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Astron. Astrophys. 343, 760-774 (1999)

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3. Luminosity function in the R band

The study of luminosity functions allows to give constraints not only on the cluster galaxy content, such as the relative abundances of the various galaxy types, but also on larger scale properties. In particular, environmental effects have recently been shown to be important in several clusters; in Coma, for example, Lobo et al. (1997) have shown that the faint end of the luminosity function is steeper in the cluster than in the field, except in the regions surrounding the two large central galaxies. This was interpreted as due to the fact that each of these two galaxies is at the center of a group falling on to the main cluster; these groups tend to accrete dwarf galaxies, and as a result the luminosity function is flatter in these regions.

We will discuss here the properties of the bright part of the luminosity function of ABCG 85 in the R band.

3.1. Description of the available samples

In order to estimate the luminosity function, we can use either our redshift catalogue or our CCD imaging catalogue.

The redshift catalogue covers a roughly circular region of [FORMULA] radius around the center of ABCG 85; it is the shallowest one: its completeness is 82% for R[FORMULA]18 in a circle of 2000 arcsec diameter around the cluster center (302 galaxies in this region). We will limit our analysis to this region hereafter.

The CCD imaging catalogue in the V and R bands was obtained from 10 minute exposures in each band, in a small region in the center of the cluster (see Fig. 4 in Durret et al. 1998a), covering an area of 246 arcmin2; 381 and 805 galaxies were detected in the V and R bands respectively.

A photographic plate catalogue (4232 galaxies) has also been obtained by scanning a plate in the [FORMULA] band (Slezak et al. 1998). It is complete down to [FORMULA]=19.75 in a [FORMULA] square region; however, since it is shallower than the CCD imaging catalogue, we will not use it here.

Background counts were kindly made available to us in the R band from the Las Campanas survey (LCRS) by H. Lin (see Lin et al. 1996) and from the ESO-Sculptor survey (ESS) by V. de Lapparent and collaborators (see e.g. Arnouts et al. 1997). Note that the LCRS is made in a wide angle and therefore has small error bars in each bin, but is limited to R[FORMULA]17.8. The ESS is meant to reach very deep magnitudes in a small beam, and therefore its number counts are small at relatively bright magnitudes (for 17[FORMULA]R[FORMULA]18).

3.2. The R band luminosity function

We have chosen to draw the luminosity function in the R band, because our CCD imaging catalogue is deeper in R than in V. In the bright part (R[FORMULA]18), we will use galaxies with redshifts in the cluster range, therefore avoiding the problem of background subtraction. For these galaxies, the R magnitude was estimated from the photographic plate [FORMULA] magnitude, as explained in Slezak et al. (1998).

Fig. 9 shows a wavelet reconstruction of the distribution of galaxies with velocities in the 13000-20000 km s-1 velocity range as a function of absolute R magnitude (with an adopted distance modulus of 37.6). The wavelet reconstruction shows features significant at a level higher than 3[FORMULA]. In the reconstruction process, we are able to use all the available scales actually determined by the number of galaxies in the sample. However, we are not interested by phenomena at very small scales, which anyway are rather noisy. We therefore excluded the two smallest scales in our density profile reconstruction. The resulting density profile shows a dip for R[FORMULA], which corresponds to an absolute magnitude [FORMULA].

[FIGURE] Fig. 9. Wavelet reconstruction of the galaxy density as a function of absolute R magnitude for galaxies belonging to ABCG 85 with velocities between 13000 and 20000 km s-1 (full line) and for galaxies in Coma (dashed line). Arbitrary units are used to allow the direct comparison of both distributions.

We also derived the luminosity function from the R CCD imaging catalogue, which is complete to R[FORMULA]22, or [FORMULA], but in this case it was necessary to subtract a "typical" background contribution in this band. The ESS and the LCRS give consistent number counts for the magnitude bins which they have in common (within poissonian error bars and both normalized to the same area). We constructed a background function as follows: for both surveys, we estimated the numbers of galaxies Nbg per square degree per magnitude bin; we merged both surveys by considering that the background was represented by the LCRS for R[FORMULA]17.8 and by the ESS for R[FORMULA]17.8; we then fit several curves (power laws or polynomials) to the points thus obtained. The best fit was reached for a power law with the following mathematical expression:

[EQUATION]

The background counts and fit are displayed in Fig. 10.

[FIGURE] Fig. 10. Best fit of the background contribution estimated from the Las Campanas redshift survey for R[FORMULA]17.8 and from the ESO-Sculptor survey for R[FORMULA]17.8 (see text).

Fig. 11 shows a wavelet reconstruction of the distribution of galaxies derived from our CCD imaging catalogue after subtraction of the background contribution as explained above. The curve has roughly the same shape as that displayed in Fig. 9 for galaxies with redshifts, but it is shifted by [FORMULA]0.6 magnitude, with a dip now at [FORMULA]. As discussed in Sect. 3.4, it was not possible to draw the luminosity function for fainter magnitudes because of the background subtraction problem.

[FIGURE] Fig. 11. Wavelet reconstruction of the number of galaxies per square degree and per magnitude bin as a function of absolute R magnitude for the CCD imaging sample after subtraction of the background contribution (see text). The bottom line indicates the background contribution.

3.3. A dip in the luminosity function?

3.3.1. How real is the dip in the luminosity function?

As seen above, the wavelet reconstruction of the galaxy distribution shows a dip. In order to illustrate the robustness of this result, we have done two calculations. First, we consider that our data are the only available realization of a parent sample. Therefore, the bootstrap technique proposed by Efron (1979, 1982) seems well adapted to estimate error bars. We perform 1000 Monte-Carlo draws and do a wavelet analysis on each of the 1000 draws. We choose as limits to the error bars the 10 and 90 percentiles of the distributions thus obtained. These are shown on the top panel of Fig. 12. The dip therefore appears to be statistically significant. However, this bootstrap technique gives too large a weight to the observed realization; in particular, if a gap is present in the data, no draw will be able to fill it.

[FIGURE] Fig. 12. Top panel: Luminosity function as in Fig. 9 with error bars obtained by a bootstrap technique (see text). Bottom panel: wavelet reconstructions of the luminosity functions at three different scales; error bars are obtained by a Monte-Carlo technique (see text). The dashed line in the bottom panel shows the luminosity function at a scale twice that of the full line.

We have therefore applied a second method. As a first step, we have wavelet reconstructed the luminosity function eliminating the three smallest scales. The distribution obtained in this way does not show any dip. We have then performed 1000 Monte-Carlo draws following this profile, and again have done a wavelet analysis on each of these draws.

The result of this method is shown in Fig. 12 (bottom panel). The dip clearly appears outside the error bar region, implying that the probability to obtain such a feature from such a parent sample (devoid of dips) is smaller than 0.001; even the luminosity function drawn at a larger scale (dashed line in Fig. 12) shows a shallower but still significant dip.

3.3.2. Physical interpretation of the dip

A comparable dip was found in the luminosity function of several clusters. We give in Table 2 the positions of the dips for R band absolute magnitudes recalculated when necessary for a Hubble constant of 50 km s-1 Mpc-1; luminosity functions drawn in the B band have been shifted to the R band assuming B-R=1.7 for all clusters except Virgo, a typical value for ellipticals, taken as the dominant cluster population. For Virgo, where spirals are dominant, we took B-R=1.4. No K-correction or Galactic absorption corrections were included, since this is only a rough comparison.


[TABLE]

Table 2. Dip positions in clusters. ABCG 85(z) and (CCD): luminosity functions derived from the reshift and CCD imaging catalogues respectively.


It is interesting to note that the dips in the luminosity functions are found at comparable absolute magnitudes in all these clusters within a range of only one magnitude. The only cluster that we found in the literature having a dip at a significantly different absolute magnitude is ABCG 496.

As mentioned above, the dip position derived from the redshift catalogue differs from that derived from the CCD imaging catalogue in ABCG 85. This apparent discrepancy is most likely accounted for by the fact that the latter corresponds to a much smaller central region, and suggests that environmental effects modify the luminosity function in this cluster (see below).

These dips do not all seem to have the same width: the dip found in the luminosity function of Shapley 8 is notably broader, while shallower dips (or at least flattenings) are found in the luminosity functions of Virgo and ABCG 963. However the methods used by these various authors are quite different from ours; we have redone the analysis described by Biviano et al. (1995) in the Coma cluster using the wavelet reconstruction technique; the corresponding luminosity function is displayed in Fig. 9 and the dip has a shape notably broader than that of ABCG 85.

The above facts suggest that the bright galaxy distributions in these clusters have roughly comparable properties, but also that they differ from a cluster to another, and even from one region to another in a given cluster. This is also the case for the relative abundances of galaxy types, which depend on the local density and/or on the global properties of each cluster.

In fact, a simple model based on the shapes of the luminosity functions of the various galaxy types and on their relative proportions (e.g. Böhm & Schmidt 1995, Jerjen & Tammann 1997, and references therein) can roughly account for the dip in the luminosity function.

By using only three kinds of luminosity functions, and adjusting the relative proportions of these three types of galaxies, it is easy to recover a luminosity function with a similar shape to that observed. As an example, such a toy-luminosity function is given in Fig. 13; in this case, we have used:

  • a Gaussian with [FORMULA]15.8 and [FORMULA]1.1, for the spiral luminosity function;

  • a Gamma density (also called Erlang density):

    [EQUATION]

    for ellipticals. This density function is asymmetric and has been used by Biviano et al. (1995) to describe the luminous part of the Coma luminosity function; [FORMULA] is a cut-off value (in the example we have chosen [FORMULA]17.2) the maximum is given by [FORMULA];
  • a power law to represent faint or dwarf galaxies, filtered by an apodisation function to account for the incompleteness for high values of R.

[FIGURE] Fig. 13. Simulated luminosity function. The bold curve is obtained by the analytical method described in the text. The dashed curve corresponds to the median of Monte-Carlo draws. Error bars are obtained by a Monte-Carlo technique as described previously.

We can see from Fig. 13 that although located at the good position, the dip is broader than the ABCG 85 dip and not as deep, and that the luminous part of the luminosity function is convex instead of concave. In fact, because the game is played with at least three functions, each of them driven by 3 or even 4 parameters, we have too many degrees of freedom and are able to modify these features in various ways. However, the location of ellipticals relative to dwarfs is well determined: the dip corresponds to the transition zone between ellipticals and dwarfs. However, it has not been possible to play the same game with two functions only.

In this hypothesis, the fact that the dip falls roughly at the same absolute magnitude for at least seven clusters suggests that in the dip region these clusters have comparable galaxy populations; however, the fact that the various dips are not located exactly at the same absolute magnitudes and have different widths raises the question of the relative positions and densities of these various populations. Lobo et al. (1997) have shown that the slope of the faint luminosity function varies with the local environment. The less rich a cluster, the more numerous are spiral galaxies; an increase in the number of spirals will modify the luminosity function only around [FORMULA]. The combination of these well established results leads to confirm, as suggested by previous authors, that in general luminosity functions depend simultaneously on type and on local density and/or global properties (such as cluster richness; see e.g. Phillipps et al. 1998). However, our data is not complete enough to allow a further analysis.

3.4. A word about the faint end of the luminosity function

We initially intended to analyze the faint end of the luminosity function derived from our CCD imaging catalogue after subtracting the background as described above. However, for [FORMULA] the luminosity function is found to decrease dramatically, rapidly reaching negative values, although the data sample appears to be complete up to R[FORMULA]. This implies that the background contribution has been overestimated, i.e. the "Universal" background counts as obtained in previous section (Fig. 10) are not representative of the background of ABCG 85.

Marginal evidence for the existence of a background larger than expected from statistical arguments has also been found in Coma. Out of 51 redshifts obtained for faint galaxies (R[FORMULA]) in a small region near the Coma cluster center, at most five galaxies were found to be cluster members (Adami et al. 1998b), while the expected number was [FORMULA]. Due to the small number of redshifts involved, this result is of course still preliminary; however, it raises the question of statistical background subtraction, which should not be as universally accepted as it is now.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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