3 The luminosity function

We have determined the R-band LF in each one of the annuli A1-A5 by counting the number of stars in bins of 0.5 mag each as a function of the R magnitude and correcting these values for photometric incompleteness (see Sect.2 above). The same procedure was applied to the comparison fields, and the number of stars found in this way was subtracted from the LF of annuli A1-A5, after having properly rescaled it so as to account for the different area covered by the annuli and by the comparison field.

Figure 5: The LFs measured in annuli A1-A3, after correction for photometric incompleteness and for field star contamination, are here compared with the LF derived by De Marchi et al. (1999) with the VLT-TC further out in the cluster (see text).

Analysis of the LF of annuli A4 and A5 suggests that, in the region covered by these annuli, the cluster population starts to become negligible with respect to the field (see Table2), and the completeness drops severely below 50% at R>22. In particular, the LF of annulus A5 oscillates statistically around zero. In the following we, therefore, focus our investigation only on the region covered by the first three annuli. The LF obtained in this way in annuli A1-A3 are listed in Table 3 and shown graphically in Fig. 5 (boxes). Error bars in Fig. 5 reflect the total error associated with each bin and include both the Poisson statistical error and the uncertainty due to the correction for incompleteness:

$$\sigma_{\rm bin} = \frac{\displaystyle \sqrt{N}}{\displaystyl... ... + \frac{\displaystyle N \sigma_{\Phi}}{\displaystyle \Phi^2}$$ (1)

where N is the number of stars before correction for incompleteness, $\Phi$ and $\sigma_{\Phi}$ are the completeness factor and the associated errors (ranging from $\sigma_{\Phi} \simeq 0.01$ to $\simeq$0.03). We note here that the equation above provides a conservative (upper limit) estimate of the uncertainty and is based on the formalism developed by Bolte (1989). To account for the errors introduced by the statistical subtraction of the comparison field, we have combined in quadrature the $\sigma_{\rm bin}$ of each annulus with that of the comparison field. It should also be noted here that the very good agreement between the field LFs shown in Fig. 4 and measured $\sim$$40^\prime$ away from one another suggests that our LFs are not likely to be affected by local fluctuations in the distribution of contaminating field stars.

 

Table 3: Luminosity functions for annuli A1, A2, A3, and for the stars belonging to the control field in the R band. In particular, mag-bin is the centre of the bin used to obtain the LF; N is the actual number of stars observed; $\Phi$ is the completeness factor in each bin; $N_{\rm C}$ is the number of stars after the correction for incompleteness; $N_{\rm S}$ is the number of stars after the subtraction for the control field, properly rescaled; and $\sigma$ is the standard deviation of the associated uncertainty.

mag-bin		A1					A2					A3
R	MR	N	$\Phi$	$N_{\rm C}$	$N_{\rm S}$	$\sigma$	N	$\Phi$	$N_{\rm C}$	$N_{\rm S}$	$\sigma$	N	$\Phi$	$N_{\rm C}$	$N_{\rm S}$	$\sigma$
19.75	4.42	530	0.91	582	484	25	462	0.92	502	379	24	359	0.88	407	244	23
20.25	4.92	536	0.84	638	517	27	566	0.86	658	507	28	485	0.88	551	350	26
20.75	5.42	585	0.78	750	609	29	496	0.84	590	415	27	486	0.85	571	338	27
21.25	5.92	514	0.69	744	593	29	515	0.79	651	463	28	478	0.79	605	354	28
21.75	6.42	433	0.58	746	573	29	465	0.72	645	430	28	477	0.77	619	332	29
22.25	6.92	333	0.52	640	453	28	382	0.69	553	321	27	413	0.72	573	263	28
22.75	7.42	315	0.50	630	415	28	359	0.60	598	331	28	415	0.63	658	302	30

Inspection of Fig. 5 immediately reveals that the three LFs, measured at different radial distances from the centre of the cluster in the range $60^{\prime\prime}$ to $105^{\prime\prime}$ ( $0.77\,r_{\rm h}$to $1.34\,r_{\rm h}$), are rather similar to one another: except for a modest increase with decreasing luminosity up to $R \simeq 20.5$, they are substantially flat or slightly decreasing down to $R \simeq 23$, where the completeness drops below $\sim$50%.

An interesting check is to compare the LF in our annuli with that derived by De Marchi et al. (1999) in a region of the cluster at a slightly larger radial distance ( $r = 135^{\prime\prime}$) and located within our zone of avoidance (see Sect.2) between annuli A3 and A4. This comparison is shown in Fig. 5, in units of absolute R-band magnitude. To convert our measurements from R to M_R we have adopted a distance modulus in the R band of (m-M)_R =15.33(see Paltrinieri et al. 2001), which differs only marginally from the value of (m-M)_R = 15.31 used by De Marchi et al. (1999). An inspection of this figure confirms that for NGC 6712 the LFs obtained at different radial distances from the centre have a similar shape and, over the magnitude range common to all of them, they are consistent with one another within the quoted errors in the region from $60^{\prime\prime}$ to $135^{\prime\prime}$.

This result is reassuring in that it confirms the validity of the inverted MF found by De Marchi et al. (1999; see Sect.5 below). One should note here that De Marchi et al. could not use a comparison field to correct their LF for field star contamination and, because of this, they suggest that the number of stars in their LF was likely to be over-estimated at the faint end, since the field LF is expected to increase steadily with magnitude. As it turns out, however, field stars account for a small fraction of the total stellar population in these regions, with cluster stars being from $\sim$4 to $\sim$2 times more numerous at $R \simeq 21.5$ (respectively in annuli A1 and A3). This fact, coupled with the slow rate of increase of the field LF with magnitude (see Fig. 4), makes field star contamination in the TC field not sufficient to significantly alter the shape of the LF.