4 Luminosity functions and incompleteness effects

   
4.1 Integrated luminosity function

In Fig. 11, we compare the Gemini with the HST luminosity function for the $K^\prime_{\rm trans}$ vs. m205 observations. For direct comparison of the observational efficiency, no colour cut has been applied, but the entire physically reasonable colour range from approximately 0 to 4 mag in $H - K^\prime$, including reddened and foreground objects, has been included in the luminosity function (LF). Therefore, these LFs are not the ones from which the mass functions have been derived. The only selection criterion that has been applied is a restriction of the photometric uncertainty in both magnitude ( $\sigma_{K^\prime} < 0.2$ mag) and colour ( $\sigma_{H-K^\prime} < 0.28$ mag, corresponding to $\sigma_{K^\prime} < 0.2$ mag and $\sigma_H < 0.2$ mag). The uncertainty selection in colour allowed us to select only those objects that have been detected with high confidence in both H and $K^\prime$images in the Gemini data, and in the F160W and F205W filters in the NICMOS data, respectively. The colour-uncertainty selection on the HST data simulates the matching of H and $K^\prime$ detections used on the Gemini data for the selection of real objects. Thus, only objects that are detected in both H and $K^\prime$ have been included in the luminosity function. This gives us some confidence that we are not picking up hot pixels or cosmic ray events. The area covered with Gemini has been selected from the HST photometry as displayed in Fig. 1.

As can be seen in Fig. 11, many objects are missed by Gemini in the fainter regime, though the actual limiting (i.e., cut-off) magnitudes are the same in both datasets. This is due to the fact that 50% of the light is distributed into a halo around each star. These halos prevent the detection of faint objects around bright sources, especially in the crowded regions. This effect is most obvious when examining the star-subtracted frames resulting from the DAOPHOT allstar task. In these frames the cluster center is marked by a diffuse background, enhanced by $\sim$20 counts in $K^\prime$ and $\sim$40 counts in H above the observational background of 2 and 4 counts in the cluster vicinity, respectively. In addition to the simple crowding problem due to the stellar density affecting both datasets, the overlap of many stellar halos hinders the detection of faint stars in the Gemini data. At larger radial distances from the cluster center, more and more faint stars are detected both in the HST as well as in the Gemini data (Fig. 12).

The fact that the incompleteness corrected Gemini LFs follow closely the shape of the HST LFs supports the results of our incompleteness calculations, which will be used to determine the incompleteness in the mass function.

Figure 11: Comparison of Gemini versus HST luminosity functions.

   
4.2 Radial variation of the luminosity function

Radial luminosity functions were calculated in $\Delta R = 5$ $^{\prime \prime }$ bins, using the same uncertainty selection as in Fig. 11 (Sect. 4.1). The resulting radial LFs are shown in Fig. 12, along with the incompleteness determined for each radial bin. In the cluster center (lowest panel), the very good match of the Gemini and HST LFs for $K^\prime_{\rm trans} < 16$ mag reveals the comparable spatial resolution obtained in both data sets. Despite the strong crowding seen already in these bright stars, the Gemini AO data resolve the sources in the cluster center nicely. When we move on to fainter magnitudes, however, we are limited by the halos of these bright stars, as discussed above. The clear decrease below $K^\prime_{\rm trans} = 16$ mag marks the point where stars are lost due to the enhanced background. When we move radially outwards, the limiting magnitude above which faint stars are lost shifts towards fainter magnitudes. The tendency to loose the faint tail of the magnitude distribution nevertheless remains clearly seen, though it becomes much less pronounced for $R > 10\hbox{$^{\prime\prime}$ }$, where the Gemini and HST LFs resemble each other. For $R > 15\hbox{$^{\prime\prime}$ }$ (upper panel) we are limited by small number statistics due to the small area in this radial bin. As it is hard to observe a well-defined LF at these radii, we will add the two upper bins when we create the radially dependent mass functions in Sect. 5.3.

The magnitude-dependent distribution of stars within the cluster is evident in these LFs. While bright stars are predominantly found in the cluster center, their number density strongly decreases with increasing radius. When we analyse the Gemini LFs more quantitatively, we find 25 (50) stars with $K^\prime_{\rm trans} < 13 (14)$ mag within $R < 5\hbox{$^{\prime\prime}$ }$, but only 8 (23) such stars with $5 < R < 10\hbox{$^{\prime\prime}$ }$, and beyond 10 $^{\prime \prime }$, we observe only 7 (11) such stars. The numbers for HST are comparable in the bright magnitude bins. On the other hand, the number of faint stars with $K^\prime_{\rm trans} > 18 (19)$ mag increases from 1 (0) to 14 (3) to 48 (16). As we see significantly more faint stars in the HST data, the corresponding numbers are higher, i.e. the number of stars with m205 > 19 mag is 0 in the innermost bin, 24 in the intermediate bin, and 81 in the outermost bin. Despite the fact that the area on the Gemini frame increases by about a factor of 3 between the inner and intermediate bin, the number of bright stars is strongly diminished beyond a few arcseconds, while the number of the faint stars increases by much more than the change in area can account for. Although for the fainter stars the effects of crowding and a real increase in the fainter population of the cluster cannot be disentangled, the decrease in the number of bright stars is a clear indication of mass segregation within the Arches cluster.

Figure 12: Radial variation of the luminosity function. The comparison of Gemini and HST LFs is shown together with the Gemini incompleteness calculation. The dependence of the magnitude limits on the distance to the cluster center is striking.