 -553.3 23.8 15.41 14.84 14.71 36.0 40.0 34.0 0.61 0.76 star
2251 -856.6 939.4 14.81 14.87 14.87 13.5 14.9 12.2 -0.17 -0.10  
1525 -292.9 501.1 15.79 15.13 14.95 11.6 13.6 12.3 0.66 0.83  
1061 -640.4 193.7 15.66 15.15 15.25 12.0 15.0 11.7 0.47 0.33  
748 -272.2 55.5 15.83 15.30 15.17 26.0 29.0 25.0 0.52 0.65 star
249 -240.6 -146.6 15.89 15.34 15.29 26.0 29.0 25.0 0.56 0.60 star
1448 -286.8 440.7 15.99 15.44 15.32 31.0 37.0 30.0 0.50 0.68 star
968 -631.8 152.6 15.98 15.48 15.42 32.0 34.0 29.0 0.50 0.58 star
114 -587.8 -198.7 16.15 15.69 15.55 27.0 30.0 27.0 0.43 0.61 star
935 47.3 138.5 16.27 15.76 15.59 18.0 20.0 19.0 0.54 0.71 star
1543 -437.4 514.5 16.11 15.80 15.80 11.0 11.6 9.9 0.33 0.33  
573 -149.6 -22.6 16.19 15.83 15.81 10.1 10.6 10.3 0.43 0.43  
1365 -418.3 373.6 16.58 15.99 15.87 20.0 21.0 18.0 0.64 0.79 star
113 159.7 -199.1 16.61 16.08 15.89 17.0 19.0 18.0 0.55 0.74 star
2257 -1077.9 942.2 16.32 16.12 16.18 9.9 10.4 9.0 0.19 0.16  
355 -61.8 -107.4 16.50 16.18 16.18 9.7 10.0 9.2 0.35 0.32  
387 -25.6 -93.0 16.81 16.27 16.10 14.0 17.0 15.0 0.55 0.73 star
632 56.8 8.0 16.91 16.38 16.19 12.2 13.6 12.9 0.56 0.74 star
71 -186.1 -217.7 17.81 16.42 15.35 7.4 9.7 11.8 1.43 2.47  
480 -25.9 -60.0 16.92 16.44 16.18 12.0 15.0 14.0 0.46 0.72 star
1919 -922.3 681.2 17.49 16.56 16.28 8.1 10.4 9.5 0.91 1.24  
1200 -339.8 251.2 17.12 16.58 16.52 17.0 19.0 16.0 0.53 0.62 star
1280 -482.8 304.7 17.74 16.59 15.89 7.8 9.3 9.6 1.18 1.89  
1522 -633.1 496.8 16.96 16.61 16.58 20.0 20.0 17.0 0.41 0.50 star
1006 -77.7 175.2 16.81 16.62 16.65 9.0 9.3 8.8 0.25 0.19  
2032 -1078.6 771.6 17.44 16.63 16.44 7.8 9.3 8.5 0.79 1.01  
781 -236.8 69.3 16.96 16.69 16.72 8.8 9.3 8.4 0.22 0.19  
2308 -908.9 987.2 17.83 16.70 15.97 7.1 9.5 9.6 1.10 1.87  
1511 -420.9 489.4 17.24 16.70 16.61 20.0 22.0 19.0 0.52 0.67 star
982 -135.0 159.3 17.88 16.74 16.06 7.1 9.0 9.9 1.15 1.81  
1082 -629.4 200.5 17.12 16.76 16.60 19.0 20.0 18.0 0.37 0.51 star
925 131.5 132.0 17.31 16.76 16.61 15.0 17.0 16.0 0.55 0.70 star
901 -516.1 121.2 17.14 16.78 16.77 8.7 9.5 8.5 0.34 0.34  
826 -67.3 90.6 17.30 16.80 16.62 12.0 13.0 13.0 0.50 0.68 star
981 -360.6 159.0 17.24 16.86 16.85 8.7 9.3 8.2 0.32 0.33  
1604 -1083.8 545.4 17.86 16.87 16.52 7.7 10.1 8.9 0.95 1.34  
130 -68.4 -189.4 18.00 16.90 16.32 6.9 9.0 9.7 1.09 1.67  
2082 -897.2 812.4 17.13 16.93 16.96 8.6 8.9 7.6 0.17 0.22  
478 -580.7 -60.5 17.40 16.97 16.95 8.2 9.2 7.9 0.39 0.41  

                     


Starting from the Poissonian statistics, due to the errors propagation law, for the magnitude uncertainty, we expect


\begin{displaymath}\sigma(m)= {\rm const}\ 10^{0.2m}~~,\end{displaymath}


where the constant is given by the characteristics of the electronics. At magnitude 22 we estimate that the uncertainty of the photometric measures $\sigma_{22}$ is 0.20, 0.19, 0.22 magnitude for filter g, r, irespectively. Then, for each magnitude m we can draw $\sigma_{m}$ as


\begin{displaymath}\sigma(m) = \sigma_{22}~10^{0.2(m-22)}~~,\end{displaymath}


which is the uncertainty of our photometric measures as function of the magnitude.

  \begin{figure}{\psfig{figure=8680_f11.ps,width=8cm} } \end{figure} Figure 11: Differences between r magnitude measurements of the same objects performed from two frames. Plotted against magnitude, they show the expected exponential slope. 1 and 2 $~\sigma_{m}$ level curves are shown


  \begin{figure}{\psfig{figure=8680_f12.ps,width=8cm} } \end{figure} Figure 12: We split up the plane in three different regions on the basis of the Colour-Magnitude Relation. Using Metcalfe et al. (1994) terminology the three regions are defined as the "blue'', "the sequence'' and the "red''. We identify 637 galaxies within the sequence zone (little squares), 371 in the red zone and 47 in the blue one. The filled circle at $r\sim 13$ refers to cD galaxy isophotal magnitude and core index colour (see nextsubsection)

4.5 Description of the catalogue

The derived catalogue consists of 2355 objects: 2076 are classified as galaxies, 956 galaxies have magnitude measures in all three filters, 1081, 2055, 1500 galaxies have magnitude values below the filter g, r, i limit, respectively. We estimate g-r colours of 1058 galaxies and g-i of 955 galaxies. The whole catalogue is available in electronic form (http://www.merate.mi.astro.it/$\sim $molinari/A496-cat.dat), while in Table 8 the list of the 40 brightest galaxies is reported, and in Table 9 we summarise the statistics of the catalogue of galaxies. In the different columns we list:
   
Table 9: Statistics of the catalogue of galaxies; 279 classified bright stars are included in the catalogue, but not in the present summary table. At faint magnitudes (>20.75) we expect $15\%$ of the entries are foreground stars. In parentheses absolute magnitude limits are reported assuming H0 = 50 km/s/Mpc
Skill Min. Max Gal. enters
$\delta_{2000}$ $-13^\circ 32' 24''$ $ -13^\circ 11' 26'' $ 2077
$\alpha_{2000}$ $ 4^{\rm h} 32' 48''$ $ 4^{\rm h} 33' 49'' $ 2077
mag g 12.64 (-22.93) 24.14 (-12.28) 1081
mag r 12.04 (-22.62) 24.47 (-11.96) 2055
mag i 11.88 (-22.55) 23.75 (-12.67) 1500
Col. g-r -0.50 1.98 1059
Col. g-i -0.61 3.28 956

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