3 Results

3.1 Spatial distribution of galaxies of various luminosities

Figure 1 shows the studied field of view. Each ellipse correspond to one galaxy. The three dotted rectangles enclose the three $K_{\rm s}$pointings. Attentive inspection of Fig. 1 shows that:

- There are two obvious galaxy overdensities: one in the center and the other one at 3 arcmin NW;

- There is another possible overdensity in the far S, at $\sim$5 arcmin away from the center, as can be appreciated by comparing the density of galaxies at similar distances from the cluster center in the Southern and Northern pointings. Galaxies in the far S have unknown redshifts, and therefore we don't know whether this overdensity is associated with the cluster or is a background group (or cluster).

Figure 2 shows the cluster radial profile,

Figure 2: Background subtracted radial profiles of the AC118 cluster. Galaxies having $K_{\rm s}<20$, $K_{\rm s}<17$, $18<K_{\rm s}<20$ and $19<K_{\rm s}<20$ mag are selected for the computation of the radial profiles in the top-left, top-right, bottom-left and bottom-righ panels, respectively. Open dots show the profile computed over all the observed field, while closed points show the profile computed excluding the southest 1 armin and the NW quadrant. Note the steepness of the radial profile of bright galaxies (upper-right panel) and the flatness of the radial distribution of faint galaxies (lower-left panel). The density of background galaxies measured in the HDF-S is 7.8, 0.3, 6.3, 4.2 gal arcmin-2 for $K_{\rm s}<20$, $K_{\rm s}<17$, $18<K_{\rm s}<20$and $19<K_{\rm s}<20$ mag respectively, when adopting our 3 arcsec aperture magnitude, and has been already subtracted.

i.e. the number of galaxies per radius bin, measured in circular annuli. It has been computed for four magnitude ranges and both including (open points) and excluding (close points) galaxies in the far S, that are possibly unrelated to the studied cluster, and in the NW quadrant, where the effect of the NW clump should be higher. The radial profiles are statistically background subtracted in order to remove interlopers by using the background galaxy density measured in the HDF-S (da Costa et al. 2002), as computed by ourselves by using their public images. Errors are assumed to be Poissonian (i.e. for the time being we neglect the intrinsic variance of galaxy counts, that are, instead, taken into account in the next sections). The area observed at each radius is shown in Fig. 3, and care should be paid to densities computed over a small area (say, much less than 2 arcmin²) and large clustercentric radii because on these small areas the intrinsic variance of the galaxy counts could be very large with respect to the low cluster galaxy density.

The cluster radial profile of all galaxies (brighter than $K_{\rm s}<20$ mag, or $M_K \raisebox{-0.5ex}{$\,\stackrel{<}{\scriptstyle\sim}\,$ }-21$ mag, upper-left panel) shows a positive galaxy density from the center to 2 Mpc away, in particular when galaxies in the far S are counted. It is centrally peaked. When the NW quadrant is included in the profile computation, a second broad peak is present at 0.7 Mpc from the cluster center, while at larger radii the profile decreases. When the quadrant including the NW clump is instead removed, the radial profile shows a flattening, instead of a second maximum, at $\sim$1 Mpc from the cluster center.

When we consider only galaxies brighter than $K_{\rm s}=17$ mag ( $M_K\sim-24$ mag), i.e. massive galaxies, the cluster radial profile (upper-right panel) is steeper in the center than in the previous case. In fact, the galaxy density increases by a factor 7 over three bins, to be compared to an increase of a factor 2 to 3 over the same radial range when all galaxies are considered. The second maximum is still there when all galaxies are counted (open points). Overall, the profile is quite flat outside the cluster core.

Figure 3: Area over which the radial profiles are computed. Open points mark the area studied when all the field of view is considered, while close points mark the area when the NW quadrant and the southest 1 armin are excised.

The evidence of a positive density at 1.8 Mpc is marginal ($2\sigma$) when discarding galaxies in the far S, while it is significant including them.

At the other end of the luminosity function, the radial profile of faint galaxies, $19<K_{\rm s}<20$ mag or $-22\raisebox{-0.5ex}{$\,\stackrel{<}{\scriptstyle\sim}\,$ }M_K\raisebox{-0.5ex}{$\,\stackrel{<}{\scriptstyle\sim}\,$ }-21$ mag, is quite flat from the center to $\sim$1-1.2 Mpc (bottom-right panel), and undetected (i.e. statistical evidence is $\sim$$1\sigma$) at large radii, even binning the data with larger bins.

The shape of spatial distributions of galaxies in the $18<K_{\rm s}<20$ mag (bottom-left panel) and $19<K_{\rm s}<20$ (bottom-righ panel) ranges are quite similar. There is a factor of two between the amplitudes of the two radial profiles, because there is a factor of two between the two considered magnitude ranges, and because the AC118 luminosity function is quite flat at these magnitudes (see Sect. 3.3). The innermost point seems higher that the ones at $r\sim1$ arcmin, but without any statistical significance.

Therefore, the radial profile of all, bright and faint galaxies are quite different in steepness. Faint galaxies shows similar radial profiles independent of the two considered magnitude ranges.

3.2 Dwarfs to giant ratio radial profile

The left panel of Fig. 4 shows the giant-to-dwarf ratio, as a function of the clustercentric distance. For the sake of clarity, galaxies brighter than $K_{\rm s}=17$ ( $M_K\sim-24$) mag are called giants, while galaxies with $18<K_{\rm s}<20$ mag are called dwarfs. The giant-to-dwarf ratio shows a maximum at the center, where there are similar numbers of giant and dwarfs in the considered magnitude range, then decreases to a much smaller value from radii as small as 300 Kpc and as far as 2.2 Mpc. Outside the cluster core, there are roughly 3 dwarfs per giant in the considered magnitude ranges. The deficit of dwarfs in the cluster core (or the excess of giants) is in agreement with that found in the previous figure and in Paper I by analysing the shape of the LF at various cluster locations (but over a restricted cluster portion) and of the giant-to-dwarf ratio at a few cluster locations. The inclusion or exclusion of the NW quadrant or of the far S region does not appreciably change the giant-to-dwarf ratio, as shown in the figure. The new data presented in this paper do not make stronger the statistical significance of the found segregation for clustercentric distance less than 1 Mpc (that it is claimed significant at >99.9% confidence level in Andreon 2001), because new data are at larger clustercentric distances. Since we divide the R<1 Mpc range in three bins, instead of the two bins as in Paper I, the statistical evidence per bin is in fact smaller here than in Paper I ($\gtrsim$90 vs. >99.9% confidence level). At the large clustercentric radii sampled by the new data, the giant-to-dwarf ratio differs from the central one at the 90% confidence level when all the field of view is considered, and at the 80% confidence level when the NW quadrant and the far S regions are excluded. In this specific calculation, we take into account the field-to-field background variance as described in Huang et al. (1997), and we propagate the errors as described in Gehrels (1986), i.e. we do not make the simplifying assumption of Gaussian errors.

Similar conclusions can be drawn defining as dwarfs $19<K_{\rm s}<20$ mag galaxies (right panel of Fig. 4), except that the absolute value of the giant-to-dwarf ratio increases by approximatively a factor of two, because the considered magnitude range for dwarfs is now half the size. The shapes of the giant-to-dwarf radial profiles in the two panel of Fig. 4 are striking similar. This similarity implies that $19<K_{\rm s}<20$mag dwarfs are not segregated with respect to $18<K_{\rm s}<20$ mag dwarfs, as directly seen in the bottom panels of Fig. 2.

Note, however, that the smaller magnitude range adopted in the right panel of Fig. 4 also decreases the number of dwarfs, and therefore increases the size of error bars.

Figure 4: Giant ( $K_{\rm s}<17$) to dwarf ( $18<K_{\rm s}<20$ in the left panel, $19<K_{\rm s}<20$ in the right panel) ratio as a function of the clustercentric distance, including and excluding the NW quadrant and the southest 1 armin (open and solid points, respectively). Error bars in the abscissa show the bin width. For display purposes error bars in the abscissa are drawn once and points are slightly displaced in x.

For the same reason the statistical significance of a variation of the giant-to-dwarf ratio is also reduced.

3.3 Luminosity function at various cluster locations

The LF is computed as the statistical difference between (crowding-corrected) galaxy counts in the cluster direction and in the control field direction. We use the HDF-S (da Costa et al. 2002) as background (control) field, and we fully take into account the field-to-field galaxy count fluctuations in the error computation (see Paper I for details).

We fitted a spline to the background counts and we use it in place of the observed data points because background galaxy counts show an outlier point at $K_{\rm s}=17$ mag when a 3 arcsec aperture is adopted.

The present AC118 sample consists of 496 members, about as many galaxies as in Paper I, but are distributed over a larger area and a narrower magnitude range. The LF has been fitted by a Schechter (1976) function by taking into account the finite bin width (details are given in Paper I). Figure 5 shows the LF computed at different cluster locations and the best fit Schechter (1976) function to the global (i.e. those measured over the whole field of view) LF,

Figure 5: Luminosity function of AC118. The global (i.e. integrated over the whole studied field) LF is shown in panel  a). Panels  b) and  c) present the LF of the main and secondary clumps, respectively. Panel d) is the LF of galaxies inside the central pointing but outside the two clumps. Panel  e) shows the LF of galaxies in the Northern and Southern pointings (not overlapping to the central pointing) and not in the far S. Finally, panel f) shows the LF of the galaxies in the far S (southern 1 arcmin). The curve is the best fit function to the whole cluster, with $\phi ^*$ adjusted to reproduce the total number of galaxies at each considered location. There are 496, 91, 86, 164, 101, 56 galaxies in panels  a-f), respectively.

whose $\phi ^*$ is scaled by the ratio between the number of members at each location and in the global LF. The global LF is shown in panel a). Panels b) and c) present the LF of the main and secondary clumps, respectively. Their exact boundary definitions are those of Paper I (for a pictorial view see Fig. 1 there). Panel d) shows the LF of galaxies inside the central pointing but outside the two clumps. Panel e) shows the LF of galaxies in the Northern and Southern pointings (not overlapping with the central pointing) and not in the far S. Finally, panel f) shows the LF of the galaxies in the far S (southest 1 arcmin).

The best fit parameters to the global LF are: $K^*_{\rm s}=16.4$ mag ( $M_{K^*_{\rm s}}\sim-24.9$ mag) and $\alpha=-0.85$, where $\alpha$ is the slope of the faint part of the LF, and M^* is the knee of the LF, i.e. the magnitude at which the LF starts to decrease exponentially. We re-state that the present 3 arcsec magnitude misses a significant part of the galaxy flux, and hence the found parameters should not be used for, say, computing the luminosity density, or for comparison with values derived from other samples using a different metric (or any isophotal) aperture. This could also be appreciated by noting that in Paper I, using magnitudes that include a large fraction of the galaxy flux, we found steeper LFs than shown in panels b)-d) for the same considered cluster and background regions. Here we use the LF as a tool for comparing the abundance of galaxies of various luminosities in different environments for a sample of galaxies all at the same redshift and whose flux is measured in one single way. A thorough discussion of the cosmological implication of lost flux from galaxies is given in Wright (2001) and Andreon (2002). Errors, quoting the projection of the $\Delta\chi^2=2.3$(68% for two interesting parameters, Avni 1976) confidence contours on the axis of measure are: 0.35 mag and 0.21, respectively, for $K^*_{\rm s}$ and $\alpha$. The conditional errors, i.e. the errors when the other parameters are kept at the best values (that has a low statistical sense, Press et al. 1992) are found to be at least half the size. The AC118 global LF is smooth and is well described by a Schechter function ( $\chi^2/\nu\sim4.2/8$).

The parameters of the global LF also describe the shape of the LF measured at other cluster locations (see panels from c) to f)), because the reduced $\chi^2$ is of the order of 1 or less, except for the LF in panel b). Galaxies considered in panel b) are in the cluster center: for the total number of observed galaxies there are a too many very bright galaxies (say, brighter than $K_{\rm s}=16$-17 mag) and too few fainter galaxies, an effect already found in Paper I for the same region and using the same data, but adopting a magnitude definition which includes a larger galaxy flux. This is the same effect shown in Figs. 2 and 4 and presented in the previous sections, measured here by looking for differences in the LF computed at several cluster locations instead of looking for a dependence between the spatial distribution of galaxies and their luminosities. Differences found in Paper I are confirmed here (by adopting a $\sim$95% confidence level threshold and using a Kolmogorov-Smirnov test, that is preferable to comparing the best fit values because of the correlation between parameters and of the need for an assumption of a given parental distribution): the LF is flatter at the main clump (panel b)) than at all the other considered regions. All the other LFs are compatible each other at better than 95% confidence level, extending at larger radii the findings in Paper I: the LF steepens going from high - to low - density environments and the steepening stops in the region considered in panel d). The new result is that the LF does not change in regions not surveyed in Paper I, i.e. for galaxies whose average clustercentric projected distance is 1.2 Mpc (for galaxies in the N and S pointings, panel e) and 1.8 Mpc (for galaxies in the far S, panel f).

The f) panel only includes galaxies in the far S (southest 1 arcmin). These galaxies are an extension of the AC118 cluster, or another group (or part of a cluster) along the line of sight. Given the small number of galaxies in this region (56 galaxies out of 535) and the similarity of their LF to the global one, their inclusion or exclusion from the global LF makes no difference.

The LFs computed thus far can be used to test whether the galaxy overdensity in the far S is at the AC118 redshift, under the assumption that the LF is a standard candle outside the cluster core. The use of the near-infrared LF as a standard candle has been exploited by de Propris et al. (1999) to study the luminosity evolution of galaxies up to $z\sim1$. There are two paths for the computation, depending on whether a parametric form is used for the LF shape (and in such a case the errors on the data points are included in the confidence level calculation) or no (that neglects errors on data points). For galaxies in the far S sample, the 68% conditional confidence range (i.e. once $\alpha$ is keep fix to the best fit value) for M^* are 16.3 and 17.9 mag, limiting the difference in distance modulus between AC118 and the far S overdensity to $\Delta (m-M)= -0.1,\ +1.5$ mag or, in redshift, -0.01, +0.3. This range in $\Delta (m-M)$ only excludes that the galaxies in the far S are in the AC118 foreground. To be precise, the high redshift constraint is broader, because our 3 arcsec aperture includes more and more galaxy flux as the redshift increases, and we have not accounted for this effect. By using the data points alone and a Kolmogorov-Smirnov test, the 68% confidence range is $\Delta (m-M)=0,\ \sim$1.7 mag, quite similar to the parametric result. Therefore, the analysis of the LF is not sufficient to say whether these galaxies belong to the cluster of are in the background of AC118. Surely, these galaxies do not lie in front of the cluster.

In conclusion, the analysis of the luminosity function shows the same luminosity segregation found in the analysis of the galaxy spatial distribution. With respect to Paper I, we extended the analysis to much larger distances (1.8 Mpc vs. 0.58 Mpc).