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6 Data evaluation


6.1 Galaxy and stellar counts

Since EIS observations were carried out under varying conditions, it is important to evaluate the degree of homogeneity of the final object catalogs. This can be done by examining, for instance, the number counts as a function of magnitude and comparing with earlier work. To evaluate the variation in the number counts due to the varying observing conditions, the area of patch A covered by both even and odd tiles, comprising 3 square degrees, has been divided into six subregions, each having an area of 0.5 square degrees. Note that these areas cover most of patch A including highly incomplete regions (see Fig. 16). The number counts for each subregion are computed and the mean is shown in Fig. 23, where the error bars correspond to the standard deviation as measured from the observed scatter in the six sub-catalogs. From the figure it can be seen that the difference between the even and odd catalogs is negligible.

\resizebox {8.7cm}{!}{\includegraphics{7652f23.eps}}\end{figure} Figure 23: The object counts as a function of magnitude as derived from the average of the counts in six odd (open squares) and even (filled squares) sub-catalogs. The error bars are the sample rms. In some cases the error bars are of the same size as the symbols

In Fig. 24 the galaxy counts derived from the EIS catalogs are compared to those of Lidman & Peterson (1996) and Postman et al. (1996). The $1\sigma$ error bars are computed as above. There is a remarkable agreement between the EIS galaxy-counts and those of the other authors. The slope of the EIS counts is found to be $0.43\ \pm \ 0.01$. Also note that the EIS counts extend beyond those of Postman et al. (1996) even for the counts derived from single frames. The galaxies have been defined to be objects with a stellarity index < 0.75 for I < 21 and all objects fainter than I = 21. At this limit galaxies already outnumber stars by a factor of $\sim$3.

\resizebox {8.7cm}{!}{\includegraphics{7652f24.eps}}\end{figure} Figure 24: The EIS galaxy counts (filled squares) with $1\sigma$ error bars compared to the galaxy counts derived by Lidman & Peterson (1996) (triangles) and Postman et al. (1996) (diamond). The data from the other authors have been converted to the Johnson-Cousins system

\resizebox {8.7cm}{!}{\includegraphics{7652f25.eps}}\end{figure} Figure 25: The EIS star counts for stellarity index $\geq$ 0.75 (filled squares) and for stellarity index $\geq$ 0.5 (open circles) compared to the model by Mèndez& van Altena (1996) (solid line)

As discussed before, the criteria adopted for classifying stars and galaxies is somewhat arbitrary. While a large value for the stellarity index is desirable to extract a galaxy catalog as complete as possible, this may not be the most appropriate choice for extracting stellar samples. This can be seen in Fig. 25 where the EIS star-counts are shown for two different choices of the stellarity index for objects brighter than I = 21. For comparison, the star counts predicted by the galactic model of Méndez & van Altena (1996) are shown. As can be seen the observed counts agree with the model for low values of the stellarity index (0.5), while higher values shows a deficiency of stars at the faint-end.

These preliminary results based on single-frame catalogs indicate that the depth of the survey is close to that originally expected and is sufficiently deep, especially after coaddition, to search for distant clusters of galaxies, one of the main science goals of EIS.

6.2 Angular correlation function

  In this section, the characteristics of the EIS catalogs are examined by computing the angular correlation function over the whole patch, for limiting magnitudes in the range I=19 to I=23. Table 4 gives the number of galaxies down to the different magnitude limits. For comparison the odd and even catalogs are treated separately. The region defined by $\alpha ~\rlap{$\gt$}{\lower 1.0ex\hbox{$\sim$}}
340^\circ$ and $\delta ~\rlap{$\gt$}{\lower 1.0ex\hbox{$\sim$}}-39.8^\circ$ has been excluded from the analysis because of its known incompleteness (Sect. 6.3). Therefore, the total area used here is 2.38 square degrees.

Table 4: Number of Galaxies in Patch A

&\multicolumn{2}{c}{Even} &\multicolumn{2}...
 ...\ $m \le 23$\space &94689 &39785.3 &95240 &40016.8 \\ \hline

To compute the angular correlation function, a random catalog is created with the same geometry as the EIS catalog. The number of random points has been chosen in order to yield an error less than $10\%$ on the measured amplitude of $w(\theta)$ at $\theta=5$ arcsec. The estimator used is that described by Landy & Szalay (1993):

w(\theta)=\frac{DD-2DR+RR}{RR},\end{displaymath} (1)

where DD, DR, and RR are the number of data-data, data-random, and random-random pairs at a given angular separation $\theta$.

In Fig. 26, $w(\theta)$ for the even and odd catalogs are compared. The error bars are 1$\sigma$ errors calculated with 10 bootstrap realizations. The angular correlation function $w(\theta)$ is well described by a power-law $\theta^{-\gamma}$ with $\gamma
\sim 0.8$ (shown by the dotted line) over the entire range of angular scales, extending out to $\theta \sim$ 0.5 degrees. In particular, there is no evidence for any feature related to the scale of the EMMI frame. Similar results are obtained when the angular correlation function is computed from counts-in-cells.

\resizebox {8.7cm}{!}{\includegraphics{7652f26.eps}}\end{figure} Figure 26: Angular two-point correlation function calculated for the whole patch (even and odd catalogs) except the region defined by $\alpha ~\rlap{$\gt$}{\lower 1.0ex\hbox{$\sim$}}
340^\circ$and $\delta ~\rlap{$\gt$}{\lower 1.0ex\hbox{$\sim$}}-39.8^\circ$ which has been removed for completeness problems. The dotted line represents a power-law with a slope of -0.8

In Fig. 27, the amplitude, $A_{\rm w}(I)$, of the correlation function at a scale of 1 arcsec is shown as a function of the limiting magnitude. The amplitude was computed from the best linear-fits over the range $\sim$ 10-200 arcsec of the $w(\theta)$, shown in Fig. 26, imposing a constant slope of $\gamma = 0.8$. For comparison, the power-law $A_{\rm w} \propto 10^{-0.27 R}$, originally determined by Brainerd et al. (1996) in the R-band is also shown corrected for the mean color difference (R-I) = 0.6 (Fukugita et al. 1995). The agreement with previous results is excellent and demonstrates the good quality of the EIS catalogs, even for a patch observed under less than ideal conditions.

\resizebox {8.7cm}{!}{\includegraphics{7652f27.eps}}\end{figure} Figure 27: Amplitude of the correlation function at 1 arcsec calculated from the best fits of Fig. 26 as a function of limiting magnitude. The dotted line is from from Brainerd et al. (1996)

In order to evaluate the contribution of large-scale clustering to the variance in the galaxy counts computed in the previous section, the correlation function has been computed for the sub-areas also used in estimating the number counts. This offers the opportunity to compare the amplitude of the cosmic variance to the bootstrap errors used above. The results are shown in Fig. 28 where the mean value over the sub-areas of $w(\theta)$ for each magnitude limit is presented. The error bars, that correspond to the rms of the six sub-areas, are consistent with those found from the bootstrap technique.

\resizebox {8.7cm}{!}{\includegraphics{7652f28.eps}}\end{figure} Figure 28: Mean value of the angular correlation function measured for 6 sub-catalogs covering the whole patch. The error bars are the standard deviations calculated between these subsamples, and are therefore an estimate of the cosmic variance on the corresponding scales. The dotted line represents a power-law with a slope of -0.8

6.3 Field-to-field variations

  The data for patch A are far from homogeneous and for some applications it is important to be able to objectively characterize the area and location of regions not suitable for analysis. A crude selection of these regions can be done by examining the distribution of the galaxy counts per frame as a function of the limiting magnitude. This is shown in Fig. 29 for the even and odd frames. The vertical lines correspond to the 3$\sigma$deviations from the mean, where $\sigma$ includes the contribution from Poisson fluctuations and the galaxy clustering. The latter was computed from counts-in-cells of galaxies on the scale of an EMMI-frame in the best available region of patch A.

\resizebox {\columnwidth}{!}{\includegraphics{7652f29.eps}}\end{figure} Figure 29: Distribution of galaxy counts for the even (dotted line) and the odd (dashed line) catalogs as a function of limiting magnitude. The vertical lines correspond to the 3$\sigma$ deviations from the mean, where $\sigma$ includes the contribution from Poisson fluctuations and the galaxy clustering

The results show that for galaxies brighter than I = 20, all frames fall within the 3$\sigma$ level, except four, and the varying observing conditions do not seem to affect the completeness of these bright object catalogs. Going to fainter magnitudes the fraction of frames with low number counts increases. This can be used to estimate the size of the homogeneous part of the patch at a given limiting magnitude as shown in Fig. 30. From the figure, it can be seen that for $I \sim
22$ and $I \sim 23 $ the useful area for analysis corresponds to $\sim$90% and $\sim$75%, respectively. Most of the rejected tiles are located in the region $\alpha ~\rlap{$\gt$}{\lower 1.0ex\hbox{$\sim$}}
340^\circ$ and $\delta ~\rlap{$\gt$}{\lower 1.0ex\hbox{$\sim$}}-40^\circ$ (Sect. 6.2).

\includegraphics [width=8.7cm]{7652f30.eps}\end{figure} Figure 30: Fraction of rejected tiles, corresponding to counts lower than 3$\sigma$ from the mean in Fig. 29, as a function of the limiting magnitudes

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