3.1. The shape of the LF
Given the counts in the Coma direction and in the field, and their errors, the computation of the Coma cluster LF is an algebrical exercise. We stress that our main sources of error on the Coma LF are Poissonian fluctuations of total counts over the Coma region and the background field to field variance. Therefore, the error on the background is not derived from the HDFS1+S2 error bars (which are not relevant for the determination of the LF errors).
The near-infrared Coma LF is presented in Fig. 2. It is characterized by a bright end (at mag), a part increasing gently down to mag, and an "outlier" point at mag, which produce, if real, a dip in the Coma LF. The LF displayed in Fig. 2 is the deepest ever measured in any near-infrared band for any type of environment for a near-infrared selected sample.
In the optical, the Coma cluster exhibits a similar shape (Godwin & Peach 1977; Secker & Harris 1996). Using Godwin et al. (1983) data, we computed the b (a photographic J-like blue filter) Coma LF in almost exactly the same area surveyed in the H band. For simplicity, we have considered a rectangular area enclosing our H band region, without taking into account the complex geometry of our region in details. Then, because the H and b band magnitudes are available for all galaxies in this area, we have computed a mean pseudo-color 2. In order to compare the two LFs, we have shifted the b band LF by the mean pseudo-color: mag. The expected values according to Bruzual & Charlot models are for elliptical galaxies and up to for blue constant star-forming systems; thus an averaged population of of ellipticals versus of blue systems gives roughly the right mean value as expected. No normalization in has been applied. The result of this exercise is shown in Fig. 2 as a dashed-line histogram. The two LFs are remarkably similar: there is a close agreement between the cut at mag, the dip location ( mag), the dip amplitude and the increase observed at fainter magnitudes (closed dots) and expected from the b LF (dashed histogram). We have performed the same exercise with the R band photometry by Secker & Harris (1996), who studied an adjacent region of the Coma cluster. The expected values according to Bruzual & Charlot models are for early-type galaxies and for blue star-forming systems, giving an averaged value of for the same weighted population taken above ( of ellipticals versus of blue systems). This time we adopt the predicted value because the R catalog is not published. The two surveyed regions are different, thus we normalize their LF to our H LF. We obtain similar results, with the difference that the importance of the dip is smaller in R than in H (see Fig. 2).
Thus, the shape and the amplitude of the Coma LF seems not to be strongly dependent on the wavelength when we compare the results in b and in H bands, and also, with less confidence, in the R band. The strong similarity of the optical and near-infrared LF implies that in the near-infrared there is no new population of galaxies which disappears in the optical band (because dust obscured, for example), down to the magnitude of dwarfs. Furthermore, if the H band LF traces the galaxy mass function in this cluster, the same holds for the blue LF. This result has been obtained in a particular region of Coma, wich is a cluster rich in elliptical and lenticular galaxies. Before any generalization, this result should be checked in other regions of the cluster, and also in other environment conditions (cluster outskirts, clusters rich in spiral galaxies, groups,...).
3.2. The dip at
Let us consider in more details the dip point. The question is: Is it really an outlier? The statistical significance of the possible outlier point must be evaluated from the galaxy counts, since they are the original source of fluctuations. The mag bin in Fig. 2 corresponds to the mag bin in Fig. 1. First of all, we exclude the possibility that we have missed some galaxies of this magnitude, because we are complete 2.5 mag fainter, and because a typical galaxy of mag has a central brightness of 100 times the sky noise. Secondly, there is no relation between the location of the dip and the discontinuity of the magnitude system adopted (Kron magnitudes for bright galaxies and aperture magnitudes for faint ones): galaxy counts do not change because the separation is set to mag. In fact, Kron and aperture magnitudes have almost the same value down to the last magnitude bin (see Fig. 10 in Paper 1). There are two galaxies in the mag bin, whereas are needed to make the counts smooth. Therefore, this point is more than 3 away from the average of adjacent bins. To be precise, according to Poissonian statistics we can reject at more than 99.95% confidence level the hypothesis that the observed number of galaxies is drawn from a parent distribution which counts galaxies in that bin. Therefore, the dip is a real feature of the Coma near-infrared LF in this region.
The dip in the Coma LF had firstly been noticed in the optical band (for example, Godwin et al. 1983) and it had been interpreted in two different ways. Biviano et al. (1995) suggested that galaxies brighter than the dip were subjected to a recent episode of star formation induced by the hostile Coma environment, which have made them brighter. Andreon (1998) has shown that the LF of the different morphological types of galaxies are equal in Coma and in much poorer environments, and that the dip is simply the combined result of the Coma cluster morphological composition together with the shape of the type-dependent LFs. If the induced star-formation interpretation by Biviano et al. (1995) was correct, the dip should be absent or at least highly attenuated in H, because the near-infrared luminosity traces the galaxy mass and it is less affected by the short timescale starbursts that make a few galaxies brighter than the magnitude of the dip. Instead the dip is observed in the H band.
A few other near-infrared LF of clusters are (poorly) known. None of the five clusters studied by Mobasher & Trentham (1998) show such a dip. However, the area sampled in each cluster includes a tiny number of galaxies, so that errors are large and the visibility of a possible dip (if present) is arguable. The cumulative LF of three clusters at (Barger et al. 1996) does not seem to show a dip, but it barely reaches the dip magnitude. The only truly comparable LF has been presented by De Propris et al. (1998), and it is reproduced here in Fig. 4 (open squares), together with the best Schechter fit LF (dotted line) to our data. The fitting machinery adopted here is discussed in the next subsection. De Propris et al. (1998) have used Kron magnitudes for faint galaxies and aperture magnitudes (within a 62 arcsec diameter) for large galaxies (De Propris 1999, private communication). The two LFs are in remarkable good agreement () on the common range ( mag), with the exception of the dip bin, wich is present in our data and absent in De Propris et al. (1998) data. It is worth noticing that the position of the dip is well within the spectroscopic sample of De Propris and collaborators, and thus it could be hardly missed. The agreement would be even better at if the De Propris et al. (1998) bright magnitudes were of Kron type, since Kron magnitudes integrate the galaxy flux inside a smaller area than those sampled by the 62 arcsec aperture they used.
Since De Propris et al. (1998) studied an almost complementary area of the Coma cluster with respect ours, and since the dip is present in our LF and absent in theirs, it is possible that the amplitude of the dip depends on the location in the cluster, as it seems to be the case in the optical (Sekiguchi 1998). Since the H band luminosity traces the galaxy mass, as stressed in the Introduction, the possible dependence of the dip amplitude on the cluster location points out a dependence of the mass function on the surveyed region, possibly due to a joint effect of morphological dependence of the LF and variation of the morphological composition over the Coma cluster. A similar trend is seen in the optical (Andreon 1998). Such differences in the LF as a function of the location in the cluster could be related to subcluster structure. Several evidences for cluster-cluster merger are present in Coma. Two main peaks appear in the X-ray flux density (White et al. 1993), in the projected distribution of galaxies (Fitchett & Webster 1987, Mellier et al. 1988) and in the radio source counts (Kim et al. 1994): a clump centered on NGC4874 and NGC 4889, and a secondary peak around NGC 4839, about 40' SW from the previous one. The field surveyed here is centered NE from the main structure, at the opposite side with respect to the cluster center. Colless & Dunn (1996) have shown the complex dynamics and multiple substructure of the Coma cluster using a large redshift catalog. According to them, the NGC 4839 group is actually falling into the main cluster, there are two subclusters in the central region (associated with the two dominant galaxies), and late type galaxies are falling into the main cluster (which is dominated by early type galaxies). These processes might be able to locally modify the LF as observed.
3.3. Fitting the LF
Let us consider now the overall shape of the LF. Usually, a method is used to fit the LF of clusters by a Schechter (1976) function:
The method is not the optimal one for fitting a function to a small number of bins, and it is even less suitable when bins are poorly populated. Furthermore, a requires to bin the data with an arbitrary bin size. Although the method is not optimal, we are forced to use it, since we do not know any other fitting method that could take into account, even roughly, background fluctuations together with Poissonian ones without binning the data. More elegant methods implemented so far, such as maximum-likelihood fitting, do not take into account the Poissonian fluctuations of the background counts, nor the field to field variance of the background, and therefore they systematically underestimate the true errors.
In order to take into account the amplitude of the bin in the fitting process (a technical detail seldom considered), we fit the data with a Schechter function convolved with the bin width (although in practice this detail makes almost no difference on the results). An additional problem arises: given the existence of a real dip in the Coma LF, the fit of the whole LF with any Schechter function is necessarily poor (and in fact we found a minimum of 14 for 4 degrees of freedom). We are therefore left with two options: flag the dip point, or use a more complex function. Disposing of a very small number of points and lacking any physically motivated more elaborate function to be fitted, we simply flag the outlier bin.
In that case, and taking into account the finite amplitude of the bin, we found mag and , but with large confidence intervals (as shown in Fig. 3). Note that our magnitude limit, mag, is roughly equivalent to mag at the Coma distance for an early-type galaxy ( mag), which is well in the dwarf regime. agrees well with the values expected from the optical photometry and usual colors for early-type galaxies ( mag and mag). The slope is steeper than the typical value in optical bands (López-Cruz et al. 1997; Garilli et al. 1999), but nevertheless it is quite similar to that found for a few well studied clusters in optical bands (Smith et al. 1997; Schechter 1976).
3.4. Comparison to previous studies of Coma and to the field LF
Mobasher & Trentham (1998) studied a very small portion of the Coma cluster and were able to build a catalog 1.5 magnitudes deeper than ours. However, their studied field is too small to make the background variance small relative to the signal (the Coma LF), so that the resulting LF is completely unconstrained, as admitted by the authors. They computed also another LF, by performing a crude color selection, i.e. assuming that Coma cluster galaxies lay, in a color-color plane, in a region different from that occupied by the fore and background galaxies. In that case, a LF with error bars of reasonable size was derived, but under an hypothesis that should be demonstrated to be true. In Fig. 5, this LF is plotted overlapped to our H LF, after having matched the two LFs in the common bins. Their K magnitudes have been changed to H assuming mag, the typical value expected for the Coma galaxies. Our error bars are smaller in the common bins, even using the same binning for the two LFs. Mobasher & Trentham (1998) points stay relatively near the extrapolation of the best Schechter fit to our data, suggesting that the LF could keep its slope even at these very faint magnitudes (roughly equivalent to mag). The same points stay near the Secker & Harris (1996) R band Coma LF shifted in the H band, as plotted in Fig. 2.
Fig. 5 also compares the Coma cluster LF to the local field LFs, as computed by Gardner et al. (1997, dashed line) and Szokoly et al. (1998, solid line). The two field surveys differ in many respects. The former is based on a sample about 5 times larger than the latter, and it is computed from a near-infrared selected sample. Instead, the latter is optically selected and no corrections have been applied for the optical selection. The slope of the field LF computed by Gardner et al. (1997) and by Szokoly et al. (1998) differ largely, with and , respectively. However, the 68% confidence contours of the two LFs cross each other (figure not shown), implying that the two LFs are compatible to , as also claimed by Szokoly et al. (1998). The two field LFs could have different slopes but they still remain compatible because they barely reach mag 3, and therefore they sample only the exponentially declining part of the LF. Therefore, the slope of the field LF is constrained by the faintest bin (see Fig. 5) which, as in all field surveys, is quite uncertain because measured on a very small volume.
Our own data for the Coma H LF are three mag deeper than the local field ones, and the overall shape, as parametrized by the Schecther parameters, agrees with the field ones: the Coma LF has and indistinguishable from the Szokoly et al. (1998) LF and a very similar to the Gardner et al. (1997) one. Its slope, , is steeper than the Gardner et al. (1997) slope, but by less than difference, due to the large confidence level intervals of the two LFs. Also, the present Coma H band LF agrees with the field LF computed by Cowie et al. (1996) in a few redshift ranges, up to .
On the one side, the overall shape of the LF is similar both in the field and in the Coma cluster. On the other side, no dip is present in near-infrared field LFs, whereas instead in our Coma LF it is quite evident. This fact, and the absence of the dip in the Coma region studied by De Propris et al. (1998) seems to suggest that the dip amplitude could be related to the morphological mix of the studied environment. The alternative possibility requires that environmental effects change the H band luminosity preferentially at a given mass (corresponding to mag), without altering too much the mass distribution for more massive galaxies. Otherwise, the Coma and the field LFs should have different bright tails.
© European Southern Observatory (ESO) 2000
Online publication: December 17, 1999